Applied Arithmetic, Weights and Measurements

It would be difficult to overestimate the extent to which mathematics enters into the conditions of everyday life. In its elementary stages, as the science of number, it teaches us the relations of magnitude, and enables us to build up a system of calculation and measurement which, applied to the relations observed to exist in nature, gives results of far-reaching importance.
The properties of number are investigated in arithmetic, and methods examined by which those engaged in practical science are able to work out their results to any degree of approximation.

With the help of algebra, we arrive at a system of logarithms by which many of these results may be reached with the minimum of labor.
The measurement of lines and angles, by methods investigated in geometry and trigonometry, enables us to calculate areas, and work out various problems met with in surveying, and is of the first importance in astronomy.
Arithmetic, which deals with the properties of numbers, forms the basis of all mathematical calculation. (For the primary treatment of numbers, see under The Child World.)

Common Fractions

A Fraction is one or more of the equal parts into which a unit has been divided. A Common Fraction is expressed by two numbers; the one written above the line is called the Numerator, the one below, the Denominator: both, called the Terms, denote the value of the fraction.
Thus, in the fraction ⁄4, the denominator 4, denotes that a unit or whole thing has been divided into four equal parts; and the numerator 3, shows that three of those parts are taken or expressed in the fraction.
A Proper Fraction is one whose numerator is less than its denominator; as ⁄2, ⁄4, ⁄8, etc. Its value is always less than 1.
An Improper Fraction is one whose numerator is equal to, or greater than its denominator, as ⁄5, ⁄7, ⁄12, etc. Its value is never less than 1.
A Mixed Number is a whole number and a fraction; as 3⁄5, 10⁄2, 6⁄3.
The mixed number means that there are whole things taken together with a fraction of another.
A Complex Fraction is one in which the numerator or denominator, or both, are fractions.

Thus 3⁄72⁄8, 1⁄6 × ⁄4, ⁄178, are complex fractions.

Simple Fractions

A very good method of learning the combinations in small fractions is by the use of paper or cardboard disks.
Cut out a large number of them, and, in order to avoid trouble later on, it might be better to have the disks all of one size—about 4 inches in diameter.

Learning the Fraction ⁄2 With Disks

Explanation:—Take a circular disk and cut it into two equal parts. Then proceed in this manner: What is this part called? What is other part called? How many halves in the whole circle? One-half and one-half are what? One-half taken away from one leaves what? If I take a half two times, what do I get? How many halves in a whole?
Now I will write these—

12 + 12 =

1 ÷ 2 =

1 less 12 =

1 - 12 =

2 × 12 =

1 divided by 12 =

Give me the answers and I will write them.
Drawings showing the “placing” of disks for number combinations can then be made; as,
Make similar drawings to tell about halves.
Proceed like this—How many halves in a pie? If a pie cost 10 cents, what will half a pie cost? Who can tell other stories about halves? etc.
Learn fourths along with halves.

Learning the Fraction ⁄3 and Others with Disks

Cut several disks into thirds and have children practice on cutting, so that they will be able to make the three parts of each disk equal. Frequently children will find pleasure in “teaching” one another.
Then proceed like this: What do you call each of these parts? Why are they called thirds? How many thirds in a circle? I am going to take a circle and cut it any way, so as to make three parts; do I call these unequal parts thirds? Why not? Let me write one-third on a piece of paper for you. (Write, ⁄3.) Draw a circle for me. Instead of cutting it, draw lines where you would cut it to make thirds. Write one-third (⁄3) on each third of a circle. I write this (⁄3 + ⁄3). Who can tell me what the answer is? Are two-thirds and two-thirds more than one? How much more? I have two-thirds of an apple and give Mary one-third, how much have I left? Who can give other story problems about thirds? Everybody try, etc.
Learn sixths along with thirds. Use disks, dots, marks, sticks, and inches to illustrate.
Remember that no advance should be made until each little part is understood. Then have fifths compared with fourths, thirds, and halves.
Teach tenths along with fifths.
When twelfths are taught, show the relations between twelfths and sixths, fourths, thirds, and halves.

Equal Fractions in Different Forms

Have the children see how fractions may differ in form but still remain the same in value.
Begin with his knowledge of smaller fractions as

12, 24, 36, 48, and 510 of an apple.

Let them show by the use of drawings that fractions may have large or small terms but be equal in value.
Write a number of proper fractions, improper fractions, and mixed numbers, and have the children pick out those of each kind; as,

38, 27⁄2, 511, 1920, 2020, 1816, 115, 3⁄2, 16⁄3

Principles of Fractions

1. A fraction’s value is the quotient obtained by dividing the numerator by the denominator.

62 = 3 3 is the value of 62

23 = 23 23 is the value of 23

2. Multiplying the denominator of a fraction divides the fraction by that number.

12 × 4 = 18 37 × 3 = 321 23 × 9 = 227

3. Dividing the denominator of a fraction multiplies the fraction by that number.

38 ÷ 4 = 32 109 ÷ 3 = 103 310 ÷ 5 = 32

4. Multiplying the numerator of a fraction multiplies the fraction by that number.

23 × 2 = 43 19 × 8 = 89 58 × 3 = 158

5. Dividing the numerator of a fraction divides the fraction by that number.

47 ÷ 2 = 27 1216 ÷ 12 = 116 37 ÷ 3 = 17

6. Multiplying both numerator and denominator of a fraction by the same number does not change the value of the fraction.

13 × 3 × 3 = 39 = 13 67 × 2 × 2 = 1214 = 67

7. Dividing both numerator and denominator of a fraction by the same number does not change the value of the fraction.

1215 ÷ 3 ÷ 3 = 45 = 1215 1827 ÷ 9 ÷ 9 = 23 = 1827

Reduction of Fractions

is the process of changing their forms without altering their values.

To reduce a fraction to its lowest terms:
Rule.—Divide both terms by their greatest common divisor.
Reduce ⁄12 to its lowest terms.

Work:4 ) ⁄12 ( ⁄3Ans. ⁄3

Four is the G. C. D. of 8 and 12; hence ⁄12 ÷ 4 = ⁄3.
Reduce ⁄56 to its lowest terms.

Work:7 ) ⁄56 ( ⁄8Ans. ⁄8

Seven is the G. C. D. of 35 and 56; hence ⁄56 ÷ 7 = ⁄8.
A fraction whose terms have no common divisor is in its lowest terms, as ⁄16.
To reduce an improper fraction to a whole or mixed number:
Rule.—Divide the numerator by the denominator; the quotient will be the whole or mixed number.
How many units in ⁄6?

Work:30 ÷ 6 = 5Ans. 5.

There are as many units in 30 sixths as 6 is contained times in 30.
Reduce ⁄4 to a mixed number.

Work:75÷ 4 = 18 + 3Ans. 18⁄4.

In 75 fourths there are 18 units, and 3 fourths over, which equals 18⁄4.
To reduce a mixed number to an improper fraction:
Rule.—Multiply the whole number by the denominator of the fraction; add the numerator to the product, and write the sum over the denominator.
Reduce 18⁄4 to an improper fraction.

Work: 18 × 4 = 72⁄4 + ⁄4 = ⁄4Ans. ⁄4.

In 18 are 72 fourths, plus the 3 fourths, equals 75 fourths.
To reduce two or more fractions to their least common denominator:
Rule.—Find the least common multiple of the given denominators for a common denominator. Then for each new numerator take such a part of this common denominator as the fraction is part of 1.
Reduce ⁄2, ⁄3 and ⁄4 to their L. C. D.

Work: 12 = 612 23 = 812 34 = 912

Ans. ⁄12, ⁄12 and ⁄12.
The L. C. M. of the denominators 2, 3 and 4 is 12. Hence, 12 is the L. C. D. to which the given fractions can be reduced. Then to change ⁄2 to 12ths, say, ⁄2 of 12 is 6, and write it over 12; to change ⁄3 to 12ths, say ⁄3 of 12 is 8, and write it over 12; to change ⁄4 to 12ths, say, ⁄4 of 12 is 9, and write it over 12.
Fractions must be reduced to a common denominator to be added or subtracted.

Addition of Fractions

If two or more fractions have the same denominator, their sum is obtained by adding the numerators.

Work: 17 + 47 + 57 = 1 + 4 + 57 = 107 = 137

If the fractions have different denominators, we must first express them as equivalent fractions with the same denominator.

Example 1: Find the value of 19 + 37 + 521 + 23

The lowest common multiple is 63. The several denominators, when divided into 63, give 7, 9, 3, 21 respectively, for quotients. Therefore, we multiply the numerators and denominators of the fractions by 7, 9, 3, 21, and add the numerators to obtain the required sum. The result must be reduced to a mixed number or to lower terms, if necessary.

Work: 19 + 37 + 521 + 23 = 7 + 27 + 15 + 4263 = ⁄63 = 1⁄63 = 1⁄9 Ans.

In adding mixed numbers, first add the whole numbers, then the fractions, finally adding the two results.

Example 2: Add together 3⁄8 + ⁄24 + 7⁄15 + 4⁄20. Given expression:

= 3 + 7 + 4 + 18 + 724 + 1115 + 320

= 14 + 15 + 35 + 88 + 18120

= 14 + 156120 = 14 + 136120 = 15310 Ans.

Subtraction of Fractions

The principle is the same as in addition. Reduce the fractions, if they have different denominators, to a common denominator, and then take the difference of the numerators. In the case of mixed numbers, subtract the whole numbers and the fractions separately.

Example 1: Take 4⁄21 from 6⁄7.

637 - 4521 = 6 - 4 + 37 - 521

637 - 4521 = 2 + 9 - 521

637 - 4521 = 2 + 421 = 2421 Ans.

If the fractional part of the number to be subtracted be greater than the fractional part of the other number, we proceed as follows:

Example 2: From 7⁄15 take 4⁄25.

7415 - 41125 = 7 - 4 + 415 - 1125

7415 - 41125 = 3 + 20 - 3375

7415 - 41125 = 2 + 75 + 20 - 3375

7415 - 41125 = 2 + 6275 = 26275 Ans.

Example 3: Simplify 3⁄9 + 4⁄7 - 5⁄21 + ⁄35 - 1⁄15. Given expression:

= 3 + 4 - 5 - 1 + 29 + 57 - 1321 + 235 - 1415

= 1 + 70 + 225 - 195 +18 - 294315

= 1 + 313 - 489315

= 628 - 489315 = 139315 Ans.

Multiplication of Fractions

(i) When the multiplier is a whole number. This, as in the case of whole numbers, means that we have to find the sum of a given number of repetitions of the fraction.

Example 1:

79 × 4 means 79 + 79 + 79 + 79, i.e., 289 or 7 × 49

Hence, to multiply a fraction by a whole number, simply multiply the numerator by that number.
Since the multiplier thus becomes a factor of the numerator, we cancel any common factors contained in the multiplier and the denominator; and this may be done before we perform the actual multiplication:

Example 2: Multiply ⁄46 by 69.

1946 × 69 = 19 × 6946 = 19 × 32 (cancelling 23), = 572 = 2812 Ans.

It follows that if the multiplier be itself a factor of the denominator, we may, to multiply a fraction by a whole number, divide the denominator by that number.
(ii) When the multiplier is a fraction.
Example: In performing the operation 7 × 9, it is plain that we do to 7 what we do to a unit to obtain 9. Similarly, ⁄5 × ⁄11 may be looked upon as doing to ⁄5 what we do to the unit to obtain ⁄11.
Now, to obtain ⁄11 from the unit, we must divide the unit into 11 equal parts and take 4 of them.
Therefore, to find the value of ⁄5 × ⁄11 we must divide ⁄5 into 11 equal parts and take 4 of them.
But ⁄5 = ⁄55 = ⁄55 × 11, so that, the eleventh part of ⁄5 is ⁄55; and, if we take 4 of these parts, we get ⁄55 × 4 or ⁄55.

Thus, 35 × 411 = 1255. Now 12 = 3 × 4, and 55 = 5 × 11.

Hence we have the following rule: To multiply two fractions together, multiply the numerators for a new numerator and the denominators for a new denominator.
As in Example 2 the work is shortened if we cancel common factors from the numerators and denominators.
Example: Multiply ⁄91 by ⁄77.

The product = 2 22 × 13 91 7 × 77 7 = 249 Ans.

Here, the 22 of the numerator and the 77 of the denominator contain a common factor, 11. Therefore, we cross out the 22 and write 2 above it, and cross out the 77 and write 7 under it. Similarly, we cancel the factor 13 from 13 and 91. There is now 2 left for numerator and 7 × 7 for denominator.
To multiply more than two fractions together, we proceed in the same way.
In multiplication of fractions, mixed numbers must first be expressed as improper fractions.
Example: Simplify 5⁄7 × ⁄27 × 1⁄24.

Given expression = 3 36 7 × 11 27 9 × 5 35 24 2 = 55 18 = 3 1 18

Division of Fractions

(i) When the divisor is a whole number. Suppose we have to divide ⁄9 by 4.
We know ⁄9 = ⁄36. This fraction means that the unit is divided into 36 equal parts, and 28 of the parts taken. If we divide the 28 parts by 4, we get 7 of them—i.e. ⁄36. Hence ⁄9 ÷ 4 = ⁄36.
Therefore, to divide a fraction by a whole number, we multiply the denominator by that number.
In the same way as already explained for multiplication, we cancel any common factors contained in the divisor and the numerator. Hence, if the numerator be exactly divisible by the divisor, we may divide a fraction by a whole number by dividing the numerator by that number.
Example 1:

2731 ÷ 18 = 3 27 31 × 18 2 = 3 62 Ans.

Example 2:

3641 ÷ 9 = 441 Ans.

(ii) When the divisor is a fraction.
In the operation 24 ÷ 3, we have to find the number which, when multiplied by 3, will give 24. Similarly, to find the value of ⁄7 ÷ ⁄9 we have to find the fraction which, when multiplied by ⁄9, will give ⁄7.

But 3 × 97 × 5 is the fraction which gives ⁄7 when multiplied by ⁄9. Therefore, 37 ÷ 59 = 3 × 97 × 5.

Hence, to divide by a fraction, invert the divisor and multiply.
As in multiplication, mixed numbers must first be reduced to improper fractions.
Example 3: Divide 3⁄14 by 5⁄42.

3114 ÷ 5542 = 4314 ÷ 21542 = 4314 × 3 42 215 5 = 35 Ans.

Decimal Fractions

Differ in form from common fractions, in not having a written denominator; and from whole numbers, by having the decimal point (.) prefixed; which also separates the integral part from the decimal. The word decimal is derived from the Latin word decem, which signifies ten. The denominator of a decimal is always 10, or some power of 10, as 100, 1000, etc.
A Complex Decimal is a decimal with a common fraction at the right, as, .12⁄2.
A Mixed Decimal is a whole number with a decimal fraction to its right, as, 34.5.
The denominations of United States money are based on the decimal system—the dollar occupying the unit’s place, the dime the tenth’s place, the cent the hundredth’s place, and the mill the thousandth’s place.
The rules given for addition, subtraction, and so on, also apply to decimals.

Addition in Decimals

Example: 27.295 + .0287 + 591.68 + 9.1846.

27 .295

.0287

591 .68

9 .1846

628 .1883 Ans.

Write the numbers so that the same powers of 10 come under one another, or, what is the same thing, write the numbers so that the decimal points come under one another. Then, adding the ten-thousandths first, 6, 13, carry 1, etc.

Subtraction in Decimals

Example: Subtract .07295 from 21.651.

21 .651

.07295

21 .57805 Ans.

Write the first number under the second, so that the point comes under the point. Remember that we may consider there are 0’s above the 9 and 5, since in 21.651 there are no ten-thousandths and no hundred-thousandths.

Say, mentally 5 and 5 make 10, carry 1.
Say, mentally 10 and 0 make 10, carry 1.
Say, mentally 3 and 8 make 11, carry 1, etc.

Multiplication in Decimals

Rule.—Multiply as in whole numbers, and point off from the right of the product as many places as there are decimal places in both multiplier and multiplicand—prefixing ciphers if necessary.
Example 1: Multiply 87.432 by 564.

87 .432

564

43716 .0

5245 .92

349 .728

49311 .648 Ans.

Place the multiplier so that its unit’s digit comes under the right-hand digit of the multiplicand. Then place the first figure of each product underneath the multiplying digit. The decimal point of the answer will then be directly under the decimal point of the multiplicand.

Example 2: Multiply 31.56 by 5.49.

31.56

5 .49

157.80

12.62 4

2.84 04

173.26 44 Ans.

As before, place the unit’s figure of the multiplier—that is, the 5—under the right-hand digit of 31.56, and proceed as above.

Note.—The number of decimal places in the product will always be equal to the sum of the number of decimal places in the multiplier and the multiplicand. Thus, in Example 2, there are two places of decimals (i.e. two figures to the right of the point) in 31.56, and two places of decimals in 5.49; and we found 2 + 2 = 4 places in the product 173.2644.
To multiply a decimal by 10, 100, etc.
Rule.—Remove the (.) as many places to the right as there are ciphers in the multiplier.

Work: 8.75 × 10 = 87. 5

8.75 × 100 = 875.

8.75 × 1000 = 8750.

Division of Decimals

Rule.—Divide as in whole numbers, annexing ciphers to the dividend, if necessary; then point off from the right of the quotient as many places as the decimal places in the dividend exceed those in the divisor—prefixing ciphers if necessary.
(a) Division of a decimal by a whole number.
Example 1: Divide 18.2754 by 4.

4 ) 18.2758

4.5688 5

We divide 4 into 18 (units) and have 4 (units) quotient and 3 units remainder. Since the 4 is the unit’s figure of the quotient, we write the decimal point immediately after it. Then, the 2 units remainder and the 2 tenths of the dividend make 22 tenths to be divided by 4, and so on. Having reached the 4 (ten-thousandths) of the dividend, we find 8 (ten-thousandths) quotient and 2 remainder. This remainder is 20 hundred-thousandths, which when divided by 4 gives 5 (hundred-thousandths) and no further remainder.

Example 2: Divide 18.2758 by 11.

11 ) 18.2758

1.6614 3636

Here we find the digits 3, 6 repeated indefinitely in the quotient. Decimals of this sort will be fully considered later.

Example 3: Divide 354.43 by 184.

184 ) 3 5 4. 4 3 ( 1.92625 Ans.

1 7 0 4

4 8 3

1 1 5 0

4 6 0

9 2 0

Here we find the first figure of the quotient is obtained by dividing 184 into 354 units. Having now reached the decimal point in the dividend we also put the decimal point in the answer, and go on as before.

(b) Division of a decimal.
Example 4: Divide 10.6603 by 7.85.
Thus:

785 ) 1 0 6 6. 0 3 ( 1.358 Ans.

2 8 1 0

4 5 5 3

6 2 8 0

Here 7.85 is 785 hundredths, and 10.6603 is 1066.03 hundredths; so that the required quotient is obtained by dividing 1066.03 by 785.
Therefore, to divide by a decimal, move the point as many places to the right as will make the divisor a whole number; move the point in the dividend the same number of places to the right. Then proceed as in Example 3.

Example 5: Divide 176.4 by .00012.

12 ) 17640000

Ans. 1470000

Here, to make the divisor a whole number, we have to move the point 5 places. Therefore we also move the point 5 places to the right in the dividend, first writing enough 0’s after the 176.4 to enable us to do so.

To divide a decimal by 10, 100, etc.
Rule.—Remove the (.) as many places to the left as there are ciphers in the divisor.

Work:

62.5 ÷ 10 = 6 .25

62.5 ÷ 100 = .625

62.5 ÷ 1000 = .0625

Expression of decimal fractions as common fractions.

Example: Express 5.375 as a common fraction.
Example: .375 = 375 thousandths.

Therefore 5.375 = 53751000 = 538 Ans.

Rule.—Take the digits of the decimal for numerator; for the denominator put down 1 followed by as many ciphers as there are digits in the decimal. Reduce this fraction to its lowest terms.
Expression of common fractions as decimals.
We have seen that a common fraction represents the quotient of the numerator divided by the denominator. Therefore, to convert a common fraction to a decimal fraction, we divide the numerator by the denominator.

Example: Express 332 as a decimal.

4 ) 3 .0

8 ) .75

.09375 Ans.

It will be found in many cases that there is always a remainder, so that the quotient can be continued indefinitely.

