From index notation and the laws of indices to zero, negative powers and powers in real life.
6 lessons · 35 questions (A–G) · 20 calculation + 15 word problems · no time pressure
Finding a fraction of an amount comes down to one idea: divide by the bottom number, then multiply by the top. This paper builds from simple unit fractions up to larger amounts, simplifying, and real-life money, measures and multi-step problems. Every worked example shows the thinking and ends with a check.
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Lesson 1
Unit fractions of an amount
Aim: Find a unit fraction of an amount by dividing.
A unit fraction has 1 on top, like 1/4. To find 1/4 of an amount you share it into 4 equal parts — that means divide by the bottom number (the denominator). One of those equal parts is your answer.
20 split into 4 equal parts of 5; one part (hatched) is 1/4 = 5.
Worked example
Find 1/4 of 20.
The bottom number is 4, so split 20 into 4 equal parts.
Dividing does the splitting: 20 ÷ 4 = 5.
One part is 5, so 1/4 of 20 = 5.
Check: 4 parts of 5 make 4 × 5 = 20, the whole amount. ✓
Watch out
‘Of’ here means divide by the bottom number. 1/4 of 20 is 5, found by 20 ÷ 4.
Aim: Find any fraction of an amount: divide by the bottom, multiply by the top.
For a fraction like 3/4, first find one part by dividing by the bottom number, then take 3 of those parts by multiplying by the top number. In short: divide by the denominator, then multiply by the numerator.
20 in 4 parts of 5; three parts (hatched) make 3/4 = 15.
Worked example
Find 3/4 of 20.
First find 1/4 by dividing: 20 ÷ 4 = 5. That is one part.
We want 3 of those parts, so multiply by the top number: 5 × 3 = 15.
So 3/4 of 20 = 15.
Check: 3 parts of 5 make 15, and the last part (5) makes 20 in total. ✓
Watch out
Divide by the bottom, then multiply by the top. Keep the two steps separate so you don’t muddle them.
Try one
Find 2/3 of 18.
Show answer
1/3 of 18 is 18 ÷ 3 = 6.
Two parts: 6 × 2 = 12.
Check: 2 parts of 6 is 12, with 1 part (6) left to make 18. ✓
Aim: Find a fraction of an amount of money or a measurement, keeping the units.
Fractions of money or measures work in exactly the same way: divide by the bottom, multiply by the top. Just remember to write the units (pounds, grams, ml, minutes) in your answer.
£35 in 5 parts of £7; two parts (hatched) make 2/5 = £14.
Worked example
Find 2/5 of £35.
Find 1/5 first: £35 ÷ 5 = £7.
We want 2 parts: £7 × 2 = £14.
So 2/5 of £35 = £14.
Check: 5 parts of £7 make £35. ✓
Watch out
Keep the units in the answer. 2/5 of £35 is £14, not just 14.
Aim: Find fractions of larger amounts, simplifying the fraction first when it helps.
The method is the same for big numbers. Sometimes the fraction can be simplified first to make the dividing easier — for example 6/10 is the same as 3/5. Simplify if you spot it, but you do not have to.
Simplify 6/10 to 3/5: 50 in 5 parts of 10, three parts (hatched) = 30.
Worked example
Find 6/10 of 50.
6/10 simplifies to 3/5 (divide top and bottom by 2). Either form works.
Find 1/5 of 50: 50 ÷ 5 = 10.
Three parts: 10 × 3 = 30.
So 6/10 of 50 = 30. Check using 6/10 directly: 50 ÷ 10 = 5, then 5 × 6 = 30 — the same answer. ✓
Watch out
Simplifying is optional — it just makes the numbers friendlier. Both forms give the same answer.
Try one
Find 4/6 of 90.
Show answer
4/6 simplifies to 2/3.
1/3 of 90 is 90 ÷ 3 = 30; two parts: 30 × 2 = 60.
Check using 4/6: 90 ÷ 6 = 15, then 15 × 4 = 60. Same. ✓
Aim: Solve real-life problems by finding a fraction of an amount.
Many real problems are just ‘find a fraction of an amount’ in disguise: a fraction of a class, a price, a distance or a time. Spot the whole amount and the fraction, then use the same divide-then-multiply method.
30 in 5 parts of 6; three parts (hatched) are the girls, 3/5 = 18.
Worked example
In a class of 30 children, 3/5 are girls. How many girls are there?
The whole amount is 30 and the fraction is 3/5.
Find 1/5: 30 ÷ 5 = 6.
Three parts: 6 × 3 = 18.
So there are 18 girls. Check: 18 girls and 12 boys make 30. ✓
Watch out
Find the whole amount and the fraction first, then divide and multiply.
Try one
2/3 of a £45 prize is given away. How much is given away?
Show answer
The whole is £45 and the fraction is 2/3. 1/3 is £45 ÷ 3 = £15.
Aim: Solve problems that need a fraction and a second step, such as finding what is left.
Some problems need an extra step: find the fraction, then subtract to find what is left, or find a part and work back to the whole. Do the fraction step first, then the second step, and check at the end.
£60 in 3 parts of £20; one part (hatched) is spent, two parts left = £40.
Worked example
Sam has £60 and spends 1/3 of it. How much is left?
Step 1 — find the amount spent: 1/3 of £60 is £60 ÷ 3 = £20.
Step 2 — subtract to find what is left: £60 − £20 = £40.
So £40 is left.
Check: spent £20 + left £40 = £60. ✓
Watch out
Do the fraction first, then the subtraction. Read whether the question wants what is left or what was used.
Try one
1/4 of a number is 9. What is the number?
Show answer
1/4 of the number is 9, so one part is 9 and there are 4 equal parts.