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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.
There is an interesting property that equivalent fractions satisfy.
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, $\frac{2}{3}=\frac{4}{6}$ .
Determine if the following pairs of fractions are equivalent.
$\frac{3}{4}\text{and}\frac{6}{8}$ . Test for equality of the cross products.
The cross products are equals.
The fractions $\frac{3}{4}$ and $\frac{6}{8}$ are equivalent, so $\frac{3}{4}=\frac{6}{8}$ .
$\frac{3}{8}\text{and}\frac{9}{\text{16}}$ . Test for equality of the cross products.
The cross products are not equal.
The fractions $\frac{3}{8}$ and $\frac{9}{\text{16}}$ are not equivalent.
Determine if the pairs of fractions are equivalent.
$\frac{1}{2}$ , $\frac{3}{6}$
, yes
$\frac{4}{5}$ , $\frac{\text{12}}{\text{15}}$
, yes
$\frac{2}{3}$ , $\frac{8}{\text{15}}$
$\mathrm{30}\ne \mathrm{24}$ , no
$\frac{1}{8}$ , $\frac{5}{\text{40}}$
, yes
$\frac{3}{\text{12}}$ , $\frac{1}{4}$
, yes
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.
$\frac{9\xf73}{\text{15}\xf73}=\frac{3}{5}$
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.
A fraction can be reduced by dividing both the numerator and denominator by the same nonzero whole number.
Consider the collection of equivalent fractions
$\frac{5}{\text{20}}$ , $\frac{4}{\text{16}}$ , $\frac{3}{\text{12}}$ , $\frac{2}{8}$ , $\frac{1}{4}$
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