Equivalent Fractions

Spot fractions that are the same amount, write a fraction in its simplest form, and fill in the missing number in a pair of equivalent fractions.

In the exam for Edexcel IGCSE Maths 4MA1, Cambridge IGCSE Maths 0580 · 25 real past-paper questions below

اقرأ هذا الدرس مشروحاً بالعربي ←

What Equivalent Fractions Are

Equivalent fractions are the same amount, cut into different sized pieces.

Times the top and the bottom by the same number. The pieces get smaller, but you take more of them.

Worked example

1
$\frac{3}{5}$ — Start with three fifths: the whole is cut into 5 equal pieces and you take 3 of them.
2
$\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$ — Times the top and the bottom by 2. Every piece is cut in half, so there are twice as many pieces and you take twice as many.
3
$\frac{6 \times 2}{10 \times 2} = \frac{12}{20}$ — Times both by 2 again. The pieces get smaller, you take more of them, and the amount stays the same.
4
$\frac{12 \div 4}{20 \div 4} = \frac{3}{5}$ — Divide both by 4 and you are back at the start. Dividing works as well as multiplying.
5
$\frac{3}{5} = \frac{6}{10} = \frac{12}{20}$ — All three cover the same amount of the whole, so they are equivalent fractions.

Rule

Times or divide the top and the bottom by the same number. The fraction keeps the same size.

Spot the Equivalent Fraction

The 1 or 2 mark question: "Which one of these is not equivalent to $\frac{3}{8}$ ?"

\frac{6}{16} \frac{30}{80} \frac{15}{48} \frac{9}{24}

For each one, ask: was the top and the bottom multiplied by the same number?

Worked example

1
$\frac{3}{8} = \frac{6}{16}$ — Bottom: 8 × 2 = 16. Top: 3 × 2 = 6. Both were times 2, so this one is equivalent.
2
$\frac{3}{8} = \frac{30}{80}$ — 8 × 10 = 80 and 3 × 10 = 30. Both times 10, so equivalent.
3
$\frac{3}{8} = \frac{9}{24}$ — 8 × 3 = 24 and 3 × 3 = 9. Both times 3, so equivalent.
4
$48 \div 8 = 6$ — Now $\frac{15}{48}$ . Find the multiplier from the bottoms: 8 went up to 48, so it was times 6.
5
$3 \times 6 = 18 \neq = 15$ — Times the top by the same 6. An equivalent fraction would need 18 on top, but this one has 15.
6
$\frac{15}{48} is the odd one$ — Its top and bottom were not multiplied by the same number, so it is not equivalent to $\frac{3}{8}$ .

Key idea

"Tick the two that are equivalent": simplify every fraction. The two that end on the same simplest form are the pair.

Common mistake

Adding the same number to top and bottom. $\frac{3}{8}$ and $\frac{4}{9}$ are not equal. Only times or divide.

Common mistake

Changing only one number, or flipping. $\frac{3}{800}$ keeps the top but makes the pieces far smaller, and $\frac{8}{3}$ is more than one whole. Neither is equivalent to $\frac{3}{8}$ .

Simplest Form

The 1 or 2 mark question: "Write $\frac{36}{60}$ as a fraction in its simplest form." Keep dividing top and bottom by the same number.

Worked example

1
$\frac{36 \div 2}{60 \div 2} = \frac{18}{30}$ — 36 and 60 are both even, so 2 divides both. Divide the top and the bottom by 2.
2
$\frac{18 \div 3}{30 \div 3} = \frac{6}{10}$ — The digits of 18 add to 9 and of 30 add to 3, so 3 divides both. Divide both by 3.
3
$\frac{6 \div 2}{10 \div 2} = \frac{3}{5}$ — 6 and 10 are still both even, so divide both by 2 again.
4
$\frac{3}{5}$ — Nothing except 1 divides both 3 and 5, so you stop. This is the simplest form.

Faster: divide by the biggest number that goes into both. For 36 and 60 that is 12.

Worked example

1
$36 = 12 \times 3, 60 = 12 \times 5$ — 12 is the highest common factor (HCF) of 36 and 60: the biggest number that goes into both.
2
$\frac{36 \div 12}{60 \div 12} = \frac{3}{5}$ — Divide the top and the bottom by 12. One step gives the same answer as the three small ones.

