Bit 1 of 5
A fraction, a decimal and a percentage can name the . To turn a fraction into a decimal, divide the top by the bottom. Then times by 100 to get a percentage. So ⅜ = 0.375 = 37.5%. Handy ones to know: ¼ = 25%, ⅛ = 12.5% and ⅕ = 20%.
FREE LESSON · Maths Foundations
Fractions, Decimals and Percentages
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teach
Teach: keep track of what's left after each share
Switch between forms. Then take each share from what's left at that step.
Some stories take a share, then a share of what's left, then another. Write down what's left after each step. Then you always know which amount the next share comes from.
A fraction, a decimal and a percentage can name the same share. To turn a fraction into a decimal, divide the top by the bottom. Then times by 100 to get a percentage. So ⅜ = 0.375 = 37.5%. Handy ones to know: ¼ = 25%, ⅛ = 12.5% and ⅕ = 20%.
'Of' tells you which amount to take the share from. A stall has 240 tickets. ⅜ of 240 is 240 ÷ 8 × 3 = 90 tickets sold. That leaves 150. Next it sells 20% of the tickets left. That's 20% of 150, which is 30. It isn't 20% of 240.
Now 120 tickets are left. Then the stall sells ¼ of these. ¼ of 120 is 30, so 90 are left. Write a short list after every step: 240, then 150, then 120, then 90. The list shows the amount for the next share.
The question may ask for a percentage of the tickets at the start. 90 out of 240 is ⅜, which is 37.5%. Use 240, because the question says 'of the original tickets'. Using 120 would answer a different question.
You can check another way. Selling ⅜ leaves ⅝. Selling 20% of the rest leaves ⅘ of it. Selling ¼ of the rest leaves ¾ of it. ⅝ × ⅘ × ¾ = ⅜, which is 37.5%. The trap is taking 37.5%, 20% and 25% away from 100%. Each share came from a different amount.
Same share
a fraction, decimal and percentage that mean the same amount, like ⅜ = 0.375 = 37.5%.Pick the form that makes the sum easiest.The whole
the amount a fraction or percentage is taken from.For '20% of the tickets left', the whole is 150, not 240.What's left
the amount you still have after the steps so far.The next share comes from this amount.Share of the start
what's left at the end, compared with the amount at the start.90 out of the 240 at the start is ⅜, or 37.5%.- What's asked240 tickets. Sell ⅜, then 20% of the tickets left, then ¼ of the tickets still left. What percentage of the 240 is not sold?
- The factsThe whole changes each time. ⅜ of 240 = 90. 20% of the 150 left = 30. ¼ of the 120 left = 30.
- Work it outKeep a list: 240, then 150, then 120, then 90. Then compare 90 with the 240 at the start. 90 out of 240 = ⅜ = 37.5%.
- The trap100% − 37.5% − 20% − 25% = 17.5%. But those shares came from three different amounts.
- The answer37.5% of the tickets are not sold. Check: ⅝ × ⅘ × ¾ = ⅜.
The moveKeep a list of what's left. Then answer with the whole the question names.
- You can turn a fraction into a decimal and a percentage.
- You can say which amount each share is taken from.
- You can compare what's left with the start, and check it another way.
show
Show: a worked example
Watch how to solve it, one step at a time.
Read the problem, then follow the steps. You'll see the answer, the trap, and what to do next time.
Full problem: three sales, three different wholes
Question: A ticket stall starts with 240 tickets. It sells ⅜ of them. Then it sells 20% of the tickets left. Then it sells ¼ of the tickets still left. What percentage of the 240 tickets is not sold?Tickets at the school fair
A stall starts with 240 tickets.
In the first hour it sells ⅜ of them.
In the second hour it sells 20% of the tickets left.
In the third hour it sells ¼ of the tickets still left.
What percentage of the 240 tickets is not sold?
A 17.5% B 37.5% C 50% D 62.5% E 90%
Step 1 — Read the question
Work out what kind of answer you need.
The question asks for a percentage of the 240 tickets at the start. It's not a number of tickets. The first share comes from 240. 'Tickets left' means the next share comes from a new amount. 'Still left' changes it again. Estimate: about a third goes first, then a bit more each hour. So under half will be left.
- What the question asksKnowing what the question asks tells you what to look for.
- What to avoidNaming the question type helps you dodge the most common mistake.
Step 2 — Work it out
Follow the steps from the facts to the answer.
Keep a list of what's left.
