This section is from the book "The Building Trades Pocketbook", by International Correspondence Schools. Also available from Amazon: Building Trades Pocketbook: a Handy Manual of reference on Building Construction.
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.
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.