Equivalent Fractions

What are equivalent fractions?

Equivalent fractions are fractions that look different on paper but show exactly the same amount. For example, 1/2 and 2/4 are equivalent — they both mean "one half" of something. If you cut a pizza into 2 slices and take 1, you have the same amount as if you cut it into 4 slices and take 2.

If fractions are new to you, it's worth reading our introduction to fractions first, where we explain the numerator (the top number, how many parts you have) and the denominator (the bottom number, how many equal parts the whole is split into).

See it with pizza

Imagine three identical pizzas.

• The first is cut into 2 slices. You take 1. That's 1/2.
• The second is cut into 4 slices. You take 2. That's 2/4.
• The third is cut into 8 slices. You take 4. That's 4/8.

Place the slices side by side and they cover the same space every time. Even though the numbers 1/2, 2/4 and 4/8 look different, the amount of pizza is identical. That is what "equivalent" means — equal in value.

1/2  =  ▓▓▓▓▓▓▓▓░░░░░░░░
2/4  =  ▓▓▓▓▓▓▓▓░░░░░░░░
4/8  =  ▓▓▓▓▓▓▓▓░░░░░░░░

The rule: do the same to top and bottom

Here is the golden rule for making equivalent fractions:

Multiply (or divide) the top and the bottom by the SAME number.

Watch how 1/2 grows into a family of equal fractions:

Start	Multiply top and bottom by	Result
1/2	× 2	2/4
1/2	× 3	3/6
1/2	× 4	4/8
1/2	× 5	5/10

Every fraction in that list equals one half.

Why does this work?

This is the most important "why" in the lesson. When you multiply the top and bottom by the same number, you are really multiplying the whole fraction by something equal to 1.

Look: multiplying 1/2 by 2/4... no — multiplying 1/2 by 2/2. And 2/2 is just 1 (two halves make a whole). Anything multiplied by 1 stays the same value. So 1/2 × 2/2 = 2/4, and the value hasn't changed — only the way we've written it.

This is also why you must never add the same number to top and bottom. Adding doesn't keep the value the same. (1+1)/(2+1) = 2/3, which is not a half!

Worked example 1: finding an equivalent fraction

Find a fraction equal to 3/5 with a denominator of 20.

Ask: what turns 5 into 20? It's × 4. So multiply both top and bottom by 4:

• Top: 3 × 4 = 12
• Bottom: 5 × 4 = 20

So 3/5 = 12/20. The key is that we did the same thing (× 4) to both numbers.

Worked example 2: simplifying a fraction

Going the other way — dividing both numbers — makes a fraction simpler. Simplify 8/12.

Find a number that divides both 8 and 12. Both divide by 4:

• Top: 8 ÷ 4 = 2
• Bottom: 12 ÷ 4 = 3

So 8/12 = 2/3. Now check: can anything divide both 2 and 3 (other than 1)? No. So 2/3 is in its simplest form.

Worked example 3: are these equal?

Are 6/9 and 2/3 equivalent?

Simplify 6/9 by dividing both by 3: 6 ÷ 3 = 2 and 9 ÷ 3 = 3, giving 2/3. Yes — they are equivalent!

A quick cross-check trick: multiply crossways. 6 × 3 = 18 and 9 × 2 = 18. Because the cross-products match, the fractions are equal. If they were different numbers, the fractions wouldn't be equivalent.

Simplest form

A fraction is in its simplest form (also called lowest terms) when the only number that divides both top and bottom is 1. To get there, keep dividing by common factors until you can't anymore. Simplest form is useful because it's the cleanest way to write a fraction and makes fractions easier to compare and add.

Fraction	Divide by	Simplest form
4/8	4	1/2
9/12	3	3/4
10/15	5	2/3
7/13	—	already simplest

Why equivalent fractions matter

Equivalent fractions are the secret behind almost everything you'll do with fractions later:

• Comparing fractions — to see whether 2/3 or 3/4 is bigger, you rewrite them with the same denominator.
• Adding and subtracting fractions — you can only add fractions once they share a denominator, which means finding equivalent ones.
• Decimals and percentages — 1/2 = 50/100 = 50%, all built on equivalence.

Master this and the rest of fractions becomes far less scary.

Try it yourself

1. Write three fractions equivalent to 1/4 by multiplying top and bottom by 2, 3 and 5.
2. Simplify these to lowest terms: 6/10, 12/16, 15/20.
3. Spot the odd one out: which of these is not equal to the others? 2/3, 4/6, 5/9, 6/9.
4. Real-life challenge: a recipe needs 2/4 of a cup of sugar. Write that amount as a simpler fraction.

Answers: 1) 2/8, 3/12, 5/20. 2) 3/5, 3/4, 3/4. 3) 5/9 is the odd one out (the rest equal 2/3). 4) 1/2 a cup.

When you're ready, see how fractions connect to whole numbers in our introduction to fractions lesson.