remainder, and multiplying each numerator by the number of times its denominator is contained in the common denominator. Thus, the fractions 4, 7, and have, as a common denominator, 16; then,

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By "reducing a fraction to its lowest terms" is meant dividing both numerator and denominator by the greatest number that each will contain without a remainder; for example, in 1, the greatest number that will thus divide 14 14 ÷ 2 , which is reduced to the 162

and 16 is 2; so that,

lowest terms.

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A mixed number is one consisting of a whole number and a fraction, as 73.

An improper fraction is one in which the numerator is equal to, or greater than, the denominator, as . This is reduced to a mixed number by dividing 17 by 8, giving 24. If the numerator is less than the denominator, the fraction is termed proper. A mixed number is reduced to a fraction by multiplying the whole number by the denominator, adding the numerator, and placing the sum over the denominator; (1 × 8) + 7 8

thus, 1

15

8'

To add fractions or mixed numbers. If fractions only, reduce them to a common denominator, add partial results, and reduce sum to a whole or mixed number. If mixed numbers are to be added, add the sum of the fractions to that of the whole numbers; thus, 17 +24 (1+2)+(+8) 41.

To subtract two fractions or mixed numbers. If they are fractions only, reduce them to a common denominator, take less from greater, and reduce result; as, in. - in.

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14- 9

16

in. If they are mixed numbers, subtract fractions and whole numbers separately, placing remainders beside one another; thus, 3 in. - 24 in. (3-2)+(-3)= 1 in. With fractions like the following, proceed as indicated: 3 in. - 11 in. (2+18+18) - 1782}}-1} = 1}8: (6+1) - 47 24 in.

14 in.; 7 in. - 4 in.

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To multiply fractions. Multiply the numerators together, and likewise the denominators, and divide the former by the

mixed numbers are to be multiplied, reduce them to fractions, and proceed as above shown; thus, 1 in. X3; in.

To divide fractions. Invert the divisor (i. e., exchange places of numerator and denominator) and multiply the dividend by it, reducing the result, if necessary; thus, } ÷ 1 = }× 1 28 = 7 = 1}. If there are mixed numbers, reduce them to fractions, and then divide as just shown; thus, 1ğ÷3 13 ÷ 13, or 13 × 1 = 52 = }.

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DECIMAL FRACTIONS.

In decimals, whole numbers are divided into tenths, hundredths, etc.; thus, is written .1; .08 is read 186, the value of the number being indicated by the position of the decimal point; that is, one figure after the decimal point is read as so many tenths; two figures as so many hundredths; etc. Moving the decimal point to the right multiplies the number by 10 for every place the point is moved; moving it to the left divides the number by 10 for every place the point is moved. Thus, in 125.78 (read 125 and 7), if the decimal point is moved one place to the right, the result is 1,257.8, which is 10 times the first number; or, if the point is moved to the left one place, the result is 12.578 which is the first number, moving the point being equivalent to dividing 125.78 by 10.

Annexing a cipher to the right of a decimal does not change its value; but each cipher inserted between the deci mal point and the decimal divides the decimal by 10; thus, in 125.078, the decimal part is of .78.

To add decimals. Place the numbers so that the decimal points are in a vertical line, and add in the ordinary way, placing the decimal point of the sum under the other points.

To subtract decimals. Place the number to be subtracted with its decimal point under that of the other number, and subtract in the ordinary way.

101.257

12.965

43.005

920.600

1,077.827

917.678 482.710

434.968

21.72

34.1

2172

8688

To multiply decimals. Multiply in the ordinary way, and point off from the right of the result as many figures as there are figures to the right of the decimal points in both numbers multiplied; thus, in the example here given, there are three figures to the right of the points, and that many are pointed off in the result. If either number contains no decimal, point off as many places as are in the number that does.

6516

740.652

If a result has not as many figures as the sum of the decimal places in the numbers multiplied, prefix enough ciphers before the figures to make up the required number of places, and place the decimal point before the ciphers. Thus, in .002 X.002, the product of 2 × 2 4; but there are three places in each number; hence, the product must have six places, and five ciphers must be prefixed to the 4, which gives .000004.

