Equivalent Fractions

Some fractions that at first glance appear to be different from one another are really the same.

For instance, suppose that we cut a pizza into 8 equal slices, and then eat 4 of the slices. The shaded portion of the diagram at the right represents the amount eaten. Do you see in this diagram that the fractions and describe the same part of the whole pizza?

We say that these fractions are equivalent .

Any fraction has infinitely many equivalent fractions. To see why, let’s consider the fraction . We can draw different diagrams representing one-third of a whole.

All the shaded portions of the diagrams are identical, so .

A faster way to generate fractions equivalent to is to multiply both its numerator and denominator by the same whole number. Any whole number except 0 will do.

So .

Can you explain how you would generate fractions equivalent to ?

To Find an Equivalent Fraction

for (b not 0), multiply the numerator and denominator by the same whole number.

, n not 0

Explain why neither b nor n can be equal to 0 here.

An important property of equivalent fractions is that their cross products are always equal.

In this case,

EXAMPLE 1

Find two fractions equivalent to .

Solution

Let’s multiply the numerator and denominator by 2 and then by 6.

We use cross products to check.

So and are equivalent.

So and are equivalent.

EXAMPLE 8

Write as an equivalent fraction whose denominator is 35.

Solution

The question is:

What number n makes the fractions equivalent?

Express 35 as 7 · 5.

Multiply both the numerator and denominator of by 5.

So n must be 3 · 5, or 15.

Therefore is equivalent to . To check, we find the cross products:

Both 3 · 35 and 7 · 15 equal 105.