- First sale: ⅜ of 240 = 90 sold. Left: 240 − 90 = 150 tickets.
- Second sale: 20% of 150 = 30 sold. Left: 150 − 30 = 120 tickets.
- Third sale: ¼ of 120 = 30 sold. Left: 120 − 30 = 90 tickets.
- List: 240, 150, 120, 90 tickets.
- Share of the start: 90 out of 240 = ⅜ = 37.5%. Choose B.
- Fact to stepEach step says which fact or rule it uses.
- What each step showsEach step adds one thing we know, until the answer is clear.
Step 3 — Spot the trap
See why a tempting answer is wrong.
Common wrong answer: A, 17.5%.
Why it tempts you: ⅜ is 37.5% and ¼ is 25%. Take them and the 20% away from 100%, and you get 17.5%.
Why it's wrong: the 20% comes from 150, and the ¼ comes from 120. They aren't shares of 240. C, 50%, stops after the second sale. E, 90, is the right number of tickets, but the question asks for a percentage.
Fix: write what's left after each sale. Then compare the final 90 with the 240 at the start.
- Why it tempts youA trap often uses real words from the question, so it looks right.
- The fixThe fix shows which fact changes the answer.
Step 4 — The answer
Say the answer and the shortest proof.
B, 37.5%. The sales leave 150, then 120, then 90 tickets. The question asks for a share of the 240 at the start. 90 out of 240 is ⅜, which is 37.5%. Check another way: ⅝ × ⅘ × ¾ = ⅜.
- The answerSay the answer in the form the question asks for.
- The proofEvery step comes from the facts.
Step 5 — The method
What to remember for questions like this.
At each step, ask: what's the whole now? You can't add the shares sold, because they come from different amounts. One way: follow 240, 150, 120, 90, then turn 90 out of 240 into 37.5%. A quicker check: times the fractions left, ⅝ × ⅘ × ¾ = ⅜. Both ways agree. Taking 37.5%, 20% and 25% from 100% gives the wrong answer, 17.5%.
- The methodThe steps you can use on any question like this.
- The trapsMistakes to watch for next time.
Step 6 — Next time
What to do on your next question.
In this question Next time Why it helps 20% came from 150, not 240. Write what's left beside every 'of'. A percentage always needs a whole. ¼ came from 120, leaving 90. Update your list after each step. You won't use an old amount by mistake. The answer was 90 out of 240 = 37.5%. Circle 'of the original'. Divide by the amount at the start. It stops you giving the number of tickets instead. ⅝ × ⅘ × ¾ = ⅜. Times the fractions left as a check. A second way catches slips.
compare
Compare: take shares from the start, or from what's left?
The same three sales give very different answers.
Two students answered the same question. One followed the facts all the way. The other didn't.
The whole response
Takes every share from the start
¼ = 25% and ⅓ ≈ 33.3%, so 100% − 25% − 25% − 33.3% ≈ 16.7% left.
Updates what's left after each sale
160, then 120, then 90, then 60. 60 out of 160 = 37.5% left.
Point-by-point breakdown3
- Second saleWeaker: Takes 25% of the 160 at the start.Stronger: Takes 25% of the 120 left: 30.'Tickets left' means a new whole.
- Third saleWeaker: Takes another third of the start away.Stronger: Takes ⅓ of the 90 still left: 30.A later share comes from what's left at that step.
- Final answerWeaker: Gives about 16.7%, using the wrong wholes.Stronger: Compares the final 60 with the 160 at the start: 37.5%.'Of the original' tells you to divide by the start.
guide
Guide: try one together
One small step with help before you go solo.
Look back at the worked problem. Now try one like it.
A library has 200 books. It lends out 30% of them. Then it keeps ¼ of the books left for a class. What percentage of the 200 books can still be borrowed? Show your working.
Want feedback on your own answer? Get started to practise with instant marking.
- Take 30% of the 200 books, then write what's left.
- Take ¼ of what's left, not ¼ of 200.
- Compare the books still free with the 200 at the start.
Your turn
A museum has 160 entry stickers. It gives one quarter to the morning group. Later, 20% of the remaining stickers are set aside for staff.
What percentage of the original stickers is still available for visitors?
- A. 40%
- B. 55%
- C. 60%
- D. 75%
- E. 96%
A coach marks 0.4 of 200 seats as occupied. Then 25% of the empty seats are booked.
What percentage of all 200 seats is still empty?
- A. 35%
- B. 40%
- C. 45%
- D. 50%
- E. 60%
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