Circulating Decimals

The learner has already discovered that some common fractions cannot be changed to exact decimal fractions, as—
⁄3 =.33333 on to infinity.
⁄3 =.66666 on to infinity.
⁄33 =.212121, etc.
These decimals are known as Circulates, Recurring or Circulating decimals.
The part which recurs is called the Repetend.
This is marked by putting a dot over the first and last figures of it. For instance, if we write the 21 in the last case above, this way: 2̊1̊, it indicates that, if written out, the result would be 21212121, etc., on to infinity.
Where a circulating decimal occurs in work, it is best to reduce it to a common fraction. If need be, it may be expressed in the result as a circulate to any number of decimal places.
To change a pure circulate to a common fraction.
Rule.—Omit the (.) and write the figures of the repetend for the numerator, and as many 9’s for the denominator as there are places in the repetend.
Examples: Change the pure circulates .3̊, .2̊7̊, .1̊42857̊, to common fractions.

.3̊, (39 = 13) Ans. ⁄3.

.2̊7̊, (2799 = 311) Ans. ⁄11.

.1̊42857̊, (142857999999 = 17) Ans. ⁄7.

To change a mixed circulate to a common fraction.
Rule.—From the whole decimal subtract the finite part, and make the remainder the numerator. For the denominator, write as many 9’s as there are figures in the repetend, and annex as many 0’s as there are finite places.
Example: Change the mixed circulates .16̊ and .416̊ to common fractions.

16 - 1 = 15, 1590 = 16. Ans. ⁄6.

416 - 41 = 375, 375900 = 512. Ans. ⁄12.

To add, subtract, multiply and divide circulates, reduce them to common fractions, then apply the respective rules.

Short Methods in Merchandising

When one of the numbers is an aliquot part of 100, the process of multiplication and division can often be very much shortened, as shown below.
Find cost of 27 yards of goods at 16⁄3c ($⁄6) per yard. At $1 per yard, 27 yards cost $27; at $⁄6, (27 ÷ 6), $4⁄2. Ans. $4⁄2.
Find cost of a bale of cotton, 528 pounds at 8⁄3c ($⁄12) per pound. At $1 per pound, 528 pounds cost $528; at $⁄12 (528 ÷ 12) $44. Ans. $44.
Find cost of 1845 pounds of iron, at 3⁄3c ($⁄30) per pound. Take ⁄30 of 1845, since 3⁄3c is ⁄30 of $1. (1845 ÷ 30 = 61⁄2). Ans. $61⁄2.
Find cost of 16 pounds of butter at 37⁄2c ($⁄8) per pound. Here we take ⁄8 of 16. Say ⁄8 of 16 is 2, and ⁄8 is (2 × 3) 6. Or say 3 times 16 is 48, and ⁄8 of 48 is 6. Ans. $6.
Find cost of 17⁄2 bushels of apples at 75c ($⁄4) per bushel. The shortest way to find ⁄4 of $17.50 is to diminish it by ⁄4 of itself.

4 ) 17 .50 at $1

3 .37⁄2 at $⁄4

13 .12⁄2 at $⁄4

Ans. $13.12⁄2.
At 6⁄4c per pound how much sugar will $5 buy? As 6⁄4c is ⁄16 of $1, evidently each dollar will buy 16 pounds. Ans. 80 pounds.
In multiplying by a fraction, write the quantity in a line with the numerator and cancel common factors.
Find cost of 72 yards of carpet, at 87⁄2c ($⁄8) a yard. Cancel 8, also 72 and write 9 instead. Ans. $63.

78 × 72 9 = 63

Of 28 pounds of coffee, at 18⁄4c ($⁄16) per pound. Cancel 28 and 16, write 7 and 4. Ans. $5⁄4.

316 ×4 28 7 = 214 or 5⁄4

At 66⁄3c ($⁄3) per bushel, how many bushel of wheat will $34 buy? Ans. 51 bushel.

32 × 34 17 = 51

In division, invert terms of fraction.
How much syrup, at 41⁄3c ($⁄12) per gallon can be bought for $15? Ans. 36 gallons.

125 × 15 3 = 36

Table of Aliquot Parts of 100

3 ⁄3 is ⁄30

6 ⁄4 is ⁄16

8 ⁄3 is ⁄12

12 ⁄2 is ⁄8

16 ⁄3 is ⁄6

18 ⁄4 is ⁄16

20 is ⁄5

25 is ⁄4

31 ⁄4 is ⁄16

33 ⁄3 is ⁄3

37 ⁄2 is ⁄8

40 is ⁄5

41 ⁄3 is ⁄12

43 ⁄4 is ⁄16

50 is ⁄2

56 ⁄4 is ⁄16

58 ⁄3 is ⁄12

60 is ⁄5

62 ⁄2 is ⁄8

66 ⁄3 is ⁄3

68 ⁄4 is ⁄16

75 is ⁄4

80 is ⁄5

81 ⁄4 is ⁄16

83 ⁄3 is ⁄6

87 ⁄2 is ⁄8

91 ⁄3 is ⁄12

93 ⁄4 is ⁄16

This table embodies all the aliquot parts of 100 and their equivalent fractions which are generally used in practical calculations.

Problems in Grain, Stock, Cotton, Coal, Hay, Lumber, etc.

To find the value of articles sold by the unit, hundred or thousand.
Rule.—Multiply the quantity by the price, or vice versa, and point off the proper number of decimal places in the result.
Find the cost of a bale (518 pounds) of cotton at 7⁄8c per pound.

518 × .07 = 36 .26

„ × .00⁄8 = 1 .94⁄4

Ans. $38 .20⁄4

At 7c (.07) per pound, 518 pounds cost $36.26; at ⁄8c, $1.94⁄4. For ⁄8 of 518, multiply by 3, and divide product by 8.

Find cost of a lot of hogs, weighing 8740 pounds, at $4.35 per hundredweight.

87.40

4.35

380.1900

The price being $4.35 per 100 pounds and as in 8740 pounds there are 87.40 hundredweight, four decimal places are pointed off. Ans. $380.19.

Find the cost of 2864 feet of lumber, at $17⁄4 per 1000 feet.
Price being dollars per 1000, point off three places. (2.864 × 17⁄4 = 49.404.) Ans. $49.40.
To find the value of articles sold by the ton (2000 pounds).
Rule.—Multiply the weight by the price and take half of the product.
Find the cost of 2680 pounds of hay, at $11⁄2 per ton.
Point off three places, when price is dollars; five if dollars and cents. (2680 × 11⁄2 = 30820; 30820 ÷ 2 = 15.410.) Ans. $15.41.
When the long ton of 2240 pounds is used.
Rule.—Multiply the weight by the price and divide the product by 2.240.
Find the cost of 4800 pounds coal, at $6⁄4 per long ton. (4800 × 6⁄4) ÷ 2.24 = $14.46, Ans.
To find the cost of grain, when the price per bushel and weight is given.
Rule.—Reduce the weight to bushels, and multiply by the price.
Find the cost of 3570 pounds of shelled corn, at 36c per bushel.

56 ) 3570 ( 63 .75 bu.

.36

Ans. $22.9500

To reduce pounds of shelled corn to bushels, divide by 56. At 36c per bushel, 63.75 bushels come to $22.95.

Find cost of 2900 pounds of wheat, at 57c per bushel.
To reduce pounds of wheat to bushels divide by 60. 2900 ÷ 60 = 48⁄3 bushels; 48⁄3 × .57 = $27.55, Ans.
In computing the value of grain, the operation can often be abbreviated by cancellation.
Rule.—Write the weight and price per bushel, on the right of a vertical line, and the number of pounds to the bushel on the left. Then cancel common factors, as explained above.
Find the cost of 3230 bushels of wheat, at 72c per bushel.

60 3230

72 12

323 × 12 = 38.76

Here we cancel the 0’s on both sides; then, 6 and 72, which leaves 323 and 12. Their product being the answer.

At 28c per bushel, what will 4080 pounds of oats cost?

32 4080 510

4 28 7

Ans. $35.70

Oats, 32 pounds to the bushel. See table, page 861. Cancel 32 and 4080, then, 4 and 28, leaving the factors 510 and 7.

Other short cuts for computing cost of merchandise, produce, etc.
Find cost of 26⁄2 dozen eggs, at 18⁄2c a dozen.

26 × 18 = 4.68

⁄2 of 44 = .22

⁄2 × ⁄2 = ⁄4 4.90 ⁄4

Ans. $4.90.

When both fractions are ⁄2. To product of the whole numbers, add ⁄2 of their sum, and annex ⁄4 to answer.

Of 53⁄4 pounds of butter, at 28⁄4c per pound.

53 × .28 = 14.84

⁄4 of 81 = .60 ⁄4

⁄4 × ⁄4 = ⁄16 15.45

Ans. $15.45⁄16.

To the product of the whole numbers, add ⁄4 of their sum, plus the square of ⁄4.

Of 13⁄4 yards of flannel, at 31⁄4c per yard.
13 × .31 = 4.03 + .11 = 4.14, Ans.
To 4.03 add .11, ⁄4 of 44 (13 + 31). The ⁄16 (⁄4 × ⁄4) is disregarded.

Denominate Numbers

Simple denominate numbers.—When we speak of measures, whether they are of money, extension, time, or weight, we use terms like 5 dollars, 4 yards, 3 hours, or 10 pounds to express the quantity we are talking about.
Sometimes we use two or more terms or names to express the measure, as 3 hours, 15 minutes, 10 seconds; 4 gallons, 3 quarts, 1 pint. These are compound denominate numbers.
The chief differences between compound numbers and simple numbers is, that with the exceptions of United States money, and the metric system of weights and measures, the denominations of compound numbers do not increase or decrease by the scale of ten.
Reduction.—Reduction of Compound Numbers is the process of changing them from one denomination to another without altering their value.
Reduction Descending is changing the denomination of a number to another that is lower, as: 2 hours = 120 minutes; 2 feet = 24 inches.
Reduction Ascending is changing the denomination of a number to another that is higher, as: 120 minutes = 2 hours; 24 inches = 2 feet.

Rules for Addition of Denominate Numbers

First.—Write the names of the different units to be used in addition, placing them in a horizontal row, the largest to the left.
Next.—Write the numbers of each unit to be added, below the names of the units, each in its proper place.
Then.—Add and place each sum below the column added.
Example: Add 7 hours 15 minutes 30 seconds, 9 hours 30 minutes 40 seconds, and 11 hours 40 minutes 32 seconds.

Work:

hours minutes seconds

7 15 30

9 30 40

11 40 32

28 26 42

Explanation: 32 seconds + 40 seconds + 30 seconds = 102 seconds. But, 102 seconds = 1 minute 42 seconds. Write the 42 below and carry the 1 minute. 1 minute (carried) + 15 minutes + 30 minutes + 40 minutes = 86 minutes. But, 86 minutes = 1 hour 26 minutes. Write the 26 and carry the 1 hour. 1 hour + 11 hours + 9 hours + 7 hours = 28 hours. Result = 28 hours 26 minutes 42 seconds.

Subtraction of Denominate Quantities

Example: Subtract 6 tons 12 cwt. 9 pounds 10 ounces from 15 tons 7 cwt. 13 pounds 9 ounces.

Work:

Tons Cwt. Pounds Ounces

15 7 13 9

6 12 9 10

8 15 3 15

Explanation: (1) Place as in addition of denominate quantities. 10 ounces cannot be taken from 9 ounces, so we must take 1 pound from the 13 pounds and add it to the nine ounces. 16 ounces + 9 ounces = 25 ounces. 25 - 10 = 15. Write the 15 below.
(2) Now there are only 12 pounds left to take the 9 from. 12 - 9 = 3. Write the 3 below.
(3) 12 is larger than 7. 1 ton + 7 cwt. = 27 cwt. 27 - 12 = 15. Write the 15 below.
(4) 14 - 6 = 8. Write the 8 below.
(5) Result = 8 tons 15 cwt. 3 pounds 15 ounces.

Multiplication of Denominate Quantities

Example: Multiply 21 yards 2 feet 11 inches by 6.

Work:

Yards Feet Inches

21 2 11

6

131 2 6

Explanation: (1) 6 × 11 inches = 66 inches = 5 feet 6 inches. Write the 6 below and carry the 5.
(2) 6 × 2 feet = 12 feet. 12 feet + 5 feet (carried) = 17 feet, or 5 yards 2 feet. Write the 2 below and carry the 5.
(3) 6 × 21 yards = 126 yards. 126 yards + 5 yards = 131 yards.
(4) Result = 131 yards 2 feet 6 inches.

Division of Denominate Quantities

Problem: Divide 3 years 9 months 4 days by 12.

Work

Years Months Days Hours

12 ) 3 9 4 0

0 3 22 20

Explanation: (1) We cannot divide 3 by 12, so we reduce 3 years to months. 3 years = 36 months. 36 months + 9 months = 45 months. 45 ÷ 12 = 3, and a remainder 9. Write the 3 and carry the remainder 9.
(2) 9 months (carried) = 270 days. 270 days + 4 days = 274 days. 274 ÷ 12 = 22, and a remainder 10. Write the 22 and carry the 10.
(3) 10 days = 240 hours. 240 ÷ 12 = 20. Write the 20.
(4) Result = 3 months 22 days 20 hours.

Reduction Ascending

Rules: 1. Divide the given denomination by the number which will reduce it to the next higher denomination. Divide the quotient in the same manner, and continue the operation until the entire quantity is reduced.
2. To the last quotient annex the several remainders in their proper order. The result will be the answer.
Example: Reduce 201458 inches to higher denominations.

Work: Solution:

12 201458 inches 201458 inches = 16788 feet 2 inches.

3 16788 feet 2 inches 16788 feet 2 inches = 5596 yards 2 inches.

5 ⁄2 5596 yards 5596 yards 2 inches = 1017 rods 2 yards 1 foot 8 inches.

2 2

11 11192 half yards

320 1017 rods 5 half yards 1017 rods 2 yards 1 foot 8 inches = 3 miles 57 rods 2 yards 1 foot 8 inches.

3 miles 57 rods |

2 yds. 1 ft. 6 in.

201458 inches = 3 miles 57 rods 2 yards 1 foot 8 inches.

Reduction Descending

Rules: 1. Write the given quantity in the order of its denominations, beginning with the highest, and supply vacant denominations with ciphers.
2. Multiply the highest denomination by the number which will reduce it to the next lower denomination, and add to the product the units of the lower denomination, if there be any.
3. Proceed in the same manner until the entire quantity is reduced to the required denomination.
Example: Reduce 10 yards 8 feet 10 inches to inches.

Work:

Yards Feet Inches

10 8 10

3

38

12

456

10

466

Solution:

10 yards = 10 × 3 feet = 30 feet. 30 feet and 8 feet are 38 feet.
38 feet = 38 × 12 inches, or 456 inches. 456 inches + 10 inches = 466 inches.

Note.—To prove the above work, use reduction ascending, beginning with the result.

Long or Linear Measure

Long or linear measure is used in measuring lines and distances.
There are two systems in use in the United States, the English System and the French System. The English system is the one commonly used, while the French system is used in making scientific measurements. (See under Metric System.)

Table of Long Measure

12 inches (in.) = 1 foot (ft.)

3 feet = 1 yard (yd.)

5⁄2 yards, or 16⁄2 feet = 1 rod (rd.)

320 rods, or 5280 feet = 1 mile (mi.)

1760 yards = 1 mile

mi. rd. yd. ft. in.

1 = 320 = 1760 = 5280 = 63360

Architects, carpenters, and mechanics frequently write ′ for foot, and ′′ for inch. Thus 8′ 7′′ means 8 feet 7 inches.
Other measures of length are:

1 hand = 4 in. Used in measuring the height of horses.

1 fathom = 6 ft. Used in measuring depths at sea.

1 knot, nautical or geographical mile = 1.1526⁄3 miles or 6086 feet.

The knot is used in measuring distances at sea. It is equivalent to 1 minute of longitude at the equator.

Surveyors’ Linear Measure

7.92 inches = 1 link (l.)

25 links = 1 rod (rd.)

4 rods or 100 links = 1 chain (ch.)

80 chains = 1 mile (mi.)

mi. ch. rd. l. in.

1 = 80 = 320 = 8000 = 63360

The linear unit commonly employed by surveyors is Gunter’s chain, which is 4 rods or 66 feet.
An engineers’ Chain, used by civil engineers, is 100 feet long, and consists of 100 links.

Measures of Length

The following measures of length are also used:

3 barleycorns = 1 inch. Used by shoemakers.

4 inches = 1 hand. Used to measure the height of horses.

6 feet = 1 fathom. Used to measure depths at sea.

3 feet = 1 pace. - Used in pacing distances.

5 paces = 1 rod.

8 furlongs = 1 mile.

1.15 statute miles = 1 geographical, or nautical mile.

3 geographical miles = 1 league.

60 geographical miles - = 1 degree - of Latitude on a Meridian,
or of Longitude on the Equator.

69.16 statute miles

The length of a degree of latitude varies. 69.16 miles is the average length, and is that adopted by the United States Coast Survey.
The standard unit of length is identical with the imperial yard of Great Britain.
The standard yard, under William IV., was declared to be fixed by dividing a pendulum which vibrates seconds in a vacuum, at the level of the sea, at 62 degrees Fahrenheit, in the latitude of London, into 391,393 equal parts, and taking 360,000 of these parts for the yard.
The following denominations also occur: The span = 9 inches; 1 common cubit (the distance from the elbow to the end of the middle finger) = 18 inches; 1 sacred cubit = 21.888 inches.

Surface Measures

Square Measure, used in measuring surfaces, such as cloth, ceilings, floors, etc.; paving, glazing, and stone-cutting, by the square foot; roofing, flooring, and slating by the square of 100 feet.
A surface has two dimensions, length and breadth.
A square is a figure that has four equal sides and four right angles.
The unit of measure for surfaces is a square, each of whose sides is a linear unit. Thus, a square inch is a square, each of whose sides is one inch long; a square foot is a square, each of whose sides is one foot long, etc.
The area of a square is the product of two of its sides. Thus, the area of a surface 3 feet square is 3 × 3 = 9 square feet.
Hence, to find the area of a rectangle:
Rule.—Multiply the length by the breadth expressed in units of the same denomination.

As the area of a rectangle is found by taking the product of the numbers representing its length and breadth, it is evident that if the area be divided by either of those numbers, the quotient will be the other number. Hence, to find either side of a rectangle when its area and the other side are given:
Rule.—Divide the area by the given side. The quotient will be the required side.

Table of Square Measure

144 square inches (sq. in.) = 1 square foot (sq. ft.)

9 square feet = 1 square yard (sq. yd.)

30⁄4 square yards = 1 square rod (sq. rd.)

160 square rods = 1 acre (A.)

640 acres = 1 square mile (sq. mi.)

Sq. ′ and sq. ′′ are frequently used for square foot and square inch. Thus, 15 sq.′ 6 sq.′′ means 15 square feet 6 square inches.
A square is 100 square feet. It is used in measuring roofing.

Practical Application of Square Measure

Papering

Facts about Wall Paper:

(1) Wall paper in this country is ⁄2 yard wide, and comes in rolls 8 yards long, or in double rolls, 16 yards long.
(2) It is sold by the roll only.
(3) Bordering is sold by the linear yard.
(4) Make liberal allowances for waste in matching figures.
(5) If the border is wide, the strips need not extend to the ceiling.

Rules for Measuring:

(1) Measure the distance around the room in feet.
(2) Deduct the width of doors and windows.
(3) Divide the difference by 1⁄2, and the quotient will be the number of strips needed.
(4) Multiply the number of strips by the number of yards in a strip, and the product is the number of yards needed, approximately.
(5) Divide the number of yards by 8, and the result is the number of single rolls needed.
Example: A room of ordinary height, 16 feet by 24 feet, has three windows and 2 doors, each 4 feet wide. How many rolls of paper are needed to paper the sides?
Solution:

Distance around the room = 80 feet

Width of doors and windows = 20 feet

After deducting for doors and windows 60 feet

60 ÷ 7 = 8⁄7, or 9 double rolls.

Carpeting

Facts about Carpets:

(1) Carpets are usually ⁄4 yard wide and are sold by the linear yard.
(2) Always draw a diagram of the floor or stairs to be covered.
(3) The number of yards required depends on which way the strips run—whether lengthwise or across the room. Sometimes by running the strips lengthwise, there is less waste in matching the pattern.
(4) The part cut off in matching patterns is charged to the purchaser.

Rules for Estimating:

The number of yards required will be the number of yards in a strip (including the waste for matching), multiplied by the number of strips.
Example: What is the cost of carpeting a room 16 feet by 24 feet at 85c per yard? The carpet is 2⁄4 feet wide and the strips run lengthwise.
Solution:
16 ÷ 2⁄4 = 7⁄9. Hence, I must buy 8 strips.
24 ÷ 3 = 8, which is the number of yards in a strip.
8 × 8 yards = 64 yards.
64 yards will cost 64 × 85c, or $54.40.
To this must be added the cost of sewing, the laying of the carpet, and the waste in matching the pattern.

