Equivalent Fractions

Defining Equivalence

Different names, same value.

Equivalent fractions look different but represent the exact same amount. For example,

\frac{1}{2}

of a pizza is the exact same amount of food as

\frac{2}{4}

of a pizza.

Key Points

They take up the exact same amount of space.
If you simplify them, they will all shrink down to the exact same basic fraction.
They are just different ways to name the same slice of the pie.

Olympiad Fact

Cross-multiplication is the ultimate test: if $a \times d = b \times c$ , then $\frac{a}{b} = \frac{c}{d}$ . This works reliably even for very large, complex fractions.

Creating Equivalents

Scale up or down.

You can create your own equivalent fraction by multiplying both the top and bottom numbers by the exact same number.

Key Points

Pick any number (except zero) and multiply it by both the top and bottom. For example, multiply $\frac{1}{3}$ by $2$ on top and bottom to get $\frac{2}{6}$ .
You can also divide both numbers by the same amount to create a simpler equivalent fraction.
Never add or subtract! Only multiplying or dividing keeps the fraction's true value the same.

Olympiad Fact

Multiplying both parts of a fraction by the same non-zero integer $k$ is equivalent to multiplying the fraction by $\frac{k}{k}$ . Since $\frac{k}{k} = 1$ , the fundamental value remains invariant.

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