Formula

Equivalent fractions

Also written: equivalent fraction rule · cross multiplication check

ab=a×nb×n\frac{a}{b} = \frac{a \times n}{b \times n}

Multiplying or dividing the top and bottom of a fraction by the same number gives an equivalent fraction — different numbers, same amount.

What each part means

aa
the numerator
bb
the denominator, never zero
nn
any number other than zero

When to use it

You need to give two fractions a shared denominator, simplify a fraction, or check whether two fractions are equal.

The rule that everything else rests on

12=24=48\frac{1}{2} = \frac{2}{4} = \frac{4}{8}

Cutting each part in half doubles how many parts there are and doubles how many you hold. The shading on a bar never moves.

Running the rule backwards — dividing both by nn — is simplifying.

Why the value survives

Any number over itself equals 11:

nn=1\frac{n}{n} = 1

So multiplying top and bottom by nn is multiplying the whole fraction by 11:

ab×nn=anbn\frac{a}{b} \times \frac{n}{n} = \frac{an}{bn}

Multiplying by 11 never changes an amount. The numbers change costume; the value does not.

Adding the same number to both parts is a different operation and does change the value: 12≠23\tfrac{1}{2} \ne \tfrac{2}{3}.

The cross product test

Two fractions are equal exactly when their cross products match:

ab=cd⟺ad=bc\frac{a}{b} = \frac{c}{d} \quad\Longleftrightarrow\quad ad = bc

This is not a separate trick. Giving both fractions the denominator bdbd turns them into adbd\tfrac{ad}{bd} and bcbd\tfrac{bc}{bd}, and fractions over the same denominator are equal only when their tops match.

Worked examples

Write 34\tfrac{3}{4} with a denominator of 1212.

4×3=124 \times 3 = 12, so multiply both parts by 33:

34=912\frac{3}{4} = \frac{9}{12}

Are 37\tfrac{3}{7} and 1228\tfrac{12}{28} equal?

3×28=8412×7=843 \times 28 = 84 \qquad 12 \times 7 = 84

The products match, so yes — and no drawing of 2828 parts was needed.

Lessons that teach this

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