Land Measure

Rule.—To find the number of acres in a tract of land, divide the number of square rods by 160, or number of square chains by 10.
Example: (1) How many square rods, also acres, in a field 80 rods long and 62⁄2 rods wide?
80 × 62⁄2 = 5000 square rods; 5000 ÷ 160 = 31⁄4 acres.
Ans. 31⁄4 acres.
(2) In tract, 79 chains 84 links (79.84 chains) by 41 chains 25 links (41.25 chains)?
79.84 × 41.25 = 3293.4 square chains; 3293.4 ÷10 = 329.34 acres. Ans. 329.34 acres.
Table showing one side of a Square Tract or Lot containing

1 acre = 208 .7 feet = 43,560 square feet

1 ⁄2 acres = 255 .6 feet = 65,340 square feet

2 acres = 295 .2 feet = 87,120 square feet

2 ⁄2 acres = 330 feet = 108,900 square feet

3 acres = 361 .5 feet = 130,680 square feet

5 acres = 466 .7 feet = 217,800 square feet

10 acres = 660 feet = 435,600 square feet

⁄10 acre = 66 feet = 4,356 square feet

⁄8 acre = 73 .8 feet = 5,445 square feet

⁄6 acre = 85 .2 feet = 7,260 square feet

⁄4 acre = 104 .4 feet = 10,890 square feet

⁄3 acre = 120 .5 feet = 14,520 square feet

⁄2 acre = 147 .6 feet = 21,780 square feet

⁄4 acre = 180 .8 feet = 32,670 square feet

Table of Surveyors’ Square Measure

272⁄2 square feet = 1 square rod

16 square rods = 1 square chain

160 square rods, or 10 square chains = 1 acre

640 acres = 1 square mile, or section

36 square miles, or 36 sections = 1 township

Texas Land Measure
(Also used in Mexico, New Mexico, Arizona, and California)

26,000,000 square varas (square of 5,099 varas) = 1 league and 1 labor = 4,605 .5 acres

1,000,000 square varas (square of 1,000 varas) = 1 labor = 177 .136 acres

25,000,000 square varas (square of 5,000 varas) = 1 league = 4,428 .4 acres

12,500,000 square varas (square of 3,535 .5 varas) = ⁄2 league = 2,214 .2 acres

8,333,333 square varas (square of 2,886 .7 varas) = ⁄3 league = 1,476 .13 acres

6,250,000 square varas (square of 2,500 varas) = ⁄4 league = 1,107 .1 acres

7,225,600 square varas (square of 2,688 varas) = 1,280 acres

3,612,800 square varas (square of 1,900 .8 varas) = 1 section = 640 acres

1,806,400 square varas (square of 1,344 varas) = ⁄2 section = 320 acres

903,200 square varas (square of 950 .44 varas) = ⁄4 section = 160 acres

451,600 square varas (square of 672 varas) = ⁄8 section = 80 acres

225,800 square varas (square of 475 varas) = ⁄16 section = 40 acres

5,645 .376 square varas (square of 75 .137 varas) = 4,840 square yards = 1 acre

To find the number of acres in any number of square varas, multiply the latter by 177 (or to be more exact, by 177⁄8), and cut off six decimals.

1 vara = 33⁄3 inches
1,900.8 varas = 1 mile.

The Measure of Solids, or Cubic Measure

Just as the rectangle is the chief surface considered in arithmetic, so the rectangular solid is the chief solid body.
A rectangular solid is bounded by six rectangular surfaces, each opposite pair of rectangles being equal and parallel to each other.
A rectangular solid thus has three dimensions—length, breadth, and thickness.
If the length, breadth, and thickness are all equal to one another, the solid is called a cube. Hence, a cubic foot, the unit of volume, is a solid body whose length, breadth, and thickness are each a linear foot. Similarly, a cubic inch measures one linear inch in length, breadth, and thickness; and a cubic yard measures one linear yard in length, breadth, and thickness.
The number of cubic feet (or inches, or yards) in the volume of a rectangular solid is equal to the number of linear feet (or inches, or yards) in the length, multiplied by the number of linear feet (or inches, or yards) in the breadth, multiplied by the number of linear feet (or inches, or yards) in the thickness.
This is usually abbreviated into
Length × breadth × thickness = volume, or cubic content.
For example, suppose the solid in the diagram is 10 feet in length, 8 feet in breadth, and 5 feet in thickness. It is clear that the solid can be cut into five slices, each 1 foot thick, by planes parallel to the bottom. But, the bottom contains 10×8 square feet and above each square foot there is a cubic foot. Thus, each slice contains 10×8 cubic feet. Therefore, since there are five slices, the whole solid contains 10×8×5, or 400 cubic feet.
Since length × breadth × thickness = cubic content, it follows that, if we know any three of these four quantities, we can find the fourth.
The student should remember that
(a) A cubic foot of water weighs 1000 ounces (avoirdupois) approximately.
(b) A gallon of pure water weighs 10 pounds (avoirdupois).
We have thus a relation between weight, capacity, and cubic content.

Table of Cubic Measure

1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)

27 cubic feet = 1 cubic yard (cu. yd.)

128 cubic feet = 1 cord (C.)

Cubic
Yard Cubic
Feet Cubic
Inches

1 = 27 = 46656

A cord of wood or stone is a pile 8 feet long, 4 feet wide, and 4 feet high.
A pile of wood 4 feet high, 4 feet wide and 1 foot long makes a cord foot. 8 cord feet = 1 cord.
A perch of stone or masonry is 16⁄2 feet long, 1⁄2 feet thick, and 1 foot high, and contains 24⁄4 cubic feet.
A cubic yard of earth is considered a load.
Brick work is commonly estimated by the thousand bricks.
Bricklayers, masons, and joiners commonly make a deduction of one half the space occupied by windows and doors in the walls of buildings.
In computing the contents of walls, masons and bricklayers multiply the entire distance around on the outside of the wall by the height and thickness. The corners are thus measured twice.
A cubic foot of distilled water at the maximum density, at the level of the sea, and the barometer at 30 inches, weighs 62⁄2 pounds or 1000 ounces avoirdupois.
By actual measurements, it has been found that a bushel, dry measure, contains about 1⁄4 cubic feet. This makes it easy to estimate about how many bushels any bin will hold.

Practical Applications of Cubic Measure

Example: An open tank made of iron ⁄4 inch thick, is 4 feet long, 2 feet 6 inches broad, and 2 feet deep, outside measurement. Assuming that iron weighs 7.8 times as much as water, find the weight of the tank.
The external volume of the tank = 2 × 2⁄2 × 4 cubic feet = 20 cubic feet.
Since the iron is ⁄4 inch thick, the inside length is ⁄2 inch less than the outside, the inside breadth is ⁄2 inch less than the outside, and the inside depth is ⁄4 inch less than the outside.
Therefore the interior volume

= 29⁄2 × 47⁄2 × 23⁄4 cubic inches

= 59 × 95 × 9516 cubic inches

= 33279⁄16 cubic inches

Therefore, volume of iron in the tank

= 20 cubic feet - 33279⁄16 cubic inches

= 1280⁄16 cubic inches.

But 1 cubic foot of iron weighs as much as 7.8 cubic feet of water, i. e., 7.8 × 1000 ounces, or 7800 ounces.

∴ Weight of tank = 1280⁄16 × 78001728 × 16 pounds.

∴ Weight of tank = 361.199 pounds, Ans.

Example: A wood pile is 8 feet high and 40 feet long. The sticks are 4 feet long. How many cords in it?
Solution: Being 8 feet high, it is 2 cords high. 40 feet in length equal 5 cords in length. Hence, the pile contains 2 × 5 cords, or 10 cords.
To estimate a bin:
(1) Find the number of cubic feet in the bin.
(2) Divide the number of cubic feet by 1⁄4.
(3) The result is the number of bushels.
Example: How many bushels will a bin hold, if its inside measurements are, length 20 feet, width 12 feet, depth 8 feet?
Solution: The number of cubic feet in a bin is 8 × 12 × 20, or 1920.
If 1 bushel contains 1⁄4 cubic feet, in 1920 cubic feet there are as many bushels as 1⁄4 is contained times in 1920, or 1536.
Work:

8 × 12 × 20 = 1920

1920 ÷ 1⁄4 = 1536.

The work may be indicated in this way as well:—

8 × 12 × 20 × ⁄5 = 1536.

To get the number of heaped bushels of corn in the ear in a crib:
(1) Multiply the length of the crib in inches by the width in inches.
(2) Multiply the product obtained, by the height of the corn in the crib in inches.
(3) Divide the result by 2748.
Example: How much corn in the ear can I put into a crib 12 feet wide, 20 feet long, and 10 feet deep?
Solution: The number of cubic inches in the crib is 144 × 240 × 120, or 4,147,200.
Since 2748 cubic inches hold 1 bushel, 4,147,200 cubic inches hold as many bushels as 2748 is contained times in 4,147,200, or 1509+ bushels.
Work:

144 × 240 × 1202748 = 1509+.

Measures of Capacity

Measures used in telling the extent of room in vessels are called measures of capacity.
There are two kinds of capacity measures, dry measures and liquid measures.
Dry measures are used to measure grain, seeds, and the like.
Liquid measures are used to measure water, milk, oils, etc.

Common Liquid Measure Table

4 gills (gi.) = 1 pint (pt.)

2 pints = 1 quart (qt.)

4 quarts = 1 gallon (gal.)

31⁄2 gallons = 1 barrel (bbl.)

2 barrels = 1 hogshead (hhd.)

Gallon Quarts Pints Gills

1 = 4 = 8 = 32

A pint, quart, or gallon, dry measure, is more than the same quantity, liquid measure; for a quart, dry measure, is ⁄32 of a bushel, or ⁄32 of 2150.4 cubic inches, which is about 67⁄5 cubic inches, while a quart liquid measure is ⁄4 of 231 cubic inches, or 57⁄4 cubic inches.

Cu. In.
in 1 Gal. Cu. In.
in 1 Qt. Cu. In.
in 1 Pt. Cu. In.
in 1 Gi.

Liquid measure 231 57 ⁄4 28 ⁄8 7 ⁄32

Dry measure 268 ⁄5 67 ⁄5 33 ⁄5 8 ⁄5

In determining the capacity of cisterns, reservoirs, etc., 31⁄2 gallons are considered a barrel (bbl.), and 2 barrels, or 63 gallons, a hogshead (hhd.). In commerce, however, the barrel and hogshead are not fixed measures.
Casks of large size, called tierces, pipes, butts, tuns, etc., do not hold any fixed quantity. Their capacity is usually marked upon them.
The standard gallon of the United States contains 231 cubic inches, and will hold a little over 8⁄3 pounds of distilled water. The imperial gallon, now adopted by Great Britain, contains 277.274 cubic inches, or 10 pounds of distilled water, temperature 62 degrees Fahrenheit, the barometer standing at 30 inches.

Table of Apothecaries’ Liquid Measure

These measures are used in mixing medicines.

60 minims (m) = 1 fluid dram (fʒ)

8 fluid drams = 1 fluid ounce (f℥)

16 fluid ounces = 1 pint (O.)

8 pints = 1 gallon (Cong.)

A minim is about 1 drop.

Table of Dry Measure

2 pints (pt.) = 1 quart (qt.)

8 quarts = 1 peck (pk.)

4 pecks = 1 bushel (bu.)

Bushel Pecks Quarts Pints

1 = 4 = 32 = 64

A common Winchester bushel (the standard of the United States) contains 2150.42 cubic inches.
A dry quart contains 67.2 cubic inches.
A liquid quart contains 57.75 cubic inches.
Example 1: Reduce 5 bushels 2 pecks 4 quarts 1 pint to pints.

Operation:

4 8 2

bu. pk. qt. pts.

5 2 4 1

4

22 pk.

8

180 qt.

2

361 pt.

Explanation: As there are 4 pecks in 1 bushel, any number of bushels is equal to 4 times that number of pecks. Then, 5 bushels = 20 pecks, and 2 pecks added make 22 pecks. As there are 8 quarts in 1 peck, any number of pecks is equal to 8 times that number of quarts. Then 22 pecks = 176 quarts, and 4 quarts added make 180 quarts. As there are 2 pints in 1 quart, any number of quarts is equal to 2 times that number of pints. Then, 180 quarts = 360 pints, and 1 pint added make 361 pints. Hence, 5 bushels 2 pecks 4 quarts 1 pint = 361 pints.

Example 2: Reduce 361 pints to bushels.

Operation:

2 361 pt.

8 180 qt. + 1 pt.

4 22 pk. + 4 qt.

5 bu. + 2 pk.

Explanation: As there are 2 pints in 1 quart, 361 pints are equal to one-half that number of quarts = 180 quarts, with a remainder of 1 pint. Also, 180 quarts are equal to one-eighth of that number of pecks = 22 pecks, with a remainder of 4 quarts. Finally, 22 pecks are equal to one-fourth of that number of bushels = 5 bushels, with a remainder of 2 pecks. Hence, 361 pints are equal to 5 bushels 2 pecks 4 quarts 1 pint.

Measures of Weight

Avoirdupois Weight

Avoirdupois Weight is used for weighing heavy articles as grain, groceries, coarse metals, etc.

16 ounces (oz.) = 1 pound (lb.)

100 pounds = 1 hundredweight (cwt.)

20 hundredweight = 1 ton (T.)

Ton Hundred-
weight Pounds Ounces

1 = 20 = 2000 = 32000

Scale.—20, 100, 16.

In weighing coal at the mines and in levying duties at the United States Custom House, the long ton of 2240 pounds is sometimes used.
The ounce is considered as 16 drams.
The unit is the pound. It contains 7000 grains.
The following denominations are also used:

14 pounds = 1 stone

100 pounds butter = 1 firkin

100 pound grain or flour = 1 cental

100 pounds dried fish = 1 quintal

100 pounds nails = 1 keg

196 pounds flour = 1 barrel

200 pounds pork or beef = 1 barrel

280 pounds salt at N. Y. works = 1 barrel

Troy Weight

Troy Weight is used in weighing gold, silver, and jewels.

TABLE

24 grains (gr.) = 1 pennyweight (pwt.)

20 pennyweights = 1 ounce (oz.)

12 ounces = 1 pound (lb.)

Pounds Ounces Penny-
weights Grains

1 = 12 = 240 = 5760

In weighing diamonds, pearls, and other jewels, the unit commonly employed is the carat, which is equal to 4 carat grains, or 3.168 Troy grains.
The term carat is also used to express the fineness of gold, and means ⁄24 part. Thus, gold that is 18 carats fine is ⁄24 gold, and ⁄24 alloy.
The standard unit of weight is the Troy pound. It is equal to the weight of 22.7944 cubic inches of distilled water at its maximum density, the barometer being at 30 inches. It is identical with the Troy pound of Great Britain.
The following are approximate avoirdupois weights of certain articles of produce according to the laws of the United States, and in the majority of States:

Table I. UNITED STATES LEGAL WEIGHTS (in pounds) PER BUSHEL
Prepared by the United States Bureau of Standards

State or Territory Apples Bar-
ley Beans Beets Blue-
Grass Seed Bran Broom-
Corn Seed Buck-
wheat Car-
rots Clover Seed Coal Coke Corn Corn Meal

Ap-
ple Dried Ap-
ples Beans Castor
Beans,
Shelled Corn in Ear Shelled Corn

United States ... ... 48 60 50 ... ... ... ... 48 ... ... 80 ... ... ... ...

Alabama ... 24 47 60 ... ... ... ... ... ... ... ... ... ... 70 56 ...

Arizona ... ... 45 a55 ... ... ... ... ... ... ... ... ... ... ... ... ...

Arkansas b50 24 48 a60 ... ... 14 20 48 52 ... 60 ... ... 70 56 48

California ... ... 50 ... ... ... ... ... ... 40 ... ... ... ... ... ... ...

Colorado ... ... 48 60 ... ... 14 ... ... 52 ... 60 80 ... 70 ... 50

Connecticut 48 25 48 60 ... c60 ... 20 ... 48 50 60 80 ... ... ... 50

Delaware ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Florida b48 24 48 d60 48 ... ... 20 ... ... ... ... ... ... 70 56 48

Georgia ... 24 47 e60 ... ... 14 f20 ... 52 ... 60 80 ... 70 56 48

Hawaii ... ... 48 ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Idaho ... 28 48 ... ... ... ... ... ... 42 ... 60 ... ... ... 56 ...

Illinois ... 24 48 e60 46 ... 14 20 ... 52 ... 60 80 ... 70 56 48

Indiana ... 25 48 60 46 ... 14 ... ... 50 ... 60 80 ... g68 56 50

Iowa 48 24 48 60 46 56 14 20 50 52 50 60 80 38 70 56 50

Kansas b48 24 48 60 46 56 h14 20 ... 50 50 60 80 ... 70 i56 ...

Kentucky ... 24 47 e60 45 ... 14 20 ... 56 ... 60 76 ... j70 56 50

Louisiana ... ... 32 ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Maine 44 ... 48 60 ... 60 ... ... ... 48 50 ... ... ... ... ... 50

Maryland ... 28 ... ... 50 ... 14 20 ... ... 50 60 80 ... ... 56 48

Massachusetts 48 25 48 k60 ... 60 ... 20 ... 48 50 60 ... ... ... l56 50

Michigan 48 22 48 60 46 ... 14 ... ... 48 ... 60 80 ... 70 56 50

Minnesota b50 28 48 60 ... 50 14 ... 57 50 45 60 80 ... 70 56 ...

Mississippi ... 26 48 e60 m46 ... 14 20 ... 48 ... 60 80 ... 72 56 48

Missouri 48 24 48 m60 46 ... 14 20 ... 52 50 60 80 ... 70 56 50

Montana 45 ... 48 60 ... 50 14 20 ... 52 50 60 76 ... 70 56 50

Nebraska 48 24 48 e60 46 ... 14 20 ... 52 ... 60 80 ... 70 56 50

Nevada b48 24 48 60 46 56 h 20 ... 50 50 60 ... ... 70 56 48

New Hampshire 48 25 48 62 ... 60 ... 20 ... 48 50 60 ... ... e50 ... 50

New Jersey 50 25 48 60 ... 60 ... ... ... 50 50 64 ... ... ... ... ...

New York 48 25 48 60 ... ... ... 20 ... 48 50 60 ... ... ... ... 50

North Carolina 48 ... 48 ... 46 ... 14 ... 46 50 ... 60 ... ... ... ... 48

North Dakota 50 ... 48 60 ... 60 ... 20 30 42 ... 60 80 ... 70 56 ...

Ohio 50 24 48 60 ... 56 ... ... ... 50 50 60 n80 40 68 56 ...

Oklahoma 48 24 48 60 ... 60 14 20 30 52 50 60 80 ... 70 56 50

Oregon 45 28 46 ... ... ... ... ... ... 42 ... 60 ... ... ... ... ...

Pennsylvania ... ... 47 ... ... ... ... ... ... 48 ... 60 o76 40 ... ... ...

Rhode Island 48 25 48 60 46 50 ... 20 ... 48 50 60 80 40 70 56 50

South Carolina ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 48

South Dakota ... ... 48 60 ... 60 ... 20 30 42 ... 60 80 ... 70 56 ...

Tennessee b50 24 48 p60 46 50 14 20 42 50 50 q60 80 40 70 56 ...

Texas 45 28 48 e60 ... ... ... 20 ... 42 ... 60 80 ... 70 56 ...

Vermont 46 ... 48 62 ... 60 ... ... ... 48 50 60 ... ... ... ... ...

Virginia 45 28 48 e60 ... ... 14 ... ... 52 ... 60 80 ... 70 56 50

Washington b45 28 48 ... ... ... ... ... ... 42 ... 60 ... ... ... ... ...

West Virginia ... 25 48 60 ... ... ... ... ... 52 ... 60 80 ... ... ... ...

Wisconsin 50 25 48 60 ... 50 ... 20 ... 50 50 60 ... ... ... ... 50

State or Territory Apples Barley Beans Beets Blue-Grass Seed Bran Broom-Corn Seed

Apple Dried Apples Beans Castor
Beans,
Shelled

United States ... ... 48 60 50 ... ... ... ...

Alabama ... 24 47 60 ... ... ... ... ...

Arizona ... ... 45 a55 ... ... ... ... ...

Arkansas b50 24 48 a60 ... ... 14 20 48

California ... ... 50 ... ... ... ... ... ...

Colorado ... ... 48 60 ... ... 14 ... ...

Connecticut 48 25 48 60 ... c60 ... 20 ...

Delaware ... ... ... ... ... ... ... ... ...

Florida b48 24 48 d60 48 ... ... 20 ...

Georgia ... 24 47 e60 ... ... 14 f20 ...

Hawaii ... ... 48 ... ... ... ... ... ...

Idaho ... 28 48 ... ... ... ... ... ...

Illinois ... 24 48 e60 46 ... 14 20 ...