Rule

A fraction is in its simplest form when no number except 1 divides both the top and the bottom.

Common mistake

Stopping at $\frac{6}{10}$ . Both are still even. On a 2 mark question this loses a mark.

Key idea

On a 2 mark question, write down your middle fractions, like $\frac{18}{30}$ . If the last step slips, a correct smaller equivalent fraction can still earn a mark for working.

Fill In the Missing Number

The 1 mark question: "Write a number in the box." Find what the known pair was multiplied or divided by.

Worked example

1
$\frac{3}{4} = \frac{□}{28}$ — Use the pair you know fully. Here it is the bottoms: 4 went up to 28.
2
$28 \div 4 = 7$ — Divide to find the multiplier: 28 ÷ 4 = 7, so the bottom was times 7.
3
$3 \times 7 = 21$ — The top must change the same way as the bottom, so times it by 7 too.
4
$\frac{3}{4} = \frac{21}{28}$ — The box is 21. Check: 21 ÷ 3 = 7 and 28 ÷ 4 = 7, the same multiplier.

Going down works the same way, with divide:

Worked example

1
$\frac{18}{24} = \frac{3}{□}$ — This time you know both tops. 18 went down to 3, so the change is a divide.
2
$18 \div 3 = 6$ — Find the divider: 18 ÷ 3 = 6, so the top was divided by 6.
3
$24 \div 6 = 4$ — Do the same to the bottom: 24 ÷ 6 = 4. The box is 4, so $\frac{18}{24} = \frac{3}{4}$ .

Key idea

"Write down a fraction equivalent to $\frac{5}{6}$ ": any answer works. Times both by 2 to get $\frac{10}{12}$ .

Common mistake

Adding instead: $4 + 24 = 28$ , so $3 + 24 = 27$ . But $\frac{27}{28}$ is not $\frac{3}{4}$ .

Shade the Same Fraction

Some questions show a shaded shape and ask you to shade another shape so both show the same fraction. It is a missing-number question in disguise.

Worked example

1
$\frac{3}{10}$ — A circle is cut into 10 equal parts and 3 are shaded. Write that fraction first.
2
$\frac{3}{10} = \frac{□}{40}$ — The rectangle has 40 squares, so 40 goes on the bottom. The box is how many squares to shade.
3
$40 \div 10 = 4$ — The bottom went from 10 to 40, so it was times 4.
4
$3 \times 4 = 12$ — Times the top by the same 4: 3 × 4 = 12.
5
$shade 12 squares$ — Any 12 squares will do, because $\frac{12}{40} = \frac{3}{10}$ . Both shapes now show the same fraction.

Key idea

This is often a 2 mark question. Write $\frac{3}{10} = \frac{□}{40}$ next to the shape. If you miscount the shading, the working can still earn a mark.

If the question gives a smaller bottom, divide instead: "12 of the 18 squares are shaded. Write the fraction in thirds."

Worked example

1
$\frac{12}{18}$ — 12 shaded squares over 18 squares in total.
2
$18 \div 6 = 3$ — The answer must be in thirds, so the bottom goes from 18 to 3: divide by 6.
3
$12 \div 6 = 2, so \frac{2}{3}$ — Divide the top by the same 6. The shaded fraction is $\frac{2}{3}$ .

Common mistake

Shading 3 squares because 3 parts of the circle were shaded. The pieces are different sizes, so match the fraction, not the count.

Which Method? Read the Question

The words in the question tell you what to do.

You see	You do
Which one is not equivalent?	check both numbers were multiplied by the same
Tick the two that are equivalent	simplify each, find the matching pair
In its simplest form	divide top and bottom until only 1 divides both
Write a number in the box	find the multiplier, use it on the other number
Shade so both shapes show the same fraction	write the shaded fraction, find the missing top
Write 30 as a fraction of 48	Fractions of an Amount
Adding fractions with different bottoms	Addition and Subtraction of Fractions
Top heavy fractions	Mixed Numbers and Improper Fractions

Key idea

Quick check: $\frac{3}{8}$ and $\frac{6}{16}$ are equivalent because $3 \times 16 = 8 \times 6$ .

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