To divide decimals. Divide in the usual way. If the dividend has more decimal places than the divisor, point off, from the right of the quotient, the number of places in excess. If it has less than the divisor, annex as many ciphers to the decimal as are necessary to give the dividend as many places as there are in the divisor; the quotient will then have 25.75 82.5 82.50

no decimal places. For example,

10.3;

2.5

2.75

2.75

To carry a division to any number of decimal places. Annex ciphers to the dividend, and divide, until the desired quotient is reached, which are Thus, 36.5 18.1 to three decimal

number of figures in the pointed off as above shown.

36.5000

places

18.1

2.016+.

(The sign + thus placed after a

number indicates that the exact result would be more than the one given if the division were carried further.)

To reduce a decimal to a common fraction. Place the decimal as the numerator; and for the denominator put 1 with as many ciphers as there are figures to the right of the decimal point; thus, .375 has three figures to the right of the point; hence, .375 in.

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in.

To reduce a common fraction to a decimal. Divide the numerator by the denominator, and point off as many places as there have been ciphers annexed; thus, in. = 3.0000+16 =.1875 in.

DUODECIMALS.

The duodecimal system of numerals is that in which the base is 12, instead of 10, as in the common decimal system. As a method of calculation it has fallen into almost entire disuse, and it is only in calculation of areas that duodecimals are now used. When the work becomes familiar, the work is practically as rapid as by using feet and decimals, and has the advantage of being absolutely accurate, which the decimal system is not, as inches cannot be expressed exactly in decimals of a foot. The principles upon which the work is based are as follows: When feet are multiplied by feet, or inches by inches, the product is, of course, square feet or square inches, respectively; but, when feet are multiplied by inches, or vice versa, the results may, for want of a better name, be termed parts (although this name was given originally to the 144th part of a square foot). Suppose a square foot to be divided into 144 sq. in.; then a strip 12 in., or 1 ft., long, and 1 in. wide will correspond to 1 part. A strip 5 ft. long and 7 in. wide will contain 7 × 5, or 35 parts, which, divided by 12 (parts to a square foot), equals 2 sq. ft. and 11 parts, or 2 sq. ft.+ (11 X 12) sq. in. Square inches may be reduced to parts by dividing by 12; thus, 54 sq. in. 4 parts 6 sq. in. To illustrate these principles, let it be required to find the area of a room 18 ft. 10 in. long, and 16 ft. 7 in. wide. Place feet under feet, and inches under inches, thus:

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Begin by multiplying 10 in. by 16 ft., which equals 160 parts, or 13 sq. ft. 4 parts. Place the 4 parts in the inches column, and carry the 13 sq. Then 18 ft. X 16 ft.

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ft.

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sq.

288 sq. ft.+ 13 sq. ft. 301 sq. ft. Next, multiply 10 in. by 7 in. 70 sq. in., or 5 parts 10 sq. in. Set the 10 sq. in. to the right of the column of parts, as shown. Then, 18 ft. x 7 in. 126 parts+ 5 parts carried 131 parts; dividing by 12, 131 parts 10 ft. 11 parts. Set these down in the proper columns and add, beginning at the right; 4 parts + 11 parts = 1 sq. ft. 3 parts. Expressed in square feet and square inches, the result is 312 sq. ft. 46 sq. in., which can be verified by reducing the given numbers to inches, multiplying them, and dividing by 144.

INVOLUTION.

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Involution is the process of multiplying a number by itself one or more times, the product obtained being called a certain power of the number. If the number is multiplied by itself, the result is called the square of the number; thus, 9 is the square of 3, since 3 X3 9. If the square of a number is multiplied by the number, the result is called the cube of the number; thus, 27 is the cube of 3, since 3X 3X3 27. The power to which a number is to be raised is indicated by a small figure, called an exponent, placed to the right and a little above the number; thus, 72 means that 7 is to be squared; 273 means that 27 is to be cubed, etc.

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The operations of involution present no difficulty, as nothing but multiplication is involved, the number of times the number is to be taken as a factor being shown by the exponent.

If the number is a fraction, raise both numerator and denominator to the power indicated.

A valuable little rule to memorize for finding the square of a mixed number in which the fraction is, as 3, 101, etc., is as follows: Multiply the next less whole number by the next greater, and add. For example, the square of 6 is 6 (the next less number) 7 (the next greater) + 424; (19) 19 X 20+ 3804; (8)2=8X9+ = 724; etc.

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