Indiana ... 25 48 60 46 ... 14 ... ...

Iowa 48 24 48 60 46 56 14 20 50

Kansas b48 24 48 60 46 56 h14 20 ...

Kentucky ... 24 47 e60 45 ... 14 20 ...

Louisiana ... ... 32 ... ... ... ... ... ...

Maine 44 ... 48 60 ... 60 ... ... ...

Maryland ... 28 ... ... 50 ... 14 20 ...

Massachusetts 48 25 48 k60 ... 60 ... 20 ...

Michigan 48 22 48 60 46 ... 14 ... ...

Minnesota b50 28 48 60 ... 50 14 ... 57

Mississippi ... 26 48 e60 m46 ... 14 20 ...

Missouri 48 24 48 m60 46 ... 14 20 ...

Montana 45 ... 48 60 ... 50 14 20 ...

Nebraska 48 24 48 e60 46 ... 14 20 ...

Nevada b48 24 48 60 46 56 h 20 ...

New Hampshire 48 25 48 62 ... 60 ... 20 ...

New Jersey 50 25 48 60 ... 60 ... ... ...

New York 48 25 48 60 ... ... ... 20 ...

North Carolina 48 ... 48 ... 46 ... 14 ... 46

North Dakota 50 ... 48 60 ... 60 ... 20 30

Ohio 50 24 48 60 ... 56 ... ... ...

Oklahoma 48 24 48 60 ... 60 14 20 30

Oregon 45 28 46 ... ... ... ... ... ...

Pennsylvania ... ... 47 ... ... ... ... ... ...

Rhode Island 48 25 48 60 46 50 ... 20 ...

South Carolina ... ... ... ... ... ... ... ... ...

South Dakota ... ... 48 60 ... 60 ... 20 30

Tennessee b50 24 48 p60 46 50 14 20 42

Texas 45 28 48 e60 ... ... ... 20 ...

Vermont 46 ... 48 62 ... 60 ... ... ...

Virginia 45 28 48 e60 ... ... 14 ... ...

Washington b45 28 48 ... ... ... ... ... ...

West Virginia ... 25 48 60 ... ... ... ... ...

Wisconsin 50 25 48 60 ... 50 ... 20 ...

State or Territory Buckwheat Carrots Clover Seed Coal Coke Corn Corn Meal

Corn in Ear Shelled Corn

United States 48 ... ... 80 ... ... ... ...

Alabama ... ... ... ... ... 70 56 ...

Arizona ... ... ... ... ... ... ... ...

Arkansas 52 ... 60 ... ... 70 56 48

California 40 ... ... ... ... ... ... ...

Colorado 52 ... 60 80 ... 70 ... 50

Connecticut 48 50 60 80 ... ... ... 50

Delaware ... ... ... ... ... ... ... ...

Florida ... ... ... ... ... 70 56 48

Georgia 52 ... 60 80 ... 70 56 48

Hawaii ... ... ... ... ... ... ... ...

Idaho 42 ... 60 ... ... ... 56 ...

Illinois 52 ... 60 80 ... 70 56 48

Indiana 50 ... 60 80 ... g68 56 50

Iowa 52 50 60 80 38 70 56 50

Kansas 50 50 60 80 ... 70 i56 ...

Kentucky 56 ... 60 76 ... j70 56 50

Louisiana ... ... ... ... ... ... ... ...

Maine 48 50 ... ... ... ... ... 50

Maryland ... 50 60 80 ... ... 56 48

Massachusetts 48 50 60 ... ... ... l56 50

Michigan 48 ... 60 80 ... 70 56 50

Minnesota 50 45 60 80 ... 70 56 ...

Mississippi 48 ... 60 80 ... 72 56 48

Missouri 52 50 60 80 ... 70 56 50

Montana 52 50 60 76 ... 70 56 50

Nebraska 52 ... 60 80 ... 70 56 50

Nevada 50 50 60 ... ... 70 56 48

New Hampshire 48 50 60 ... ... e50 ... 50

New Jersey 50 50 64 ... ... ... ... ...

New York 48 50 60 ... ... ... ... 50

North Carolina 50 ... 60 ... ... ... ... 48

North Dakota 42 ... 60 80 ... 70 56 ...

Ohio 50 50 60 n80 40 68 56 ...

Oklahoma 52 50 60 80 ... 70 56 50

Oregon 42 ... 60 ... ... ... ... ...

Pennsylvania 48 ... 60 o76 40 ... ... ...

Rhode Island 48 50 60 80 40 70 56 50

South Carolina ... ... ... ... ... ... ... 48

South Dakota 42 ... 60 80 ... 70 56 ...

Tennessee 50 50 q60 80 40 70 56 ...

Texas 42 ... 60 80 ... 70 56 ...

Vermont 48 50 60 ... ... ... ... ...

Virginia 52 ... 60 80 ... 70 56 50

Washington 42 ... 60 ... ... ... ... ...

West Virginia 52 ... 60 80 ... ... ... ...

Wisconsin 50 50 60 ... ... ... ... 50

Table I. UNITED STATES LEGAL WEIGHTS (in pounds) PER BUSHEL—Continued

State or Territory Corn Meal, Bolted Corn Meal, Un-
bolted Cotton Seed Cran-
berries Flaxseed (Linseed) Goose-
berries Plas-
tering Hair Hemp Seed Herdes Grass Hun-
garian Grass Seed Millet Oats Onions Orchard Grass Seed Osage Orange Seed Pars-
nips Peaches

Cotton Seed Sea-
Island Onions Onion Sets

United States ... ... ... ... ... 56 ... ... ... ... ... ... 32 57 ... ... ... ... ...

Alabama ... ... 32 ... ... ... ... ... ... ... ... ... 32 ... ... ... ... ... ...

Arizona ... ... ... ... ... ... ... ... ... ... ... ... 32 ... ... ... ... ... ...

Arkansas ... ... 33⁄3 ... ... 56 ... ... ... ... ... 50 32 57 ... 14 ... ... ...

California ... ... ... ... ... ... ... ... ... ... ... ... 32 ... ... ... ... ... ...

Colorado ... ... ... ... ... ... ... ... 44 ... ... ... 32 57 ... ... ... ... ...

Connecticut ... ... 30 44 ... 55 ... ... ... 45 ... ... 32 52 ... ... ... 45 ...

Delaware 44 48 ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Florida ... ... 32 46 ... ... ... ... ... ... ... 50 32 56 ... ... ... ... 54

Georgia ... ... 30 ... ... 56 ... 8 44 ... ... ... 32 57 ... ... ... ... ...

Hawaii ... ... ... ... ... ... ... ... ... ... ... ... 32 ... ... ... ... ... ...

Idaho ... ... ... ... ... 56 ... ... ... ... ... ... 36 ... ... ... ... ... ...

Illinois ... ... ... ... ... 56 ... 8 44 ... ... ... 32 57 ... ... ... ... ...

Indiana ... ... ... ... 33 ... ... ... 44 ... ... 50 32 48 ... 14 33 55 ...

Iowa ... ... ... ... ... 56 40 ... 44 ... 50 50 32 57 ... 14 32 42 48

Kansas ... ... ... ... ... 56 ... r8 44 ... 50 50 32 57 ... ... ... 52 48

Kentucky ... ... ... ... ... 56 ... 8 44 ... 50 50 32 57 36 14 ... ... ...

Louisiana ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

Maine ... ... ... ... ... ... ... 11 ... 45 ... ... 32 52 ... ... ... 45 ...

Maryland ... ... ... ... ... 56 ... ... 44 45 50 50 32 57 ... 14 ... ... 40

Massachusetts ... ... 30 44 32 55 ... ... ... 45 ... ... 32 52 ... ... ... 45 48

Michigan ... ... ... ... 40 56 ... ... 44 ... 50 50 32 54 ... 14 33 ... ...

Minnesota ... ... ... ... 36 ... 40 r8 50 ... 48 48 32 52 ... 14 ... 42 ...

Mississippi 44 48 32 ... ... 56 ... ... 44 ... 50 50 32 57 ... ... ... ... ...

Missouri ... ... 33 ... ... 56 ... ... 44 ... 48 50 32 57 28 14 36 44 48

Montana ... ... ... ... ... 56 ... ... 44 ... 50 ... 32 57 ... ... ... 50 ...

Nebraska ... ... ... ... ... 56 ... 8 44 ... 50 50 32 57 25 ... 32 ... ...

Nevada ... ... ... ... ... 56 ... ... 48 ... 50 50 32 57 ... ... ... 50 48

New Hampshire ... ... ... ... 32 56 ... ... ... 45 ... ... 32 52 ... ... ... 45 48

New Jersey ... ... ... ... ... 55 ... ... ... ... ... ... 30 57 ... ... ... ... 50

New York ... ... 30 44 ... 55 ... ... ... 45 ... ... 32 57 ... ... ... ... ...

North Carolina 48 48 30 44 ... 55 ... ... 44 ... ... 50 32 57 ... 14 ... ... ...

North Dakota ... ... ... ... ... 56 ... ... ... ... ... 50 32 52 ... ... ... ... ...

Ohio ... ... ... ... ... 56 ... ... 44 ... 50 50 32 55 ... ... ... ... 48

Oklahoma ... ... 32 ... ... 56 ... ... 44 ... ... 50 32 57 28 14 36 44 48

Oregon ... ... ... ... ... ... ... ... ... ... ... ... 32 ... ... ... ... ... ...

Pennsylvania ... ... ... ... ... ... ... ... ... ... ... ... 32 50 ... ... ... ... ...

Rhode Island ... ... 30 44 ... 56 ... ... 44 ... 50 50 32 50 ... ... ... 50 48

South Carolina 48 48 30 ... ... ... ... ... ... ... ... ... ... 52 ... ... ... ... ...

South Dakota ... ... ... ... ... 56 ... ... ... ... ... ... 32 52 ... ... ... ... ...

Tennessee 50 48 28 ... ... 56 48 8 44 ... 48 50 32 56 28 14 33 50 50

Texas ... ... 32 ... ... 56 ... ... 44 ... 48 50 32 57 ... ... ... ... 50

Vermont ... ... ... ... ... ... ... ... ... 45 ... ... 32 52 ... ... ... ... ...

Virginia ... ... 32 ... ... 56 ... 8 44 12 48 50 30 57 28 14 34 ... ...

Washington ... ... ... ... ... 56 ... ... ... ... ... ... 32 ... ... ... ... ... ...

West Virginia ... ... ... ... ... 56 ... ... ... ... ... ... 32 ... ... ... ... ... ...

Wisconsin ... ... 30 44 ... 56 ... 8 44 ... 48 50 32 57 ... ... ... 44 ...

State or Territory Corn Meal, Bolted Corn Meal, Unbolted Cotton Seed Cranberries Flaxseed (Linseed) Gooseberries Plastering Hair Hemp Seed

Cotton Seed Sea-Island

United States ... ... ... ... ... 56 ... ... ...

Alabama ... ... 32 ... ... ... ... ... ...

Arizona ... ... ... ... ... ... ... ... ...

Arkansas ... ... 33⁄3 ... ... 56 ... ... ...

California ... ... ... ... ... ... ... ... ...

Colorado ... ... ... ... ... ... ... ... 44

Connecticut ... ... 30 44 ... 55 ... ... ...

Delaware 44 48 ... ... ... ... ... ... ...

Florida ... ... 32 46 ... ... ... ... ...

Georgia ... ... 30 ... ... 56 ... 8 44

Hawaii ... ... ... ... ... ... ... ... ...

Idaho ... ... ... ... ... 56 ... ... ...

Illinois ... ... ... ... ... 56 ... 8 44

Indiana ... ... ... ... 33 ... ... ... 44

Iowa ... ... ... ... ... 56 40 ... 44

Kansas ... ... ... ... ... 56 ... r8 44

Kentucky ... ... ... ... ... 56 ... 8 44

Louisiana ... ... ... ... ... ... ... ... ...

Maine ... ... ... ... ... ... ... 11 ...

Maryland ... ... ... ... ... 56 ... ... 44

Massachusetts ... ... 30 44 32 55 ... ... ...

Michigan ... ... ... ... 40 56 ... ... 44

Minnesota ... ... ... ... 36 ... 40 r8 50

Mississippi 44 48 32 ... ... 56 ... ... 44

Missouri ... ... 33 ... ... 56 ... ... 44

Montana ... ... ... ... ... 56 ... ... 44

Nebraska ... ... ... ... ... 56 ... 8 44

Nevada ... ... ... ... ... 56 ... ... 48

New Hampshire ... ... ... ... 32 56 ... ... ...

New Jersey ... ... ... ... ... 55 ... ... ...

New York ... ... 30 44 ... 55 ... ... ...

North Carolina 48 48 30 44 ... 55 ... ... 44

North Dakota ... ... ... ... ... 56 ... ... ...

Ohio ... ... ... ... ... 56 ... ... 44

Oklahoma ... ... 32 ... ... 56 ... ... 44

Oregon ... ... ... ... ... ... ... ... ...

Pennsylvania ... ... ... ... ... ... ... ... ...

Rhode Island ... ... 30 44 ... 56 ... ... 44

South Carolina 48 48 30 ... ... ... ... ... ...

South Dakota ... ... ... ... ... 56 ... ... ...

Tennessee 50 48 28 ... ... 56 48 8 44

Texas ... ... 32 ... ... 56 ... ... 44

Vermont ... ... ... ... ... ... ... ... ...

Virginia ... ... 32 ... ... 56 ... 8 44

Washington ... ... ... ... ... 56 ... ... ...

West Virginia ... ... ... ... ... 56 ... ... ...

Wisconsin ... ... 30 44 ... 56 ... 8 44

State or Territory Herdes Grass Hungarian Grass Seed Millet Oats Onions Orchard Grass Seed Osage Orange Seed Parsnips Peaches

Onions Onion Sets

United States ... ... ... 32 57 ... ... ... ... ...

Alabama ... ... ... 32 ... ... ... ... ... ...

Arizona ... ... ... 32 ... ... ... ... ... ...

Arkansas ... ... 50 32 57 ... 14 ... ... ...

California ... ... ... 32 ... ... ... ... ... ...

Colorado ... ... ... 32 57 ... ... ... ... ...

Connecticut 45 ... ... 32 52 ... ... ... 45 ...

Delaware ... ... ... ... ... ... ... ... ... ...

Florida ... ... 50 32 56 ... ... ... ... 54

Georgia ... ... ... 32 57 ... ... ... ... ...

Hawaii ... ... ... 32 ... ... ... ... ... ...

Idaho ... ... ... 36 ... ... ... ... ... ...

Illinois ... ... ... 32 57 ... ... ... ... ...

Indiana ... ... 50 32 48 ... 14 33 55 ...

Iowa ... 50 50 32 57 ... 14 32 42 48

Kansas ... 50 50 32 57 ... ... ... 52 48

Kentucky ... 50 50 32 57 36 14 ... ... ...

Louisiana ... ... ... ... ... ... ... ... ... ...

Maine 45 ... ... 32 52 ... ... ... 45 ...

Maryland 45 50 50 32 57 ... 14 ... ... 40

Massachusetts 45 ... ... 32 52 ... ... ... 45 48

Michigan ... 50 50 32 54 ... 14 33 ... ...

Minnesota ... 48 48 32 52 ... 14 ... 42 ...

Mississippi ... 50 50 32 57 ... ... ... ... ...

Missouri ... 48 50 32 57 28 14 36 44 48

Montana ... 50 ... 32 57 ... ... ... 50 ...

Nebraska ... 50 50 32 57 25 ... 32 ... ...

Nevada ... 50 50 32 57 ... ... ... 50 48

New Hampshire 45 ... ... 32 52 ... ... ... 45 48

New Jersey ... ... ... 30 57 ... ... ... ... 50

New York 45 ... ... 32 57 ... ... ... ... ...

North Carolina ... ... 50 32 57 ... 14 ... ... ...

North Dakota ... ... 50 32 52 ... ... ... ... ...

Ohio ... 50 50 32 55 ... ... ... ... 48

Oklahoma ... ... 50 32 57 28 14 36 44 48

Oregon ... ... ... 32 ... ... ... ... ... ...

Pennsylvania ... ... ... 32 50 ... ... ... ... ...

Rhode Island ... 50 50 32 50 ... ... ... 50 48

South Carolina ... ... ... ... 52 ... ... ... ... ...

South Dakota ... ... ... 32 52 ... ... ... ... ...

Tennessee ... 48 50 32 56 28 14 33 50 50

Texas ... 48 50 32 57 ... ... ... ... 50

Vermont 45 ... ... 32 52 ... ... ... ... ...

Virginia 12 48 50 30 57 28 14 34 ... ...

Washington ... ... ... 32 ... ... ... ... ... ...

West Virginia ... ... ... 32 ... ... ... ... ... ...

Wisconsin ... 48 50 32 57 ... ... ... 44 ...

Table I. UNITED STATES LEGAL WEIGHTS (in pounds) PER BUSHEL—Continued

State or Territory Dried Peaches Peeled Peanuts Pears Peas Potatoes Sweet Potatoes Redtop Seed Rough Rice Ruta-
bagas Rye Meal Rye Shorts Sorghum Seed Tomatoes Timothy Seed Turnips Wheat

Green Peas Un-
shelled Peas

United States ... ... ... ... 60 60 ... ... ... ... ... 56 ... ... ... ... ... 60

Alabama 38 ... ... ... 60 60 55 ... ... ... ... 56 ... ... ... ... 55 60

Arizona ... ... ... ... ... ... ... ... ... ... ... 56 ... ... ... ... ... 60

Arkansas 33 ... ... ... 60 60 50 14 ... ... ... 56 ... 50 ... 60 57 60

California ... ... ... ... ... ... ... ... ... ... ... 54 ... ... ... ... ... 60

Colorado ... ... ... ... ... 60 ... ... ... ... ... 56 ... ... ... 45 ... 60

Connecticut 33 ... ... ... 60 60 54 ... 45 60 50 56 20 ... ... ... 50 60

Delaware ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... 60

Florida 33 22 60 ... ... 60 60 ... ... ... ... 56 ... 56 ... ... 54 60

Georgia 38 ... ... ... 60 60 55 ... 43 ... ... 56 ... ... ... 45 55 60

Hawaii ... 25 ... ... ... ... ... ... ... ... ... 56 ... ... ... ... ... 60

Idaho ... ... 45 ... ... 60 ... ... ... ... ... 56 ... ... ... ... ... 60

Illinois ... ... ... ... ... 60 50 ... ... ... ... 56 ... ... ... 45 55 60

Indiana ... ... ... ... ... 60 55 ... ... ... ... 56 ... ... ... 45 55 60

Iowa ... 20 ... 50 60 60 46 14 ... 50 ... 56 ... 50 50 45 55 60

Kansas ... ... ... ... ... 60 50 ... ... ... ... 56 ... 50 56 45 55 60

Kentucky 39 24 ... ... 60 60 55 ... ... ... ... 56 ... ... ... 45 60 60

Louisiana ... ... ... ... ... ... ... ... ... ... ... 32 ... ... ... ... ... 60

Maine ... ... ... ... 60 60 ... ... ... 60 50 50 ... ... ... ... 50 60

Maryland ... 22 ... ... 60 60 60 14 ... ... ... 56 ... 50 60 45 50 60

Massachusetts 33 20 58 ... 60 60 54 ... 45 ... 50 56 20 ... 56 45 55 60

Michigan 28 ... ... ... 60 60 56 14 ... ... ... 56 ... ... ... 45 58 60

Minnesota 28 ... ... ... 60 60 55 14 ... 52 ... 56 ... 57 ... 45 ... 60

Mississippi 33 24 ... ... 60 60 60 ... ... ... ... 56 ... 42 ... 45 55 60

Missouri 33 ... 48 56 60 60 56 14 ... 50 ... 56 ... 42 45 45 42 60

Montana ... ... 45 ... 60 60 ... ... ... ... ... 56 ... ... ... 45 50 60

Nebraska 33 ... ... ... 60 60 50 ... ... ... ... 56 ... 50 ... 45 55 60

Nevada ... ... 58 ... 60 60 50 ... ... ... ... 56 ... 50 56 45 56 60

New Hampshire ... 20 ... ... 60 60 54 ... ... ... 50 56 20 ... 56 45 55 60

New Jersey 33 ... ... ... 60 60 54 ... ... ... ... 56 ... ... ... 45 ... 60

New York 33 ... ... ... 60 60 54 ... 45 ... 50 56 20 ... ... 45 ... 60

North Carolina ... 22 ... ... 60 60 56 14 44 ... ... 56 ... 50 ... 45 50 60

North Dakota ... ... ... ... 60 60 46 ... ... ... ... 56 ... ... ... 45 60 60

Ohio 33 ... ... ... 60 60 50 ... ... ... ... 56 ... ... 56 45 60 60

Oklahoma ... 22 48 56 60 60 55 14 ... 50 ... 56 ... 50 45 45 60 60

Oregon 28 ... 45 ... ... 60 ... ... ... ... ... 56 ... ... ... ... ... 60

Pennsylvania ... ... ... ... ... 56 ... ... ... ... ... 56 ... ... ... ... ... 60

Rhode Island 33 ... ... ... 60 60 54 ... ... ... 50 56 20 ... 56 45 50 60

South Carolina ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...

