It is taken for granted that the reader knows how to perform the common operations of addition, subtraction, etc., where only whole numbers are used; but, when there are mixed or fractional numbers, a little refreshment of the memory may be desirable to some; hence, a little space is devoted to this elementary branch of arithmetic.

## Common Fractions

The numerator of a fraction is the number above the bar; and the denominator is the number beneath it; thus, in the fraction 3/4, 3 is the numerator and 4 is the denominator. Two or more fractions having the same denominator are said to have a common denominator. By "reducing fractions to a common denominator" is meant finding such a denominator as will contain each of the given denominators without a remainder, and multiplying each numerator by the number of times its denominator is contained in the common denominator. Thus, the fractions 1/4, 7/8, and ft have, as a common denominator, 16; then, 1/4=4/16; 7/8=14/16; 9/16=9/16.

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 14/16, the greatest number that will thus divide 14 and 16 is 2; so that, (14/2)/(16/2)=7/8, which is 14/16 reduced to the lowest terms.

A mixed number is one consisting of a whole number and a fraction, as 7 3/8.

An improper fraction is one in which the numerator is equal to, or greater than, the denominator, as 17/8. This is reduced to a mixed number by dividing 17 by 8, giving 2 1/8. 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; thus 1 7/8=[(1x8)+7] / 8=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, 1 7/8+2 1/4=(1 + 2)+(7/8+2/8)= 4 1/8.

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, 7/8 in. - ft in. = (14-9)/16

=5/16 in. If they are mixed numbers, subtract fractions and whole numbers separately, placing remainders beside one another; thus, 3 7/8 in. - 2 1/4 in.= (3 - 2)+ (7/8-2/8)= 1 5/8 in. With fractions like the following, proceed as indicated: 3 7/16 in.-1 13/16 in.= (2+16/16 +7/16)-1 13/16=2 23/16 - l 13/16 = l 10/16 = 1 5/8in.; 7in.- 4 3/4in. = (6 + 4/4)- 4 3/4 = 2 1/4in.

To multiply fractions. Multiply the numerators together, and likewise the denominators, and divide the former by the latter; thus, 1/2 in. x3/4 in x5/8 in.=(1x3x5)/(2x4x8)=15/64 cu. in. If mixed numbers are to be multiplied, reduce them to frac-tions, and proceed as above shown; thus, l 1/2 in. x 3 1/4 in. = 3/2x13/4 =39/8 = 4 7/8 sq. 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, (7/8)/(3/4) =(7/8)x(4/3)=28/24=7/6=1 1/6. If there are mixed numbers, reduce them to fractions, and then divide as just shown; thus, (l 5/8)/(3 1/4)=(13/8)/(13/4),or(13/8)x(4/13) =52/104 =1/2.