Algebra Know-It-ALL

90 Fractions Built of Integers

Once you’ve “prepped” the fractions and you want to multiply them together, multiply

the individual numerators to get the numerator of the product, and multiply the individual

denominators to get the denominator of the product:

(a/b)(c/d)=ac/bd

If either of the original fractions is not in lowest terms, the product won’t be either. You can

reduce the product to lowest terms after you’ve multiplied two fractions, but you don’t neces-

sarily have to. If the multiplication is part of a multiple-step calculation process, you might as

well wait until the entire process is complete before you worry about lowest terms.

The reciprocal of a fraction

Thereciprocal of an integer, also called the multiplicative inverse, is the quantity by which you

must multiply the original integer to get 1. The reciprocal of any integer is equal to 1 divided

by that integer. That means 0 has no reciprocal, 1 is its own reciprocal, and −1 is also its own

reciprocal. The reciprocal of every other integer lies somewhere between (but not including)

0 and 1, or else somewhere between (but not including) −1 and 0.

Suppose you have a fraction or ratio a/b, where a and b are integers and neither of them is

equal to 0. Then the reciprocal of a/b is equal to b /a. It’s easy to see why this is true when you

multiplya/b times b /a. To do that, multiply the numerators and the denominators:

(a/b)(b /a)=ab/ba

The commutative law for multiplication can be used to switch around the factors in the

denominator in the right-hand side of this equation, so you get

(a/b)(b/a)=ab /ab

Any nonzero quantity divided by itself is equal to 1. You have already been assured that a≠ 0

andb≠ 0, so you know that ab≠ 0. Therefore

(a/b)(b /a)=ab /ab= 1

This shows that the reciprocal of a/b is equal to b/a.

A fraction divided by a fraction

When you want to divide a fraction by another fraction, you can find the reciprocal of the

second fraction (the divisor) and then multiply the first fraction (the dividend) by it. Suppose

a is an integer, c is a nonzero integer, and b and d are positive integers. If you want to divide

the quantity a/b by the quantity c/d, you can do it like this:

(a/b)/(c/d)= (a/b)(d/c)

=ad/bc

The original expression in this equation, (a/b)/(c/d), is called a compound fraction. It gets that

name from the fact that it’s a fraction made of other fractions! You can also think of it as a

ratio of ratios.