South Dakota ... ... ... ... 60 60 46 ... ... ... ... 56 ... ... ... 42 60 60

Tennessee 26 23 56 30 60 60 50 14 ... ... ... 56 ... 50 56 45 50 60

Texas 28 ... ... ... ... 60 55 ... ... ... ... 56 ... ... 55 45 55 60

Vermont ... ... ... ... 60 60 ... ... ... ... ... 56 ... ... ... 45 60 60

Virginia 40 22 ... ... 60 56 56 12 ... ... ... 56 ... ... ... 45 55 60

Washington 28 ... 45 ... ... 60 ... ... ... ... ... 56 ... ... ... ... ... 60

West Virginia 33 ... ... ... ... 60 ... ... ... ... ... 56 ... ... ... 45 ... 60

Wisconsin 33 ... ... ... 60 60 54 ... 45 56 50 56 20 ... ... 45 42 60

State or Territory Dried Peaches Peeled Peanuts Pears Peas Potatoes Sweet Potatoes Redtop Seed Rough Rice

Green Peas Un-
shelled Peas

United States ... ... ... ... 60 60 ... ... ...

Alabama 38 ... ... ... 60 60 55 ... ...

Arizona ... ... ... ... ... ... ... ... ...

Arkansas 33 ... ... ... 60 60 50 14 ...

California ... ... ... ... ... ... ... ... ...

Colorado ... ... ... ... ... 60 ... ... ...

Connecticut 33 ... ... ... 60 60 54 ... 45

Delaware ... ... ... ... ... ... ... ... ...

Florida 33 22 60 ... ... 60 60 ... ...

Georgia 38 ... ... ... 60 60 55 ... 43

Hawaii ... 25 ... ... ... ... ... ... ...

Idaho ... ... 45 ... ... 60 ... ... ...

Illinois ... ... ... ... ... 60 50 ... ...

Indiana ... ... ... ... ... 60 55 ... ...

Iowa ... 20 ... 50 60 60 46 14 ...

Kansas ... ... ... ... ... 60 50 ... ...

Kentucky 39 24 ... ... 60 60 55 ... ...

Louisiana ... ... ... ... ... ... ... ... ...

Maine ... ... ... ... 60 60 ... ... ...

Maryland ... 22 ... ... 60 60 60 14 ...

Massachusetts 33 20 58 ... 60 60 54 ... 45

Michigan 28 ... ... ... 60 60 56 14 ...

Minnesota 28 ... ... ... 60 60 55 14 ...

Mississippi 33 24 ... ... 60 60 60 ... ...

Missouri 33 ... 48 56 60 60 56 14 ...

Montana ... ... 45 ... 60 60 ... ... ...

Nebraska 33 ... ... ... 60 60 50 ... ...

Nevada ... ... 58 ... 60 60 50 ... ...

New Hampshire ... 20 ... ... 60 60 54 ... ...

New Jersey 33 ... ... ... 60 60 54 ... ...

New York 33 ... ... ... 60 60 54 ... 45

North Carolina ... 22 ... ... 60 60 56 14 44

North Dakota ... ... ... ... 60 60 46 ... ...

Ohio 33 ... ... ... 60 60 50 ... ...

Oklahoma ... 22 48 56 60 60 55 14 ...

Oregon 28 ... 45 ... ... 60 ... ... ...

Pennsylvania ... ... ... ... ... 56 ... ... ...

Rhode Island 33 ... ... ... 60 60 54 ... ...

South Carolina ... ... ... ... ... ... ... ... ...

South Dakota ... ... ... ... 60 60 46 ... ...

Tennessee 26 23 56 30 60 60 50 14 ...

Texas 28 ... ... ... ... 60 55 ... ...

Vermont ... ... ... ... 60 60 ... ... ...

Virginia 40 22 ... ... 60 56 56 12 ...

Washington 28 ... 45 ... ... 60 ... ... ...

West Virginia 33 ... ... ... ... 60 ... ... ...

Wisconsin 33 ... ... ... 60 60 54 ... 45

State or Territory Ruta-
bagas Rye Meal Rye Shorts Sorghum Seed Tomatoes Timothy Seed Turnips Wheat

United States ... ... 56 ... ... ... ... ... 60

Alabama ... ... 56 ... ... ... ... 55 60

Arizona ... ... 56 ... ... ... ... ... 60

Arkansas ... ... 56 ... 50 ... 60 57 60

California ... ... 54 ... ... ... ... ... 60

Colorado ... ... 56 ... ... ... 45 ... 60

Connecticut 60 50 56 20 ... ... ... 50 60

Delaware ... ... ... ... ... ... ... ... 60

Florida ... ... 56 ... 56 ... ... 54 60

Georgia ... ... 56 ... ... ... 45 55 60

Hawaii ... ... 56 ... ... ... ... ... 60

Idaho ... ... 56 ... ... ... ... ... 60

Illinois ... ... 56 ... ... ... 45 55 60

Indiana ... ... 56 ... ... ... 45 55 60

Iowa 50 ... 56 ... 50 50 45 55 60

Kansas ... ... 56 ... 50 56 45 55 60

Kentucky ... ... 56 ... ... ... 45 60 60

Louisiana ... ... 32 ... ... ... ... ... 60

Maine 60 50 50 ... ... ... ... 50 60

Maryland ... ... 56 ... 50 60 45 50 60

Massachusetts ... 50 56 20 ... 56 45 55 60

Michigan ... ... 56 ... ... ... 45 58 60

Minnesota 52 ... 56 ... 57 ... 45 ... 60

Mississippi ... ... 56 ... 42 ... 45 55 60

Missouri 50 ... 56 ... 42 45 45 42 60

Montana ... ... 56 ... ... ... 45 50 60

Nebraska ... ... 56 ... 50 ... 45 55 60

Nevada ... ... 56 ... 50 56 45 56 60

New Hampshire ... 50 56 20 ... 56 45 55 60

New Jersey ... ... 56 ... ... ... 45 ... 60

New York ... 50 56 20 ... ... 45 ... 60

North Carolina ... ... 56 ... 50 ... 45 50 60

North Dakota ... ... 56 ... ... ... 45 60 60

Ohio ... ... 56 ... ... 56 45 60 60

Oklahoma 50 ... 56 ... 50 45 45 60 60

Oregon ... ... 56 ... ... ... ... ... 60

Pennsylvania ... ... 56 ... ... ... ... ... 60

Rhode Island ... 50 56 20 ... 56 45 50 60

South Carolina ... ... ... ... ... ... ... ... ...

South Dakota ... ... 56 ... ... ... 42 60 60

Tennessee ... ... 56 ... 50 56 45 50 60

Texas ... ... 56 ... ... 55 45 55 60

Vermont ... ... 56 ... ... ... 45 60 60

Virginia ... ... 56 ... ... ... 45 55 60

Washington ... ... 56 ... ... ... ... ... 60

West Virginia ... ... 56 ... ... ... 45 ... 60

Wisconsin 56 50 56 20 ... ... 45 42 60

* Not defined.

a Small white beans, 60 pounds.

b Green apples.

c Sugar beets and mango-wurzels.

d Shelled beans, 60 pounds; velvet beans, 78 pounds.

e White beans.

f Wheat bran.

g Corn in ear, 70 pounds until December 1st next after growth; 68 pounds thereafter.

h English blue-grass seed, 22 pounds; native blue-grass seed, 14 pounds.

i Rice corn.

j Corn in ear from November 1st to May 1st following, 70 pounds; 68 pounds from May 1st to November 1st.

k Soy beans, 58 pounds.

l Cracked corn, 50 pounds.

m Green unshelled beans, 30 pounds.

n Cannel coal, 70 pounds.

o Standard weight in borough of Greensburg, 75 pounds.

p Dried beans; green unshelled beans, 30 pounds.

q Red and white.

r Unwashed plastering hair, 8 pounds; washed plastering hair, 4 pounds.

Table II. LEGAL WEIGHTS PER BUSHEL FIXED IN BUT ONE OR TWO STATES

Article Weight States

Pounds

Blackberries 30 Iowa. Tennessee, 48 pounds; dried 28 pounds.

Blueberries 42 Minnesota.

Canary seed 60 Tennessee.

Cantaloupe melon 50 Tennessee.

Cement 80 Tennessee.

Cherries 40 Iowa. Tennessee, with stems 56 pounds; without stems, 64 pounds.

Chestnuts 50 Tennessee. Virginia, 57 pounds.

Cotton seed, staple 42 South Carolina.

Cucumbers 48 Iowa, Tennessee, Missouri. Wisconsin, 50 pounds.

Currants 40 Iowa and Minnesota.

Grapes 40 Iowa. Tennessee, with stems, 48 pounds; without stems, 60 pounds.

Hickory nuts 50 Tennessee.

Hominy 60 Ohio. Tennessee, 62 pounds.

Horse-radish 50 Tennessee.

Kaffir corn 56 Kansas.

Kale 30 Tennessee.

Land plaster 100 Tennessee.

Mustard 30 Tennessee.

Plums 40 Florida. Tennessee, 64 pounds.

Plums, dried 28 Michigan.

Popcorn 70 Iowa, Indiana, Tennessee. Ohio, in the ear, 42 pounds; Iowa, shelled, 56 pounds.

Prunes, dried 28 Idaho; green, 45 pounds.

Quinces 48 Florida, Iowa and Tennessee.

Rape seed 50 Wisconsin.

Raspberries 32 Iowa, Kansas, Tennessee, 48 pounds.

Rhubarb 50 Tennessee.

Salads 30 Tennessee.

Sand 130 Iowa.

Spinach 30 Tennessee.

Strawberries 32 Iowa. Tennessee, 48 pounds.

Sugar-cane seed 57 New Jersey.

Velvet-grass seed 7 Tennessee.

Walnuts 50 Tennessee, Iowa.

Apothecaries’ Weight

Apothecaries’ Weight is used by apothecaries and physicians weighing medicines for prescriptions.

TABLE

20 grains (gr.) = 1 scruple (sc., or ℈)

2 scruples = 1 dram (dr., or ʒ)

8 drams = 1 ounce (oz., or ℥)

12 ounces = 1 pound (lb., or ℔)

Pound Ounces Drams Scruples Grains

1 = 12 = 96 = 288 = 5760

1. In writing prescriptions, physicians express the number in Roman characters, using j instead of i final. They also write the symbol first; thus: ℥v, ʒvj, ℈ij.

Medical Signs and Abbreviations

℞ (Lat. Recipe), take; āā, of each; ℔, pound; ℥, ounce; ʒ, drachm; ℈, scruple; ♏, minim, or drop; O or o, pint; f℥, fluid ounce; fʒ, fluid drachm; as, ℥ss, half an ounce; ℥i, one ounce; ℥iss, one ounce and a half; ℥ij, two ounces; gr. grain; Q. S., as much as sufficient; Ft. Mist., let a mixture be made; Ft. Haust., let a draught be made; Ad., add to; Ad lib., at pleasure; Aq., water; M., mix; Mac., macerate; Pulv., powder; Pil., pill; Solv., dissolve; St., let it stand; Sum., to be taken; D., dose; Dil., dilute; Filt., filter; Lot., a wash; Garg., a gargle; Hor. Decub., at bed time; Inject., injection; Gtt., drops; ss, one-half; Ess., essence.

Comparison of Weights

TABLE

1 pound avoirdupois = 7000 grains

1 ounce avoirdupois = 437⁄2 grains

1 pound Troy, or apothecary = 5760 grains

1 ounce Troy, or apothecary = 480 grains

Circular Measures

Circular or Angular Measures are used in surveying, navigation, astronomy, geography, reckoning latitude and longitude, and computing differences in time.
A Circle is a plane figure bounded by a curved line, every point of which is equally distant from a point within, called the center.
The Circumference is the bounding line of a circle.
The Radius of a circle is a straight line drawn from the circumference to the center.
The Diameter is a straight line drawn through the center, with the ends terminating in the circumference.
An Arc of a circle is any portion of the circumference.
An Angle is the difference in direction between two straight lines which meet.
If two diameters divide a circle into four equal parts, these diameters make right angles with each other.
An angle less than a right angle is an acute angle.
The circumference of a circle may be divided into 360 equal parts, called degrees. If the circle is large, the degree is large, and if the circle is small, the degree is small, but the degree is always ⁄360 part of the circumference, whatever the size of the circle.
An angle at the center of a circle is measured by the arc which bounds it.
If the angle is a right angle, it is measured by ⁄4 of 360 degrees, or 90 degrees; hence, any angle of 90 degrees is a right angle.
An acute angle is always less than 90 degrees.
An obtuse angle is always more than 90 degrees.

Table of Circular Measure

60 seconds (′′) = 1 minute (′)

60 minutes = 1 degree (°)

360 degrees = 1 circumference (cir.)

Circum-
ference Degrees Minutes Seconds

1 = 360 = 21,600 = 1,296,000

A quadrant is ⁄4 of a circumference, or 90°; a sextant is ⁄6 of a circumference, or 60°.
The length of a degree of longitude on the earth’s surface at the Equator is 69.16 miles.
In astronomical calculation 30° are called a sign, and there are therefore 12 signs in a circle.

Longitude and Time

The earth’s circumference (which has the form of a circle) at the equator is (3.1416 × 7926), 24900 miles; which divided by 360, gives 69.17 miles for 1 degree of longitude at the equator. Leaving the equator, degrees of longitude gradually diminish, since all meridians converge at the poles. Thus, 1 degree of longitude, at 10 degrees of latitude, is 68.1 miles; at 20 degrees 65 miles; at 30 degrees 59.9 miles; at 40 degrees 53 miles; at 50 degrees 44.5 miles; at 60 degrees 34.6 miles; at 70 degrees 23.7 miles; at 80 degrees 12 miles; at 90 degrees 0.
Imaginary lines running north and south, through these degrees, from pole to pole, are called meridians. Those east and west, parallels.
One meridian which runs through Greenwich, near London, England, is called the prime meridian, and all the other meridians are reckoned as east or west of it.
Longitude is distance east or west of the prime meridian. When we say that the longitude of Paris is 2° 20′ East, we mean that the meridian running through Paris is 2° 20′ east of the prime meridian that runs through Greenwich, England. The longitude of Washington, D. C., is 77° 7′ West. That means that the meridian which passes through Washington is 77° 7′ west of the prime meridian.
The longitude of a place tells in degrees, minutes, and seconds, the distance it is east or west of the prime meridian.

Rule.—To find the difference of time between two places, when the difference of longitude is known, or vice versa, multiply the given longitude, expressed in degrees, by 4. This gives the equivalent time in minutes. Dividing the given time, expressed in minutes, by 4, gives the equivalent longitude in degrees.
Example: The difference of longitude between Boston and San Francisco is nearly 51⁄4°, what is the difference of time?
51⁄4 × 4 = 205. 205 minutes equal 3 hours and 25 minutes. Ans. 3 hours and 25 minutes.
The difference of time between London and New York is about 4 hours and 55⁄2 minutes, what is the difference of longitude?
4 hours and 55⁄2 minutes equals 295⁄2 minutes. 295⁄2 ÷ 4 = 73⁄8. Ans. 73⁄8°.

Measures of Time

The unit of time measurement is the same among all nations. Practically it is 1-86400 of the mean solar day, but really it is a perfectly arbitrary unit, as the length of the mean solar day is not constant for any two periods of time. There is no constant natural unit of time.
Time measures are used in telling the time of day, in problems in longitude and time, in figuring interest on notes and bills, and in numerous other ways.

TABLE OF THE DIVISIONS OF TIME

60 seconds (sec.) = 1 minute (min.)

60 minutes = 1 hour (hr.)

24 hours = 1 day (da.)

7 days = 1 week (wk.)

30 days = 1 commercial month (mo.)

52 weeks = 1 year (yr.)

12 months = 1 year

360 days = 1 commercial year

365 days = 1 common year

366 days = 1 leap year

100 years = 1 century

Century Years Months Days Hours Minutes

1 = 100 = 1,200 = 36,500 = 876,000 = 52,560,000

Centennial years exactly divisible by 400, and other years exactly divisible by 4, are Leap Years.

Why We Have Leap Year

The average time it takes the earth to revolve once around the sun (one year) is 365 days, 5 hours, 48 minutes, 47.8 seconds, or about 365⁄4 days.
The change in the length of the mean sidereal day, i.e., of the time of the earth’s rotation upon its axis, amounts to 0.01252 seconds in 2400 mean solar years.
Instead of reckoning this part of a day each year, it is disregarded, and an addition is made when this amounts to one day, which is very nearly every fourth year. This addition of one day is made to the month of February. Since the part of a day that is disregarded when 365 days are considered as a year, is a little less than one quarter of a day, the addition of one day every fourth year is a little too much, and, to correct this excess, addition is made to only every fourth centennial year.

Standard Time

By this is meant time which differs from Greenwich mean time by whole hours.
The earth revolves on its axis from west to east, nearly 17.3 miles in 1 minute at the equator; at the latitude of New Orleans, nearly 15 miles in 1 minute; at Memphis, 14 miles; at Chicago, 13 miles; at London, 10.8 miles; at St. Petersburg, 8.6 miles. That is, a watch would gain one minute going west, or lose one minute going east that distance, in the latitudes of the respective cities.
The globe is divided into zones of 15 degrees or one hour breadth, the Greenwich meridian being in the center of the zero zone. Thus Belgium and Holland (since 1892) keep Greenwich time; Denmark, Sweden, Switzerland (1894), Austrian railroads, Germany, and Italy (1893) keep the time of longitude 15 degrees East—i.e. one hour earlier than Greenwich. In North America again five zones are distinguished. The corresponding times are distinguished as Eastern (67⁄2 to 82⁄2 degrees), Central (82⁄2 to 97⁄2 degrees), Mountain (97⁄2 to 112⁄2 degrees), and Pacific (112⁄2 to 127⁄2 degrees) times.
New York people are in the Eastern Time Belt. If they rise at six o’clock in the morning, they will be up a whole hour before Chicago people, who get up at the same hour.

The clock at Greenwich, near London, England, from which the standard time of the world is reckoned.

Thus, each day begins an hour sooner in New York than in Chicago, two hours before Denver, and three hours before San Francisco.
Standard time in Japan is nine hours earlier than Greenwich time.
In the western parts of Canada the twenty-four hour system has been adopted, under which four P. M. becomes sixteen o’clock and so on. Steps are being taken to introduce it generally in India, Belgium, and the United States. It is of special convenience in the construction of railroad time tables; and it has long been used by the Italians and by astronomers.

SIMULTANEOUS TIME IN LONDON, NEW YORK, AND BOMBAY

This diagram illustrates the curious fact that a telegram despatched from London may be delivered in New York apparently before the time it was sent off, and why a telegram apparently takes so long to reach Bombay.

COMPARATIVE TIME ALL OVER THE WORLD WHEN NOON AT GREENWICH

The Calendar

The reckoning of time among the ancients was very inaccurate. This was owning to their ignorance of astronomy, and also to changes that were made from time to time for political reasons. The calendar was reformed by Julius Cæsar, 46 B. C., who made the year consist of 365⁄4 days, adding one day every fourth year. In 1582, the error in the calendar established by him had increased to 10 days; that is, too much time had been reckoned as a year, until the civil year was 10 days behind the solar year. To correct this error, Pope Gregory XIII. decreed that 10 days should be stricken from the calendar, that the day following the 3d day of October, 1582, should be made the 14th, and that henceforth only those centennial years should be leap years which are divisible by 400.
Most Catholic countries adopted the Gregorian Calendar soon after it was established. Great Britain did not adopt it until 1752, when the error amounted to 11 days. By Act of Parliament, the 3d of September was called the 14th. The civil year by the same act was made to commence on the 1st of January, instead of the 25th of March, as was previously the case.

THE COMPARATIVE TIME ZONES OF THE WORLD

Dates reckoned by the Julian calendar are called Old Style (O.S.), and those reckoned by the Gregorian calendar are called New Style (N.S.). The difference now amounts to 12 days.

Perpetual Calendar

To find the day of the week for any given date.
1. Take the last two figures of the year, add one-fourth of them (neglecting remainder). Thus: 1949 = 49 + 12 = 61.
2. Add for the month, if for January or October, 1; May, 2; August, 3; February, March, or November, 4; June, 5; September or December, 6; April or July, 0; if leap year (that is, if it be divisible by 4 without remainder) January 0; February 3.
3. Add day of month.
Divide the sum of these three by 7, and remainder gives the number of the day of the week.
Thus:—What day of the week is 15th July, 1908?

