1.3 Fractions: equivalent fractions

Section overview

• Equivalent Fractions
• Reducing Fractions to Lowest Terms
• Raising Fractions to Higher Terms

Equivalent fractions

Let's examine the following two diagrams.

Notice that both $\frac{2}{3}$ and $\frac{4}{6}$ represent the same part of the whole, that is, they represent the same number.

Equivalent fractions

There is an interesting property that equivalent fractions satisfy.

A test for equivalent fractions using the cross product

If the cross products are equal, the fractions are equivalent. If the cross products are not equal, the fractions are not equivalent.

Thus, $\frac{2}{3}$ and $\frac{4}{6}$ are equivalent, that is, $\frac{2}{3}=\frac{4}{6}$ .

Sample set a

Determine if the following pairs of fractions are equivalent.

Practice set a

Determine if the pairs of fractions are equivalent.

Reducing fractions to lowest terms

It is often very useful to conver t one fraction to an equivalent fraction that has reduced values in the numerator and denominator. We can suggest a method for doing so by considering the equivalent fractions $\frac{9}{\text{15}}$ and $\frac{3}{5}$ . First, divide both the numerator and denominator of $\frac{9}{\text{15}}$ by 3. The fractions $\frac{9}{\text{15}}$ and $\frac{3}{5}$ are equivalent.

(Can you prove this?) So, $\frac{9}{\text{15}}=\frac{3}{5}$ . We wish to convert $\frac{9}{\text{15}}$ to $\frac{3}{5}$ . Now divide the numerator and denominator of $\frac{9}{\text{15}}$ by 3, and see what happens.

$\frac{9÷3}{\text{15}÷3}=\frac{3}{5}$

The fraction $\frac{9}{\text{15}}$ is converted to $\frac{3}{5}$ .

A natural question is "Why did we choose to divide by 3?" Notice that

$\frac{9}{\text{15}}=\frac{3\cdot 3}{5\cdot 3}$

We can see that the factor 3 is common to both the numerator and denominator.

Reducing a fraction

A fraction can be reduced by dividing both the numerator and denominator by the same nonzero whole number.

Consider the collection of equivalent fractions

$\frac{5}{\text{20}}$ , $\frac{4}{\text{16}}$ , $\frac{3}{\text{12}}$ , $\frac{2}{8}$ , $\frac{1}{4}$

Reduced to lowest terms