Equivalent Fractions and Simplifying to Simplest Form

Why cutting pieces smaller does not change how much you have, how to make equivalent fractions, and how to simplify a fraction to simplest form in one step with the greatest common factor.

3 min read Supports 4 lessons

Take half a bar of chocolate. Now snap every piece in two. You have more pieces, but exactly the same amount of chocolate. Half became two quarters, and two quarters can become four eighths.

\tfrac{1}{2}

\tfrac{2}{4}

\tfrac{4}{8}

Three names for one amount: $\frac{1}{2} = \frac{2}{4} = \frac{4}{8}$ .

Fractions like these are called equivalent. They look different but sit at exactly the same point on the number line.

The rule

Equivalent fractions

\frac{a}{b} = \frac{a \times k}{b \times k} \qquad (k \neq 0)

Multiply or divide the top and the bottom by the same non-zero number.

Why does it work? Multiplying the top and bottom by $k$ is the same as multiplying the whole fraction by $\frac{k}{k}$ , which is just $1$ . And multiplying by $1$ changes nothing.

\frac{2}{3} = \frac{2}{3} \times \frac{4}{4} = \frac{8}{12}

Watch out

Adding the same number to top and bottom does not work: $\frac{1}{2}$ and $\frac{1 + 1}{2 + 1} = \frac{2}{3}$ are different amounts.

Simplifying

Simplifying runs the rule backwards: divide the top and bottom by a common factor to get smaller numbers. A fraction is in simplest form (or lowest terms) when the top and bottom share no factor except $1$ .

$\frac{18}{24}$
Both numbers are even, but halving would take two steps. Find the biggest shared factor instead.

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Tip

You can also simplify in stages: $\frac{18}{24} \to \frac{9}{12} \to \frac{3}{4}$ . Using the greatest common factor just gets there in one go.

Making equivalent fractions on purpose

Often you want a particular denominator, for example to compare or add two fractions. Ask what you must multiply the old denominator by to reach the new one, then do the same to the numerator.

Worked example

Write $\frac{3}{5}$ as a fraction with denominator $35$ .

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$5 \times 7 = 35$ , so the scale factor is $7$ .

Cancelling in algebra

The same rule simplifies algebraic fractions. You may divide the top and bottom by a common factor, something multiplied by everything above and below the line:

\frac{6x}{9x} = \frac{2}{3} \qquad \frac{(x + 3)(x - 1)}{(x + 3)} = x - 1

Watch out

You can only cancel factors, never terms joined by $+$ or $-$ . $\frac{x + 3}{3}$ is not $x$ .

Common questions

How do you know if a fraction is in simplest form?: Check whether the numerator and denominator share any factor other than $1$ . If their greatest common factor is $1$ , the fraction cannot be simplified further.
Do equivalent fractions have the same decimal?: Yes. They are the same number, so dividing top by bottom gives the same decimal: $\frac{3}{4}$ , $\frac{6}{8}$ and $\frac{75}{100}$ are all $0.75$ .

Practise equivalent fractions

The Equivalent Fractions & Simplifying lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

How do you know if a fraction is in simplest form?

Check whether the numerator and denominator share any factor other than $1$ . If their greatest common factor is $1$ , the fraction cannot be simplified further.

Do equivalent fractions have the same decimal?

Yes. They are the same number, so dividing top by bottom gives the same decimal: $\frac{3}{4}$ , $\frac{6}{8}$ and $\frac{75}{100}$ are all $0.75$ .