1. 8 + 2 = 10

2. July = 0

3. 15th = 15

25 = 7 × 3 + 4

4th day of the week = Wednesday.
What day of the week was December 25th, 1905?

1. 5 + 1 = 6

2. Dec. = 6

3. 25th = 25

37 = 7 × 5 + 2

2nd day of the week = Monday.
The above only applies to 20th Century. For 19th Century, add 2; for 21st Century, add 6; 18th Century, 4; but before 1752 the “old style” was used.

Where the Day Begins

The day begins earlier as you go east until you meet the 180th meridian. This is where the day begins. Starting here, it travels westward, giving the whole world a new day. The 180th meridian is called the International Date Line (I. D. L.) but in reality, the date line is a crooked line which zigzags across the 180th meridian.
From the time the day starts at the International Date Line, until the sun again reaches that line, the same day is in progress the world over.
As marked now, the International Date Line passes southward through Behring Sea, then westerly, then returns to the 180th meridian at about 40 degrees north. It then follows the 180th meridian to 10 degrees south, where it swerves east but returns again to the 180th meridian at about 50 degrees south. It then follows that meridian.

Time on Shipboard.—The twenty-four hours are divided on board ship into seven parts, and the crew is divided into two parts or watches, designated port and starboard watches. Each watch is on duty four hours, except from four to eight p. m., which time is divided into two watches of two hours each, called dog watches, by means of which the watches are changed every day, and each watch gets a term of eight hours’ rest at night. First watch, eight p. m. to midnight; middle watch, midnight to four a. m.; morning watch, four to eight a. m.; forenoon watch, eight a. m. to noon; afternoon watch, noon to four p. m.; first dog watch, four to six p. m.; second dog watch, six to eight p. m. The bell is struck every half-hour to indicate the time, as follows:

1 bell 12: 30 A.M. 1 bell 12: 30 P.M.

2 bells 1: 00 A.M. 2 bells 1: 00 P.M.

3 bells 1: 30 A.M. 3 bells 1: 30 P.M.

4 bells 2: 00 A.M. 4 bells 2: 00 P.M.

5 bells 2: 30 A.M. 5 bells 2: 30 P.M.

6 bells 3: 00 A.M. 6 bells 3: 00 P.M.

7 bells 3: 30 A.M. 7 bells 3: 30 P.M.

8 bells 4: 00 A.M. 8 bells 4: 00 P.M.

1 bell 4: 30 A.M. 1 bell 4: 30 P.M.

2 bells 5: 00 A.M. 2 bells 5: 00 P.M.

3 bells 5: 30 A.M. 3 bells 5: 30 P.M.

4 bells 6: 00 A.M. 4 bells 6: 00 P.M.

5 bells 6: 30 A.M. 1 bell 6: 30 P.M.

6 bells 7: 00 A.M. 2 bells 7: 00 P.M.

7 bells 7: 30 A.M. 3 bells 7: 30 P.M.

8 bells 8: 00 A.M. 4 bells 8: 00 P.M.

1 bell 8: 30 A.M. 1 bell 8: 30 P.M.

2 bells 9: 00 A.M. 2 bells 9: 00 P.M.

3 bells 9: 30 A.M. 3 bells 9: 30 P.M.

4 bells 10: 00 A.M. 4 bells 10: 00 P.M.

5 bells 10: 30 A.M. 5 bells 10: 30 P.M.

6 bells 11: 00 A.M. 6 bells 11: 00 P.M.

7 bells 11: 30 A.M. 7 bells 11: 30 P.M.

8 bells 12: 00 noon 8 bells 12: 00 night

How the Months Got Their Names

January, from Janus, was the sacred month of the year to the Romans. To them, Janus was the god of the year. During the 18th century, the Europeans started to recognize it as the first month, but previous to this, March was considered the first.
February comes from februa, the name of a Roman festival celebrated on the 15th of the second month.
March is from Mars, the god of war. March was the first month of the year to the Romans.
April, from the Latin aperire, “to open,” was probably so called because during this month buds begin to open.
May is from Maia, the mother of Mercury. The Romans offered sacrifices to this goddess on the first day of May.
The sixth month in our calendar, June, got its name from Juno, the wife of Jupiter.
July was so named in honor of Julius Cæsar, who was born in this month.
Emperor Augustus Cæsar commanded that the eighth month be named August after him.
September is from the Latin septem, meaning seven. At the time when March was the first month of the year, September was the seventh.
October, November, and December were originally the eighth, ninth and tenth months. Octo, novem, and decem are Latin numerals for eighth, ninth, and tenth.

How the Days Got Their Names

Sunday (that is, day of the sun, like Monday day of the moon), the first day of the week, the Lord’s day, was sacred to Sol or the Sun.
Monday (that is, moon-day; Anglo-Saxon, Monandæg, German, Montag), the second day of our week, was formerly sacred to the moon.
Tuesday, the third day of the week, is so called from Tiwesdæg, the day of Tiw or Tiu, the old Saxon name for the god of war. The day bears a corresponding name in the other Germanic dialects.
Wednesday, the fourth day of the week, the Dies Mercurii of the Romans, the Mittwoch of the modern Germans. The name Wednesday is derived from the Northern mythology, and signifies Woden’s or Odin’s day. The Anglo-Saxon form was Wôdanesday, the Old German Woutanestac. The Swedish and Danish is Onsdag.
Thursday, (Swedish Thorsdag, German Donnerstag), the fifth day of the week, is so called from Donar, or Thor (see Dictionary of Myths), who, as god of the air, had much in common with the Roman Jupiter, to whom the same day was dedicated. (Latin Jovis dies, French Jeudi).
Friday, the sixth day of the week, from the Anglo-Saxon Frige-dæg, is the day sacred to Frigga or to Freya, the Saxon Venus.
Saturday (Anglo-Saxon Sæterdæg, Sæterndæg—Sæter, Sætern, for Saturn, and dæg, a day—the day presided over by the planet Saturn), is the seventh or last day of the week; the day of the Jewish Sabbath.

Measures of Value

The common measure of value is Money.
It is also called Currency, and is of two kinds, viz.: coin and paper money.
Stamped pieces of metal having a value fixed by law are Coin or Specie.
Notes and bills issued by the government and banks, and authorized to be used as money, are Paper Money.
All moneys which, if offered, legally satisfy a debt are a Legal Tender.

United States Money

The unit of United States or Federal money is the Dollar.
The dollar mark is probably a combination of U. S., the initials of the words “United States.”
The standard of United States money is the gold dollar. Gold is used because in itself it has great worth and little bulk, and because it varies very little in value.

Names of United States Coins

Bronze: Nickel:

One-cent piece Five-cent piece

Silver: Gold:

Dollar = $1.00 Double eagle = $20.00

Half-dollar = 0.50 Eagle = 10.00

Quarter-dollar = 0.25 Half eagle = 5.00

Dime = 0.10 Quarter eagle = 2.50

It may be interesting to know that the word dollar is supposed to have come from Dale, the name of a small town where dollars were first coined.
Dime is from the French word disme, which means tenth.
Cent comes from the Latin word centum, meaning hundred.
Mill is also from the Latin, coming from mille, a thousand.
Eagles were named after our national bird.

Weights of the United States Coins
And the Amounts for Which They are Legal Tender

GOLD

Denominations Weight
Grains Amount for Which
a Legal Tender

Double eagle, $20 516. Gold coins of denomination are
legal tenders for any amount.

Eagle, $10 258.

Half eagle, $5 129.

Three dollars 77. 4

Quarter eagle, $2.50 64. 5

Dollars 25. 8

SILVER

Denominations Weight
Grains Amount for Which
a Legal Tender

Standard dollar 412. 5 Unlimited.

Trade Dollar 420. Demonetized—Not a legal tender.

Half dollars 192. 9 Ten dollars.

Quarter dollars 96. 45 Ten dollars.

Twenty-cent pieces 77. 16 Five dollars.

Dimes 38. 58 Ten dollars.

Half-dimes 19. 29 Five dollars.

Three-cent pieces 11. 52 Five dollars.

MINOR COINS

Denominations Weight
Grains Amount for Which
a Legal Tender

Five cents 77. 6 Twenty-five cents.

Three cents 30. Twenty-five cents.

Two cents 96. Twenty-five cents.

Cents 48. Twenty-five cents.

Besides the coins there is paper money, founded on credit. It represents value, but in itself has no value.
This paper money is made up of paper promises to pay the amounts named, in gold or silver, on demand.
It includes bank bills, United States treasury notes, government bonds, etc. They represent the values $1, $2, $5, $10, $20, $50, $100, $500, $1,000 and $10,000.

Notation of United States Money

Dollars and cents are written together. Thus, two dollars and sixteen cents is written, $2.16.
The dollars are separated from the cents by a period. If the number of cents is less than ten, the tens’ place is filled by a 0. Thus, we write twenty dollars and two cents, $20.02.
Mills, or tenths of a cent, are written to the right of the cents. Five dollars, six cents, four mills is written, $5.064.
Note.—The rules and processes of decimals apply to the addition, subtraction, multiplication, and division of United States money.

English or Sterling Money

Sterling Money is currency of Great Britain and Ireland.

Table of Sterling Money

4 farthings (far.) = 1 penny (d.)

12 pence (not pennies) = 1 shilling (s.)

20 shillings = 1 pound (£), or sovereign

5 shillings = 1 crown

21 shillings = 1 guinea

The standard unit of Sterling Money is 1 pound or sovereign, whose value in our money is $4.8665.
The coins of Great Britain in general use are:—
Gold: Sovereign, half-sovereign, and guinea, which is equal to 21 shillings.
Silver: The crown (equal to 5 shillings), half-crown, florin (equal to 2 shillings), shilling, six-penny and three-penny pieces.
Copper: Penny and half-penny.
Example: I have £5 sterling. What is the value in United States money?

Solution:
The value is 5 × $4.8665, or $24.33

French Money

In France the currency is decimal. The unit is the Franc.

TABLE

10 centimes (ct.) [pronounced son-teems] = 1 decime (de.)

10 decimes [pronounced des-seems] = 1 franc (fr.)

Scale.—Decimal

The value of the franc, as determined by the Secretary of the Treasury, is $.193 in United States money.
The coins of France are of gold, silver, bronze, and copper. The gold coins are the hundred, forty, twenty, ten, and five franc pieces; the silver coins are the five, two, and one franc pieces; also the fifty and twenty-five centime pieces. The bronze coins are the ten, five, two, and one centime pieces. There are also copper coins in ten and five centime pieces.
Example: When in France, I bought goods as follows:—

3 books at 2 francs,
⁄2 dozen pipes at 1 franc,
2 pictures at 4 francs.

What was the cost in United States money?

Work:

3 books at 2 francs cost 6 francs

⁄2 dozen pipes at 1 franc cost 6 francs

2 pictures at 4 francs cost 8 francs

Cost of all 20 francs

20 francs = 20 × 19.3 cents, or $3.86

German Money

German money is legal currency of the German Empire.

TABLE

100 pfennigs = 1 mark

Scale.—Decimal

1. The unit is the mark. Its value is $.2385 in United States money.
2. The coins of the German Empire are of gold, silver, nickel, and copper. The gold coins are the 20-mark piece, the 10-mark piece, and the 5-mark piece. The silver coins are the two and one mark pieces; the nickel coins are the ten and five pfennig pieces; and the copper coins are the two and one pfennig pieces.

Philippines Weights and Measures

1 pulgada (12 linea) = .927 inch

1 pie = 11 .125 inches

1 vara = 33 .375 inches

1 gantah = .8796 gallon

1 caban = 21 .991 gallons

1 libra (16 onzo) = 1 .0144 pounds average

1 arroba = 25 .360 pounds average

1 catty (16 tael) = 1 .394 pounds average

1 pecul (100 catty) = 139 .482 pounds average

Paper Measure

24 sheets = 1 quire (qr.)

20 quires = 1 ream (rm.)

2 reams = 1 bundle

5 bundles = 1 bale

Although a ream contains 480 sheets, 500 sheets are usually sold as a ream.

Number Table

12 units = 1 dozen

12 dozen = 1 gross

12 gross = 1 great gross

20 units = 1 score

Percentage and Its Business Applications

The expression “per cent,” which is an abbreviation of the Latin words “per centum,” means “for each hundred.”
The symbol % is often used to denote “per cent.” Thus, 7 per cent, or 7%, means 7 parts out of every 100 parts, i.e., ⁄100 of the whole.
Since per cent means hundredths, we may write any fraction whose denominator is 100 as so many per cent. In some cases the corresponding common fractions are so simple that it is advisable to remember them. For example:

25% = 25100 = 14, 50% = 50100 = 12,

75% = 75100 = 34, 33⁄3% = 33⁄3100 = 13,

66⁄3% = 66⁄3100 = 23, 5% = 5100 = 120,

2⁄2% = 2⁄2100 = 140, 12⁄2% = 12⁄2100 = 18,

and so on.

The number per cent is called the rate per cent.

Table of Additional Values

Symbol Decimal Common
Fraction

1% = .01 = ⁄100

2% = .02 = ⁄100 = ⁄50

3% = .03 = ⁄100

4% = .04 = ⁄100 = ⁄25

5% = .05 = ⁄100 = ⁄20

6% = .06 = ⁄100 = ⁄50

7% = .07 = ⁄100

8% = .08 = ⁄100 = ⁄25

9% = .09 = ⁄100

10% = .10 = ⁄100 = ⁄10

20% = .20 = ⁄100 = ⁄5

25% = .25 = ⁄100 = ⁄4

50% = .50 = ⁄100 = ⁄2

100% = 1 .00 = ⁄100 = 1

Here are a few others that should be learned:—

6⁄4% = ⁄16 of 100% 16⁄3% = ⁄6 of 100%

8⁄2% = ⁄12 of 100% 33⁄3% = ⁄3 of 100%

12⁄2% = ⁄8 of 100% 66⁄3% = ⁄3 of 100%

A Decimal as Per Cent

Write the decimal as hundredths, and the number expressing the number of hundredths is the per cent.

Examples:

.4 = .40 = 40100 = 40%

.8 = .80 = 80100 = 80%

.25 = 25100 = 25%

.33 ⁄3 = 33⁄3100 = 33⁄3%

.50 = 50100 = 50%

.87 ⁄2 = 87⁄2100 = 87⁄2%

If the decimal has more than two decimal places, the figures after the second one are written as a fraction of a per cent, as,—

.255 = 25⁄2100 = 25⁄2%.

.163 = 16⁄10100 = 16⁄10%.

To change a common fraction to per cent:
1. Change the fraction to a decimal.
2. Express the decimal as hundredths.
3. The result is the per cent desired.

Examples:

⁄2 = .5 = .50 = 50%

⁄4 = .75 = 75%

⁄3 = .66⁄3 = 66⁄3%

⁄10 = .90 = 90%

⁄9 = .88⁄9 = 88⁄9%

⁄8 = .87⁄2 = 87⁄2%

⁄26 = .96⁄13 = 96⁄13%

Or, they may be written this way:

⁄4 = ⁄4 of 100100 = 75100 = 75%

⁄3 = ⁄3 of 100100 = 66⁄3100 = 66⁄3%

⁄2 = ⁄2 of 100100 = 50100 = 50%

Terms Used in Percentage

In Percentage, there are five terms or quantities considered; namely, the Base, Rate per cent, Percentage, Amount and Proceeds or Difference; any two being given, a third one may be found.
The base and rate given, to find the percentage.
Rule.—Multiply the base by the rate per cent expressed decimally.

Example: How many dollars is 6% of $50?

$50, the Base, or number on which percentage is computed.

.06, the Rate, or term denoting number of hundredths taken.

$3.00, the Percentage, or the product of the base and rate per cent.

$53.00, the Amount, or the base increased by the percentage.

$47.00, the Proceeds, or Difference, the base less the percentage.

Ans. $3.00.

When the rate per cent is an aliquot part of 100, the percentage is readily found by taking such a part of the base as the rate per cent is part of 100. Thus, at 10%, take ⁄10 of base; at 12⁄2%, ⁄8; at 16⁄3%, ⁄6, etc.
The base and percentage given, to find the rate.
Rule.—Divide the percentage by 1% of the base
Example: Bought a watch for $15 and sold it for $18; what per cent did I make?

.15 ) 3.00

20

Ans. 20%

Here, $15.00 is the base, and ($18 - $15) $3.00, the gain or percentage. Now, as 1% of 15.00 is .15, it is evident that 3.00 is as many per cent of 15.00, as .15 is contained times is 3.00, which is 20.

Proof: 20% or ⁄5 of $15 = $3.

The percentage and rate given, to find the base.
Rule.—Divide the percentage by the rate per cent expressed decimally.
Example: Received $6.40, percentage or interest, for money loaned at 4%, what was the base or principal?

.04 ) 6.40

Ans. $160

If $1 produces .04 (4 cents) in a certain time, $6.40 must be the percentage of as many dollars as .04 is contained times in $6.40, which is 160.

Proof: 4% of $160 (160 × .04) = $6.40.

The amount and rate given, to find the base.
Rule.—Divide the given amount by 1.00 plus the rate per cent.
Example: Bought a horse at a certain price, and sold him for $84, making 12% on cost; what did he cost?

1.12 ) 84.00

Ans. $75

If I made 12% on cost, every dollar invested gained 12 cents; hence, the horse cost as many dollars as 1.12 is contained times in 84.00, which is 75.

Proof: 12% of $75 (75 × .12) = $9; $75 + $9 = $84.

The proceeds and the rate given, to find the base.
Rule.—Divide the given proceeds by 1.00 minus the rate per cent.
Example: Sold a wagon for $51, which is 40% less than it cost; what did it cost?

.60 ) 51.00

Ans. $85

If I lost 40%, or 40 cents on the dollar, I received only 60 cents for every dollar the wagon cost; hence, it cost as many dollars as .60 is contained times in 51.00, which is 85.

Proof: 40% of $85 (85 × .40) = $34; $85 - $34 = $51.

Note.—The principles of percentage, in one form or another, enter into nearly all commercial calculations, besides many others. It is therefore of the utmost importance to business men, clerks, accountants, bookkeepers, and others, to become expert in percentage, and to adopt the easiest, simplest and shortest methods in computing interest, partial payments, trade discount, profit and loss, commission, insurance, stocks, bonds, taxes, exchange, etc.

Profit and Loss

When a thing is sold for more than it cost the seller, it is said to be sold at a profit. If it is sold for less than the cost, it is sold at a loss. Hence,

Profit = Selling Price - Cost Price.

Loss = Cost Price - Selling Price.

A profit or loss is generally reckoned as a percentage.
It is always understood that the percentage is reckoned on the cost price.
Example: I buy wheat at 60 cents and sell it for 75 cents. What per cent do I gain?
Solution: I gain the difference between 75 cents and 60 cents, or 15 cents. 15 cents is 25% of the cost. Hence, I gain 25%.

Work:

75 cents - 60 cents = 15 cents.

15 cents ÷ 60 cents = .25, or 25%.

Example: I bought flour at $3.50 per barrel. For what must I sell it to gain 20%?
Solution: I must sell it for 100% of the cost plus 20% of the cost, or 120% of the cost.

120% of $3.50 = $4.20.

∴ I must sell it at $4.20.

Example: I sold my carriage for 80% of its cost and received $90 for it. What was the cost?

Solution:

1% of the cost is ⁄80 of $90, or $1.125.

100% of the cost = 100 × $1.125, or $112.50.

Commission

is a percentage paid for buying or selling real estate, goods, etc. A consignment is a quantity of goods, sent to an agent, broker or commission merchant, for sale. The consignor is the one who sends the goods, the consignee the one to whom they are sent.

Principles:

1. The commission is some number or per cent of the price of what is bought or sold.
2. The proceeds equal the selling price minus the commission.
3. The amount equals the selling price plus the commission.
Commission presents two classes of problems. One of these classes may be called “buying problems.” The other may be called “selling problems.”
Buying Problem: I sent my agent $1977.60 to buy wild farm lands in northern Wisconsin, at $3 per acre. He was to receive 3% for his work. How many acres did he buy?
Work and Explanation:

3% of $3 = $.09.
Cost to me of 1 acre is $3 + .09 = $3.09

For $1977.60 he buys as many acres as $3.09 is contained times in $1977.60, or 640. Hence, he buys 640 acres.
Selling Problem: My agent sells 360 pounds of butter for me at 20 cents. He pays $4.20 freight charges and $9.60 for storage. His commission is 5%. What does he send me?
Work and Explanation:

360 pounds at 20 cents = $72.00

Freight is $4.20

Storage is 9.60

Commission is 5% of $72, or 3.60

Total charges = 17.40

He sends me the difference, or $54.60

Trade Discount

is an allowance made by manufacturers and jobbers from their list or marking prices. When the market varies, they change the discount accordingly, or make several discounts instead of changing the list.

Trade discount is a certain per cent off, or from list or marking price; while profit and loss is computed on the cost or purchase price.
The amount of the discount allowed depends sometimes upon the amount of order, and sometimes upon the terms of settlement. Very often two or more discounts are deducted in succession. Thus, 10% and 5% off; or, as it is generally expressed in business, 10 and 5 off, means a discount of 10%, and then 5% from what is left; 20, 10, and 5 off, means three successive discounts. A retailer’s profit is smaller when he is allowed 10 and 5 off, than if he were allowed 15 off. The result is not affected by the order in which the discounts are taken.
Example: I receive a bill of goods amounting to $100, 20% off. What is the net cost?
First Way:

20% of $100 =$20
$100 - $20 = $80

Second Way:

100% - 20% = 80%
80% of $100 = $80

Example: A merchant receives two bills of $200 each. On one there is a discount of 25%; on the other, 15% and 10%. What must he pay on each, net?
First Bill:

100% - 25% = 75%, or ⁄4
⁄4 of $200 = $150.

Second Bill:

100% - 15% = 85%
100% - 10% = 90%
90% of 85% = 76.5%
.765 × $200 = $153.

Promissory Notes

A note is a written promise to pay a specified sum at a certain time.
The person who promises is called the maker, and the person to whom he promises is called the payee.
The FACE of a note is the sum of money promised.
A negotiable note is one which is made payable to the bearer, or to the order of the payee. A negotiable note can be sold or transferred.
A note is non-negotiable when it is payable only to the person or persons named in the note.
An indorser of a note is a person who writes his name on the back of it. The person who indorses, by so doing guarantees its payment. An indorsement in blank is simply the signature of the indorser written across the back of the note or draft. When indorsed in this way the note or draft is made payable without further indorsement to any person holding.
A note or draft is indorsed in full when the indorser states, over his signature, the person to whose order the note or draft is to be paid. If an indorser does not wish to guarantee the payment of a note or draft, he writes “Without recourse” over his name when indorsing it.
A protest of a negotiable note or draft is a formal statement by a notary public that said note or draft was presented for payment or acceptance and refused.
A note, when due, must be presented at the place at which it is made payable. The day of maturity is the day on which a note becomes due.
The days of grace are the three days beyond the specified time for payment. Days of grace are now practically abolished throughout the United States.
Kinds of Notes.—There are three principal kinds of notes—Time Notes, Joint Notes, and Joint and Several Notes.
A Time Note must be paid in a specified time.
A Joint Note is one signed by two or more persons who are jointly liable for its payment.
A Joint and Several Note is a note signed by two or more persons who are both jointly and individually liable for its payment. Each man who signs the note is as much responsible for the payment of the whole sum as if he had signed alone.

Legal Rules that Apply to Notes

A note made out on Sunday is void.
If a note does not state that interest is to be paid, it does not bear interest until after it is due.
If anyone obtains a note by fraud or from an intoxicated person, he cannot collect.
To be negotiable an instrument must be in writing and signed by the maker (of a note) or drawer (of a bill or check).
It must contain an unconditional promise or order to pay a certain sum in money.
Must be payable on demand, or at a fixed future time.
Must be payable to order or to bearer.
In a bill of exchange (check), the party directed to pay must be reasonably certain.
Every negotiable instrument is presumed to have been issued for a valuable consideration, and want of consideration in the creation of the instrument is not a defense against a bona-fide holder.
An instrument is negotiated, that is completely transferred, so as to vest title in the purchaser, if payable to bearer, or indorsed simply with the name of the last holder, by mere delivery; if payable to order, by the indorsement of the party to whom it is payable and delivery.
One who transfers an instrument by indorsement warrants to every subsequent holder that the instrument is genuine, that he has title to it, and that if not paid by the party primarily liable at maturity, he will pay it upon receiving due notice of non-payment.
To hold an indorser liable the holder upon its non-payment at maturity must give prompt notice of such non-payment to the indorser and that the holder looks to the indorser for payment. Such notice should be sent within twenty-four hours.
When an indorser is thus compelled to pay he may hold prior parties, through whom he received the instrument, liable to him by sending them prompt notice of non-payment upon receiving such notice from the holder.
One who transfers a negotiable instrument by delivery, without indorsing it, simply warrants that the instrument is genuine, that he has title to it, and knows of no defense to it, but does not agree to pay it if unpaid at maturity.
The maker of a note is liable to pay it, if unpaid at maturity, without any notice from the holder or indorser.
Notice to one of several partners is sufficient notice to all.
When a check is certified by a bank the bank becomes primarily liable to pay it without notice of its non-payment, and when the holder of a check thus obtains its certification by the bank, the drawer of the check and previous indorsers are released from liability, and the holder looks to the bank for payment.
A bona-fide holder of a negotiable instrument, that is, a party who takes an instrument regular on its face, before its maturity, pays value for it and has no knowledge of any defenses to it, is entitled to hold the party primarily liable responsible for its payment, despite any defenses he may have against the party to whom he gave it, except such as rendered the instrument void in its inception. Thus, if the maker of a note received no value for it, or was induced to issue it through fraud or imposition, that does not defeat the right of a bona-fide holder to compel its payment from him.

The dates and amounts of partial payments on a note, before it is finally paid in full, are placed on the back.
The place of payment, if not mentioned, is at the maker’s place of business or residence, during reasonable business hours.
If a note or a check received in payment of a debt is dishonored, the debt revives.
Ignorance of the law does not excuse anyone. No contract is good unless there be a consideration. No consideration is good that is illegal.
The maker of an accommodation note is not bound to the person accommodated; but he is bound to any other person receiving the note for value.

Bank Discount

The sum charged by a bank for cashing a note or time draft is called bank discount. This discount is the simple interest, paid in advance, for the number of days the note has to run. Wholesale business houses usually sell goods on time and take notes from the retailers in payment. These notes are not often for a longer period than three months. Some are placed in the banks for collection, others are discounted. When a note is discounted at a bank the payee indorses it, making it payable to the bank. Both maker and payee are then responsible to the bank for its payment. If the note is drawing interest the discount is reckoned on and deducted from the amount due at maturity. Most notes discounted at banks do not draw interest. The time in bank discount is always the number of days from the date of discounting to the date of maturity.
Example: A note of $250, dated July 7, payable in 60 days, is discounted July 7 at 6%; find the proceeds.
Explanation: This note is due in 63 days, or September 8. The accurate interest of $250 for 63 days at 6% is $2.59. The proceeds, then, will be $250-$2.59, or $247.41.
The Present Worth of a note or debt is a sum, which, if put at interest, will amount to that debt in the given time.
The True Discount is the difference between the debt at maturity and its present worth.

Remember:

1. To allow three days of grace, if the debt discounted is a note.
2. To add the interest due at maturity to the principal, before discounting, if the note bears interest.
Examples: Case I.—Note not bearing interest.
What is the present worth and true discount on a note of $200, if paid 6 months before due, the discount being 6%.
Solution: Amount of $1 for 6 months at 6% = $1.03. If $1.03 = amount of $1, $200 is the amount of as many dollars as ⁄1.03, or $194.17+.
$194.17 is the present worth. $200 - $194.17 = $5.83 true discount.
The following rule can be deduced from the foregoing solution:—
Rule: 1. To find the present worth, divide the debt by the amount of $1 for the given time.
2. To find the true discount, subtract the present worth from the debt.
Case II.—Note bearing interest.
What is the present worth of a note of $300, bearing 6% interest, due in 2 years 4 months, if money is worth 10%.
Solution: Interest on $300 for 2 years 4 months at 6% = $42.
$300 + $42 = $342. Amount due at maturity.
Amount of $1 for 2 years 4 months at 10% = $1.23⁄3.

If $1.23⁄3 = amount of $1, then $3.42 is the amount of $3421.23⁄3, or $277.29.

$277.29 = present worth.

Interest

If a person borrows money, he usually pays something for the loan.
The sum of money he borrows is called the Principal; the money he pays for the use of the principal is called Interest. Interest is generally reckoned at so much for the use of each $100 for one year. This amount is called the Rate per cent per Annum.
Thus, if we say that $200 is borrowed for three years at 4 per cent per annum, we mean that the borrower, at the end of each year, pays the lender $4 for each $100 borrowed—i.e., $8 interest for each year.
In the above example the interest is supposed to be paid to the lender at the end of each year. Interest thus reckoned is called Simple Interest.
The sum obtained by adding the interest for any given time to the principal is called the Amount in that time.

Common Interest Methods

If we were to find the interest on a sum of money for 3 years 4 months 5 days, we would find the interest for 1 year, then for 1 month (⁄12 of a year), then for 1 day (⁄360 of a year). Having the interest for 1 year 1 month 1 day, it is a simple matter of multiplication to get it for 3 years 4 months 5 days.
Example:
What is the interest on $520 for 1 year 3 months at 6%?

Work:

1 year 3 months = 1⁄4 year

$520 principal

.06

4 ) $31.20 interest 1 year

$7.80 interest ⁄4 year

$39.00 interest 1 ⁄4 year

The 60-Day Interest Method

In what is called the 60-Day Method, 360 days are considered one year, and 30 days one month. Upon this basis the interest for 60 days, or two months, at any rate, will be ⁄6 of the interest for one year; and when the rate is 6% the interest for 60 days is one per cent or ⁄100 of the principal. Thus, the interest of $247 for 60 days at 6% is $2.47.
Example: Find the interest of $1728 for 80 days at 6%.

Work:

$17 28 = interest for 60 days.

5 76 = interest for 20 days.

$23 04 = interest for 80 days.

Explanation:
The interest of $1728 for 60 days at 6% is 1% of $1728, or $17.28; and the interest for 20 days (⁄3 of 60) is ⁄3 of $17.28, or $5.76. Hence for 80 days it will be $17.28 plus $5.76, or $23.04.

Methods of Reckoning Time

The Common Method.—When the time is long, generally 30 days are considered a month.
The Exact Method.—When the time is short, the exact number of days is generally counted but we sometimes find the exact number of days also when the time is long.
The Bankers’ Method.—Bankers get the exact number of days between two dates, but each day is reckoned as ⁄360 of a year.

Problem, when the time is long.

Find the time between April 12, 1895, and September 22, 1899.

Best Method

From April 12, 1895, to April 12, 1899, is 4 years.
From April 12, 1899, to Sept. 12, 1899, is 5 months.
From Sept. 12, 1899, to Sept. 22, 1899, is 10 days.
Time between dates = 4 years 5 months 10 days.

Another Method

1899 9 22

1895 4 12

4 5 10

Problem, when the time is short. Find the difference in time between April 12 and July 15, 1902.
Work:

Number of days left in April = 18

in May = 31

in June = 30

in July = 15

Total number of days = 94

Note.—If the rate and principal are given, it is a simple matter to find the interest, now that we have the time.
Example of the use of Table: What is the time from February 10 to October 18, in the same year. February 10 is numbered 41, and October 18 is numbered 291; 291 - 41 = 250, Ans. This includes the last day, but not the first. If both days are taken, subtract 40 from 291 = 251, Ans. When February 29 occurs in a term, count an additional day. The day of the date of a note is not included in its term; thus, required the last day of grace of a note dated March 24, at 90 days. March 24 = 83; 83 + 93 = 176 = June 25, Ans.

TABLE OF TIME, IN DAYS
The following table gives the exact time, in days, between two dates.

Jan. Feb. March April May June July Aug. Sept. Oct. Nov. Dec.

1 32 60 91 121 152 182 213 244 274 305 335

2 33 61 92 122 153 183 214 245 275 306 336

3 34 62 93 123 154 184 215 246 276 307 337

4 35 63 94 124 155 185 216 247 277 308 338

5 36 64 95 125 156 186 217 248 278 309 339

6 37 65 96 126 157 187 218 249 279 310 340

7 38 66 97 127 158 188 219 250 280 311 341

8 39 67 98 128 159 189 220 251 281 312 342

9 40 68 99 129 160 190 221 252 282 313 343

10 41 69 100 130 161 191 222 253 283 314 344

11 42 70 101 131 162 192 223 254 284 315 345

12 43 71 102 132 163 193 224 255 285 316 346

13 44 72 103 133 164 194 225 256 286 317 347

14 45 73 104 134 165 195 226 257 287 318 348

15 46 74 105 135 166 196 227 258 288 319 349

16 47 75 106 136 167 197 228 259 289 320 350

17 48 76 107 137 168 198 229 260 290 321 351

18 49 77 108 138 169 199 230 261 291 322 352

19 50 78 109 139 170 200 231 262 292 323 353

20 51 79 110 140 171 201 232 263 293 324 354

21 52 80 111 141 172 202 233 264 294 325 355

22 53 81 112 142 173 203 234 265 295 326 356

23 54 82 113 143 174 204 235 266 296 327 357

24 55 83 114 144 175 205 236 267 297 328 358

25 56 84 115 145 176 206 237 268 298 329 359

26 57 85 116 146 177 207 238 269 299 330 360

27 58 86 117 147 178 208 239 270 300 331 361

28 59 87 118 148 179 209 240 271 301 332 362

29 — 88 119 149 180 210 241 272 302 333 363

30 — 89 120 150 181 211 242 273 303 334 364

31 — 90 — 151 — 212 243 — 304 — 365

Compound Interest

Interest computed, at regular intervals, on the sum of the principal and any unpaid interest, is called compound interest. In other words, as soon as interest becomes due and is unpaid, it begins to draw interest at the same rate as the principal. Compound interest is generally paid on the deposits in savings banks and is used in calculating amortization and sinking funds.
Interest may be compounded quarterly, semi-annually, annually, or at the end of any other period agreed upon. In some States the collection of compound interest is not permitted.
Example: Find the amount and the compound interest of $1200 at 6% for two years, interest compounded semi-annually.

Solution:

$1200. 00 First principal

36. Interest for 6 months

1236. Principal at beginning of second 6 months

37. 08 Interest for second 6 months

1273. 08 Principal at beginning of third period

38. 19 Interest for third period

1311. 27 Principal at beginning of fourth period

39. 34 Interest for fourth period

$1350. 61 Amount at end of two years

$1350. 61 Amount at end of two years

1200. 00 Principal

150. 61 Compound interest.

Exchange

in commerce is a method of making payments in distant places, without the actual transmission of money, but by a Bill of Exchange called Draft, which is a written request upon one person to pay a certain sum to another or to his order. The person who orders the money to be paid, is called the Drawer; the one who is directed to pay it, the Drawee, and the one to whom it is directed to be paid, the Payee.

Domestic or Inland Exchange is exchange between places in the same country: Foreign Exchange, between different countries.
If, for every little business transaction, money had to be sent from one business center to another, much needless inconvenience and expense would be incurred.
A man in Chicago owes a man in New York City a sum of money. He can send it to him in one of five ways:—

• 1. By Check.
• 2. By Post-office Order
• 3. By Express Order
• 4. By Bill of Exchange
• 5. By Telegraph

Suppose Mr. White of Chicago owes Mr. Brown of Boston $200 for groceries and Mr. Allen of Boston owes Mr. Warner of Chicago $200 for rent. Wouldn’t it save expense and trouble if Mr. White should go to Mr. Warner and Mr. Allen to Mr. Brown? Thereby two debts are cancelled by two city transactions and no money need be sent from one city to another.
This is all there is to Exchange, only in business life banks instead of individuals transact the business.
Only a small percentage of the money really passes from one city to another.
Exchange in the United States is carried on mostly by banks located in the large cities, which charge a small fee for transacting the business.

TABLE OF COMMERCIAL LAW IN THE STATES

INTEREST LAWS Statutes of Limitation Exemption Laws

Legal Rate Per Cent Per Cent Allowed by Contract and Penalty for Usury Judg-
ments Notes Open
Ac-
counts Personal
Property,
Exempt Homestead,
Exempt

Alabama 8 8; Forfeit interest 20 yrs. 6 yrs. 3 yrs. $1,000 $2,000

Arizona 6 12; No provision 4 yrs. 4 yrs. 3 yrs. 500 2,500

Arkansas 6 10; Forfeit principal and interest 10 yrs. 5 yrs. 3 yrs. 500 2,500

California 7 Any; No provision 5 yrs. 4 yrs. 4 yrs. ... 5,000

Colorado 8 Any; No provision 6 yrs. 6 yrs. 6 yrs. ... 2,000

Connecticut 6 15; Fine or imprisonment, or both 7 yrs. 6 yrs. 6 yrs. ... 1,000

Delaware 6 6; Principal and interest forfeited 10 yrs. 6 yrs. 3 yrs. 200 ...

Dist. of Col. 6 6; Forfeit interest 12 yrs. 3 yrs. 3 yrs. 300 ...

Florida 8 10; Forfeit interest 20 yrs. 5 yrs. 3 yrs. 1,000 160 acres

Georgia 7 8; Forfeit excess of interest 7 yrs. 6 yrs. 4 yrs. 300 1,600

Idaho 7 12; Forfeit interest and 10% of principal 6 yrs. 5 yrs. 4 yrs. ... 5,000

Illinois 5 7; Forfeit interest 20 yrs. 10 yrs. 5 yrs. 400 1,000

Indiana 6 8; Excess interest forfeited 20 yrs. 10 yrs. 6 yrs. 600 or 600

Iowa 6 8; Forfeit interest and 8% of principal 20 yrs. 10 yrs. 5 yrs. 200 or 40 acres

Kansas 6 10; Forfeit of double amount of usurious interest 5 yrs. 5 yrs. 3 yrs. ... 160 acres

Kentucky 6 6; Forfeit of interest 15 yrs. 15 yrs. 5 yrs. 250 1,000

Louisiana 5 8; Forfeit interest 10 yrs. 5 yrs. 3 yrs. ... Total 2,000

Maine 6 Any; No provision 20 yrs. 20 yrs. 6 yrs. ... 500

Maryland 6 6; Forfeit interest 12 yrs. 3 yrs. 3 yrs. 100 ...

Massachusetts 6 Any; No provision 20 yrs. 6 yrs. 6 yrs. ... 800

Michigan 5 7; Forfeit interest 6 and 10 6 yrs. 6 yrs. 500 1,500

Minnesota 6 10; Forfeit interest 10 yrs. 6 yrs. 6 yrs. 500 80 acres

Mississippi 6 10; Forfeit interest 7 yrs. 6 yrs. 3 yrs. ... 2,000

Missouri 6 8; Forfeiture or misdemeanor 10 yrs. 10 yrs. 5 yrs. 300 1,500 (min.)

Montana 8 Any; No provision 10 yrs. 8 yrs. 5 yrs. ... 2,500

Nebraska 7 10; Forfeit interest 5 yrs. 5 yrs. 4 yrs. 500 or 2,000

Nevada 7 Any; No provision 6 yrs. 6 yrs. 4 yrs. ... 5,000

New Hampshire 6 6; Forfeit three times excess 20 yrs. 6 yrs. 6 yrs. ... 500

New Jersey 6 6; Forfeit interest and costs 20 yrs. 6 yrs. 6 yrs. 200 1,000

New Mexico 6 12; Forfeit of twice the amount of interest 7 yrs. 6 yrs. 4 yrs. 500 1,000

New York 6 6; Forfeit of principal and interest; misdemeanor 20 yrs. 6 yrs. 6 yrs. 250 1,000

North Carolina 6 6; Forfeit interest 10 yrs. 3 yrs. 3 yrs. 500 1,000

North Dakota 7 12; Forfeit interest 10 yrs. 6 yrs. 6 yrs. 1,000 5,000

Ohio 6 8; Forfeit interest over 6% 5 yrs. 15 yrs. 6 yrs. 100 1,000

Oklahoma 6 10; Forfeit interest 5 yrs. 5 yrs. 3 yrs. ... 5,000

Oregon 6 10; Forfeit principal and interest 10 yrs. 6 yrs. 6 yrs. ... 1,500

Pennsylvania 6 6; Forfeit excess of interest 20 yrs. 6 yrs. 6 yrs. 300 ...

Rhode Island 6 Any; No provision 20 yrs. 6 yrs. 6 yrs. 800 ...

South Carolina 7 8; Forfeit interest 20 yrs. 6 yrs. 6 yrs. 500 1,000

South Dakota 7 12; Misdemeanor 20 yrs. 6 yrs. 6 yrs. 750 5,000

Tennessee 6 6; Forfeit of excess interest 10 yrs. 6 yrs. 6 yrs. ... 1,000

Texas 6 10; Forfeit interest 10 yrs. 4 yrs. 2 yrs. 500 5,000

Utah 8 12; Forfeit excess interest 8 yrs. 6 yrs. 4 yrs. ... 2,000

Vermont 6 6; Forfeit of excess interest 8 yrs. 6 yrs. 6 yrs. 200 500

Virginia 6 6; Forfeit interest 20 yrs. 5 yrs. 2 yrs. ... 2,000

Washington 6 12; Forfeit of double accrued interest and costs 6 yrs. 6 yrs. 3 yrs. 1,000 2,000

West Virginia 6 6; Forfeit excess interest 10 yrs. 10 yrs. 5 yrs. 200 1,000

Wisconsin 6 10; Forfeit treble amount of usurious interest paid 20 yrs. 6 yrs. 6 yrs. 200 5,000

Wyoming 8 12; Forfeit interest 5 yrs. 5 yrs. 8 yrs. 500 1,500

Note.—In many of the States it is impossible to place a fixed amount on personal property exempt. In the table above these states have no amount given in the personal property column. Days of grace have been abolished in all states except the following: Arkansas, Mississippi, South Carolina and Texas.

If the drawee accepts the draft, he writes across the face of it “Accepted” with the date and his signature. This is called an Acceptance.
Once accepted, the draft becomes a note, with the same laws regulating it. If the draft is not accepted, it is not binding and we say that it has been “dishonored.”
A bill of exchange is entitled to days of grace, if it is payable in a State where grace is allowed, unless a particular day is named in the draft. In most States, no grace is allowed on sight drafts.

Principles of Exchange

To find the cost of a draft, the face and rate per cent of exchange being given.
Rule.—Find the percentage of the given rate per cent of exchange and add it to, or subtract it from the amount of draft.
Example: What is the cost, in Chicago, of a sight draft on Denver for $400, if exchange is ⁄4% premium; and how much if ⁄2% discount?

$400 × .00⁄4 = $3; $400 + $3 = $403, at ⁄4% premium.
$400 × .00⁄2 = $2; $400 - $2 = $398, at ⁄2% discount.

To find the face of a draft, cost and rate per cent of exchange given.
Rule.—Divide by the cost of a draft for $1, at given rate per cent of exchange.
Example: Find face of draft that can be bought for $1000 at 1% premium; at 1% discount.

$1000 ÷ 1.01 = $990.10, at 1% premium.
$1000 ÷ .99 = $1010.10, at 1% discount.

Time Drafts, when negotiated before maturity, are subject to discount which is computed on the face of the draft, the same as interest.
Example: What is the proceeds of a 60-day draft for $800, at ⁄8% premium, and discounted at 7%?

$805.00, face + ⁄8% premium

9.33, interest (7%, 60 days)

$795.67, proceeds. Ans.

Foreign Drafts are usually made payable in the money of the country on which they are drawn.
To find the equivalent of foreign money in United States money and vice versa.
Rule.—Multiply, or divide (as the case may require) the given sum, by the equivalent of a unit in United States money.
Example: What is the cost of a draft on London for £125, reckoning exchange at $4.8665?

125 × 4.8665 = 608.31. Ans. $608.31.

Wishing to remit $182.50 to Ireland, for what amount must I buy a draft on London?

182.50 ÷ 4.8665 = 37.5. Ans. £37⁄2.

How many francs in $100?

100.000 ÷ .193 = 518.13. Ans. 518.13 francs.

How many dollars in 7500 German marks?

7500 × .238 = $1785, Ans.

How many Swedish crowns in $750?

750 ÷ .268 = 2798⁄2 crowns, Ans.

How many dollars in 4635 rubles?

4635 × .772 = $3578.32, Ans.

A simple method to reduce pounds sterling to United States money, and vice versa; exchange being at $4.8665.
Rule.—Multiply pounds sterling by 73, and divide the product by 15. Or multiply dollars by 15 and divide the product by 73.
Examples: How many dollars in £85?

85 × ⁄15 = 413.67. Ans. $413.67.

How many £’s in $748.25?

748.25 × ⁄73 = 153⁄4. Ans. £153⁄4.

Another method to change pounds sterling, shillings and pence, to dollars and cents.
Rule.—Reduce pounds sterling to shillings, add the shillings, and multiply the sum by .24⁄3—the product will be cents. Add 2 cents for each pence, if any.
Example: Change £46, 13s. 9d. to United States money.

46 × 20 = 920 933 × .24⁄3 = 227.03

13 9d. = .18

933 Ans. $227.21

Tourists of today patronize express companies for Foreign Money Orders. These are made out similar to regular express money orders and may be cashed in any of the larger cities of all foreign countries. They take the place, to a large extent, of Letters of Credit, which are letters from banking houses in one country to those in another, allowing sums to be drawn not to exceed a total named in the letter.

Stocks and Bonds

Stocks is a general name given to the capital of incorporated companies. They are divided into equal parts, usually of $100 each, called Shares, the owners of which are called Stockholders. A Dividend is a part of the net income of the company, divided among the stockholders.
A certificate of stock is a written paper signed by the proper officers of the corporation, naming the number of shares to which the person named therein is entitled, and the original value of the same.
Preferred stock is stock which is given a preference over the common stock. Ordinarily, a dividend is paid on the preferred stock before any is paid on the common shares.
Common stock is the ordinary stock of a corporation, which has no preference, in the payment of dividends, over any other.
The par value of a share of stock is the value named in the certificate of stock.
When a corporation is prosperous, its shares of stock often sell for more than the value named in the certificate of stock. They are then said to be above par, or at a premium. In times of business depression, often these shares of stock sell below their face value. They are then said to be below par, or at a discount.
The market value of a share of stock is the value for which it sells in the open market.
A stock broker is one who makes a business of buying and selling stocks and bonds. He charges a commission for this which is called brokerage.
A surplus is a part of the earnings of a corporation.
The gross earnings of a corporation are its total receipts from all sources.
The net earnings are the profits remaining when all expenses, losses, interest and debts due are paid.
An assessment is a sum levied proportionate to stock held by stockholders, to help out the business when it is not prospering, or when more money is needed to carry it on. It is levied as so many dollars on each share at its par value.
The directors are those shareholders elected by all to manage the affairs of the corporation.
A bond is, in form, a carefully drawn interest-bearing promissory note. Ordinarily, it runs for a period of years with interest often payable semi-annually. It is more formal than the ordinary promissory note. Bonds are usually issued by national, state, county, or local governments, or by corporations, when they wish to raise large sums of money for immediate use.

Difference Between Stocks and Bonds

A bond runs for a specified time. It bears a specified interest, and is an absolute promise to pay the face of it at maturity. It matures at a definite time, and at that time the holder is paid its face value and no more, by the organization that issued it, unless such organization is insolvent, or has repudiated its debts.
Stocks.—A certificate of stock is no promise to pay. It simply shows that the holder owns as much stock in the corporation as is shown by the face of the certificate. It bears no interest and has no date of maturity.
The interest returns of the bondholder are certain and definite. The returns of the stockholder, dividends, are uncertain and depend on the profits of the business.
Consequently, no table can be arranged to show at what rate stocks can be bought to yield a definite return; but with bonds, tables may be made which show at a glance what the return will be from a purchase made at any rate.

Application of Percentage to Stocks

1. To find the value of stocks, when above or below par.
Rule.—Multiply the price per share, by the number of shares.
Example: Find cost of 65 shares of bank stock, at $107 per share, or 7% premium. Also of 48 shares of railroad stock, at $87⁄2 per share, or 12⁄2% discount.

(1) 65 × 107 = 6955. Ans. $6955.
(2) 48 × 87⁄2 = 4200. Ans. $4200.

2. To find what rate per cent is realized by investing in stocks or bonds when above or below par.
Rule.—Annex two ciphers to the fixed rate per cent, and divide by the cost per share. Or by proportion: As the cost per share is to the fixed rate, so is 100 to the required rate.
Example: Mr. Warren bought ten shares of Illinois Central Railroad stock at 96. What does he get when a dividend of 6% is declared? What per cent is that on his investment?
Work and Explanation:

(1) 1 share at 6% yields $6
(1) 10 shares yield 10 × $6 = $60.
(2) Each share at 96 costs $96.
(2) Each share yields $6.
Query? $6 is what per cent of $96?
Query?$6 is ⁄96 of 100%, or 6⁄4%.
Query?∴ the investment yields 6⁄4%.

3. To find which is the more profitable investment.
Rule.—Find the rate per cent that each investment yields, by rule, under item 2; then compare rates.
Example: Which is the better investment; 6% mortgages at 10% premium, or 5% bonds at 10% discount?
(1) 110)600(5⁄11%.
(2) 90)500(5⁄9%. ⁄9 - ⁄11 = ⁄99, practically ⁄10.
Ans. The latter, by ⁄10 of 1%, nearly.

Taxes and Taxation

A tax is a contribution levied on persons, property, incomes, or business, for public purposes.
Some Uses for Taxes.—The National Government requires money to support the army and navy, to pay the salaries of government employes, to pay pensions, and to finance other activities carried on by the nation.
The State Governments require money for the expense of their officers, and to support their various institutions, schools, universities, asylums, and penitentiaries.
The counties require money for the building of bridges, the trial of criminal cases, the salaries of officers, the relief of the poor, etc.
Cities must pay for police and fire protection, care of streets, etc.
School districts contribute to the support of the public schools.
The money required for all these expenses is raised by taxes, licenses, fees, assessments, and fines.
State and Local Taxes.—The amount of tax paid by any individual to state and local governments depends upon the value of the property which he owns and the tax rate. In many places the adult male citizen pays a poll tax.
The tax levied on property is called a property tax.
The tax levied on persons is called a poll tax. This is sometimes called a capitation (by the head) tax.
Sometimes a man’s income is taxed. This is an income tax.
After the amount of money to be raised by tax is decided upon, a man, called the assessor, examines each piece of taxable personal property and real estate, and places a value upon it. This is taken as a basis for proportioning the tax among the property owners.
A tax collector is one who collects the taxes. He is sometimes paid a salary. Sometimes he gets only a percentage of the money he collects.
The treasurer receives and takes care of the money collected by the tax collector. He is paid a salary.
The Tax Rate.—Sometimes the rate is fixed by law or by vote of the citizens. More often the lump sum to be raised is named, and the assessor determines the rate.
When the assessor is to determine the rate, he proceeds in this way: First, he assesses each piece of property, usually not at its full market value. Then he determines the total value of all the property in his district. Next, he divides the total tax to be raised by the total value of the property in his district. The result is the rate of tax on the dollar.
Use of the Mill in Taxes.—When a tax is apportioned, it is usually found that if a few mills are paid on each dollar’s worth of property in the district, the aggregate amount is equal to the whole sum of tax needed. Consequently, we often hear of tax levies of so many mills on the dollar, as, 2 mills on the dollar, 5 mills on the dollar, etc.
The denomination of our money system called the mill has practically its only use in the levy of taxes.
Assessors make use of a table like the one given on the following page. This table is based on a tax levy of 9 mills on the dollar.
The following tax rates are equivalent:

16 mills (on the dollar);

1.6%;

$1.60 (on each hundred dollars).

Explanation of Table. The second column shows the tax at nine mills on the dollar, for values of $1 to $30. The fourth column shows the tax for values of $40 and multiples of ten, to $600. The sixth column shows the tax for values of $700 and multiples of one hundred, to $10,000.

Tax Table

Prop-
erty
Value Tax Prop-
erty
Value Tax Prop-
erty
Value Tax

$ 1 $ 0.009 $ 40 $ 0.36 $ 700 $ 6.30

2 0.018 50 0.45 800 7.20

3 0.027 60 0.54 900 8.10

4 0.036 70 0.63 1,000 9.00

5 0.045 80 0.72 2,000 18.00

6 0.054 90 0.81 3,000 27.00

7 0.063 100 0.90 4,000 36.00

8 0.072 200 1.80 5,000 45.00

9 0.081 300 2.70 6,000 54.00

10 0.09 400 3.60 7,000 63.00

20 0.18 500 4.50 8,000 72.00

30 0.27 600 5.40 9,000 81.00

The Amount of Tax.—To find the amount of tax to be paid by any property owner.
Rule.—Multiply the assessed value of the property by the tax rate.
Example: Taylor’s property is assessed at $3800. The rate is 24 mills.

Solution: $3800 assessed valuation

.024 tax rate in mills

$91.20 tax.

Example: The town of Grant is to raise $4725 in tax. The property in the town has an assessed valuation of $395,140. What is the rate?
If on $395,140 a tax of $4725 is to be raised, on $1 as much tax must be raised as $395,140 is contained times in $4725, which is .0119+, or about $.0119. This would be called $0.012, or 12 mills on the dollar.
Example: Finch’s property is assessed at $5470. The tax rate is $1.95.

Solution:

$1.95 the rate per hundred dollars

54.70 the number of hundreds of dollars assessed value

$106.67 the tax.

Indirect Taxes are taxes placed upon goods by the national government, and collected before the goods are sold to the consumer.
The national government needs this money to pay:—
1. Interest on the public debt.
2. To support an army and navy, to build vessels, and keep up arsenals and forts.
3. To pay pensions.
4. To improve the rivers and harbors.
5. To pay the salaries of its officers; as, the president, cabinet officers, judges, ministers to foreign countries, congressmen, etc.
Indirect taxes are of two kinds, customs or duties, and excises or internal revenue.
Excises or internal revenue are taxes levied on certain domestic goods, as, manufactured tobacco, liquors, and the like.
Indirect taxes levied by the government on imported goods or merchandise are called duties or customs.
A custom house is a government office where duties are collected and where vessels are entered and cleared. Nearly every seaport of consequence has a custom house. So also have important towns near the Canadian and Mexican boundaries.
Duties are of two kinds, specific and ad valorem.
A specific duty is one levied at a specified sum per yard, gallon, ton, etc.
An ad valorem duty is one levied at a certain percentage of the value of the goods, at the port of export.
Tare is an allowance made for the weight of bags, barrels, or cases, in which merchandise is shipped.
Leakage is an allowance made for loss of liquids from casks, barrels, etc., in shipping.
Breakage is an allowance made for the loss of liquids from bottles in shipping.
Example: Find the duty on 4 dozen bottles of cologne, allowing 4% for leakage and 3% for tare. The invoice value is 90 cents a bottle and the duty is 25% ad valorem and 20 cents specific. Find the total cost per bottle.
Work and Explanation:

Leakage and tare are 4% + 3% = 7%.

4 dozen bottles = 48 bottles.

The invoice value of 48 bottles is 48 × 90 cents = $ 43 .20

Tare and leakage are 7% of $43.20 = 3 .024

Value on which duty is paid $ 40 .176

Ad valorem duty is 25% of $40.176 = $ 10 .044

Specific duty is 48 × 20 cents = 9 .60

Total duty $ 19 .644

The total cost is

Invoice value $ 43 .20

Ad valorem duty 10 .04

Specific duty 9 .60

$ 62 .84

The total cost per bottle is ⁄48 of $62.84, or $1.31-.

Square Root and Cube Root

Powers and Roots.—When a product consists of the same factor repeated any number of times it is called a power of that factor.
7 × 7 is the second power, or the square of 7.
7 × 7 × 7 is the third power, or the cube of 7.
A power of a number is generally expressed by writing the number only once, and placing after it, above the line, a small figure to show how many factors are to be taken. The small figure is called an index.
Thus, 7 = 49; 7 = 343; 7 = 2401.
A number is called the square root of its square.
Since 7 = 49, the square root of 49 is 7.
The “square root of 49” is written √49.
Again, a number is called the cube root of its cube. 7 = 343. Therefore, the cube root of 343 is 7.
The “cube root of 343” is written ∛343.
A perfect square is a number whose square root is a whole number. A perfect cube is a number whose cube root is a whole number.
Square Root.—If a number can be put into prime factors, its square root can be written down by inspection.
Example: Find the square root of 27225.

Since 27225 = 3 × 5 × 11.

∴ √27225 = 3 × 5 × 11 = 165 Ans.

Rule for Digits.—We know that √1 = 1, and √100 = 10. Therefore, the square root of any number which lies between 1 and 100 lies between 1 and 10; i.e., if a number contains one or two digits, its square root consists of one digit.
Similarly, since √100 = 10 and √10000 = 100, the square root of a number between 100 and 10000 lies between 10 and 100. That is, if a number contains three or four digits, its square root consists of two digits.
Proceeding in this way, we obtain a general result—viz., the square of a number has either twice as many digits as the number, or one less than twice as many.
Hence, to ascertain the number of digits in the square root of a perfect square, mark off the digits in pairs, beginning from the right. Each pair marked off gives a digit in the square root; and, if there is an odd digit remaining, that digit also gives a digit in the square root.
Examples: There are three digits in the square root of 546121, and four in the square root of 5774409.
For, marking off the digits from the right, we get in the first case 54,61,21, giving three digits in the square root, and in the second case 5,77,44,09, the odd digit giving the fourth in the square root.
The method of finding the square root of a given number depends on the form of the square of the sum of two numbers.
Explanation: The square root of 144 is 12. Let us see how we found it.

12 = 1 ten + 2 units.

12 is the same as (10 + 2).

Let us square (10 + 2), that is, multiply 10 + 2 by 10 + 2.

10 + 2

10 + 2

10 + (10 × 2)

+ (10 × 2) + 2

10 + 2(10 × 2) + 2

Then, 12 = 10 + 2(10 × 2) + 2

Rule.—The square of any number made up of tens and units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units.
Another Explanation: Find the square root of 45369.

Solution:

4·53·69 ) 213

4

41 53

41

423 1269

1269

(1) Point off the number into periods of two figures each, as before.
(2) The square root of the first period is 2. 2 × 2 = 4. Write the 2 in the root and subtract the 4 from 4. Bring down the next period, 53.
(3) 2 × 2 = 4. (Remember the 4 is to be used as a trial divisor, being 2 × the tens.)
4 is contained in 5 about 1 time. Place 1 in the root, also on the right of the 4 in the divisor. Multiply 41 by 1. Subtract and bring down the next period.
(4) 2 × 21 = 42. 42 is the trial divisor. 126 ÷ 42 = about 3 times. Place the 3 in the root also at the right of the 42 in the divisor. Multiply out.
Square root = 213.

Cube Root.—The cube root of a number is one of the three equal factors of that number.
Thus, 5 is the cube root of 125, because 5 × 5 × 5 = 125.
The radical sign with a figure 3 over it (∛ ) means that the cube root of the number following it is to be taken.
∛125 reads, “The cube root of 125.”
If we can find the prime factors of any perfect cube, we can write down its cube root by inspection.
Example: Find the cube root of 74088.

8 74088

9 9261

3 1029

7 343

7 49

7

∴ 74088 = 8 × 9 × 3 × 7 × 7 × 7

= 2 × 3 × 7

∴ ∛74088 = 2 × 3 × 7

= 42 Ans.

Rule for Digits.—Since 1 = 1 and 10 = 1000, therefore the cube of a number which lies between 1 and 10 lies between 1 and 1000, i. e., the cube of a number of one digit contains either one, two or three digits.
Again, since 10 = 1000 and 100 = 1000000, the cube of a number of two digits contains either four, five, or six digits.
Proceeding in this way, we see that the cube of a number contains three times, or one less or two less than three times, as many digits as the number.
Hence, to find the number of digits in the cube root of a given number, we mark off the digits in sets of three, beginning at the decimal point, and marking both to the right and to the left.

FIGURES REPRESENTING THE PROCESSES OF FINDING CUBE ROOT

Thus, 289383 will be pointed off into two periods—289·383—and we readily see there will be only 2 figures in the root.
The simplest method of finding the cube root of numbers whose prime factors are not known is analogous to the method of finding square root, being based upon the form of the cube of the sum of two numbers.
Explanation: The cube root of 1728 is 12. Let us see how we found it.

12 = 1 ten + 2 units

12= (10 + 2)

(10 + 2) means 10 + 2 × 10 + 2 × 10 + 2

10 + 2

10 + 2

10 + (10 × 2)

+ (10 × 2) + 2

10 + 2(10 × 2) + 2

10 + 2

10 + 2(10 × 2) + (10 × 2)

+ (10 × 2) + 2(10 × 2) + 2

10 + 3(10 × 2) + 3(10 × 2) + 2

That is, the cube of any number made up of tens and units equals—
The cube of the tens + three times the product of the square of the tens by the units + three times the product of the tens by the square of the units + the cube of the units,
or
tens + 3(tens × units) + 3(tens × units) + units.
For graphic illustration the geometrical representation of the cube of units and tens in the drawings is helpful.
After the process is understood, this short method of writing the work may be used by the pupil:
Example: Find the cube root of .0163956, carrying the root to 3 decimal places.

Work:

.016 ·395·600 ) .254+

8

1200 8395

300

25

1525 7625

770600

187500

3000

16

190516 762064

8536