Increase with a multiplier
Increase £80 by 20%
100% + 20% = 120%: × 1.2
80 × 1.2 = £96
Percentages and multipliers Why doesn't adding 20% back undo a 20% discount?
Foundation and Higher · AQA · 45 minutes
Students judge a tempting wrong answer (20% off £120: add 20% back?), then see every percentage question as one line: new = multiplier × original. Increase, decrease, change and reverse in one idea. Slides, a vote that shows who's stuck, practice and homework.
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The sloping line is new price = multiplier × original price, and the multiplier is m ÷ 10: m = 8 is × 0.8 (20% off), m = 10 is no change, m = 12 is × 1.2 (20% more). The flat line is a new price of £120. Move m, and read the original price below the crossing: is it £144 for 20% off?
They cross at (150, 120)
The idea in one minute
The original amount is always 100%. A percentage change makes a new percentage of it, and that percentage ÷ 100 is the multiplier: 20% more is 120%, so × 1.2; 20% off is 80%, so × 0.8. Every question is then one equation, new = multiplier × original. To find the new amount, multiply. To find a percentage change, divide the change by the original. To find the original (a reverse percentage), divide the new amount by the multiplier: never take the percentage of the new amount.
The lesson starts with a wrong answer that sounds right: a coat is £120 after 20% off, so add 20% of £120 back on. On a graph, new = multiplier × original is a straight line through the origin whose gradient is the multiplier; reading it backwards from the new price gives the original. Percentage bars show where 100% is. The core is for Foundation and Higher; the stretch asks what percentage undoes a 25% rise.
By the end, students can
- find a percentage of an amount, with and without a calculator
- write the multiplier for any percentage increase or decrease
- work out a percentage change, dividing by the original
- find the original amount by dividing by the multiplier (a reverse percentage)
Key words
- percentage
- multiplier
- original amount
- percentage increase
- percentage decrease
- percentage change
- reverse percentage
Before this lesson
Students should multiply and divide decimals (with a calculator), and know that 50% is a half and 10% is a tenth. Multipliers are taught here.
Taught in Fractions: the four operations.
The lesson, step by step
45 minutes in total. The stretch is optional.
6 min 1 · Hook · students think
Was it £144?
Students judge a reverse percentage that sounds right, then check it on a line.
Say“A coat is 20% off, and now it costs £120. Jo adds 20% of £120 back on and says it was £144. Vote first: is Jo right?”
Ask- If the coat was £144, what is 20% off £144?
- Which amount is 100% here: £120 or the price before?
Watch forStudents who agree with Jo: ask them to take 20% off £144 and see whether they get £120.
For the teacher
Show the coat: 20% off, now £120. Jo's method: 20% of 120 is 24, so it was £144. Everyone votes before anyone calculates.
Then check: 20% off £144 is 144 × 0.8 = £115.20, not £120. So Jo is wrong; the price was more than £144.
Why? Jo took 20% of the wrong amount: the 20% was of the original price, not of £120. Show the line new = multiplier × original and move the multiplier: the original is where it meets £120.
Take the vote with Ask the class, then let students check Jo's answer by taking 20% off £144 themselves. Show the line slide; leave the 'what was it?' for the explain step.
Shorter: Vote and check Jo's answer; skip the line slide (−2 min).
14 min 2 · Explain · you explain
One line: new = multiplier × original
The original is 100%; the multiplier; percentage change; and reverse percentages as dividing.
Say“Every percentage question today is one equation: new = multiplier × original. The original is always 100%.”
Ask- A 20% increase: what percentage of the original is the new amount?
- What is the multiplier for a 5% increase? For a 5% decrease?
- £50 goes up to £62. Which amount do you divide by?
Watch for1.5 for a 5% increase: 5% more is 105%, so × 1.05.
Percentage change
£50 goes up to £62
Change: 12. 12 ÷ 50 = 0.24
A 24% increase
Reverse: divide by the multiplier
20% off, now £120
0.8 × original = 120
original = 120 ÷ 0.8 = £150
For the teacher
Percentage bars: each 1 along the axis is 10%. The top bar is the original, £80: 100%, so 10% is £8. The bottom bar is 120%: 12 tenths of £80, so 12 × £8 = £96. In one step, 80 × 1.2 = 96.
The multiplier is the new percentage ÷ 100: up 20% is × 1.2, down 20% is × 0.8, up 5% is × 1.05, down 35% is × 0.65, and 15% of it is × 0.15. Without a calculator, build it from 10%, 5% and 1%.
Percentage change: the change ÷ the original × 100. £50 to £62 is up £12, and 12 ÷ 50 = 0.24, a 24% increase. Or 62 ÷ 50 = 1.24, the multiplier: up 24%. Always divide by the original.
Reverse: the same equation, with the original missing. 0.8 × original = 120, so original = 120 ÷ 0.8 = £150. Check by going forwards: 150 × 0.8 = 120. On the graph, start at £120 on the new-price axis, across to the line, and down: £150.
On the last slide move m to 12 (a 20% rise to £120): the original is £100. Ask why it isn't £96.
Shorter: Skip the bars; percentage change waits for Your turn (−5 min).
Longer: Percentages over 100%: a house worth 250% of its price in 2000 (+5 min).
9 min 3 · Your turn · students write
Worked, then yours
Each worked example beside a twin that students solve themselves.
Say“Watch me do one, then you do its twin. Say the equation first: new = multiplier × original.”
Ask- Which amount is the original here?
- Is the multiplier more or less than 1?
Watch forIn the reverse twin: taking 35% of £52. The £52 is 65% of the original.
Increase and decrease
Increase £64 by 15%
× 1.15
64 × 1.15 = £73.60
Then theirs
Decrease £85 by 30%
Percentage change
£40 goes up to £46
Change: 6. 6 ÷ 40 = 0.15
A 15% increase
Then theirs
£250 goes down to £210. Percentage change?
Reverse percentage
After a 20% rise, rent is £540. Before?
1.2 × original = 540
540 ÷ 1.2 = £450
Then theirs
After 35% off, a coat is £52. What was it before?
For the teacher
Work each example aloud, saying the equation first. Then students solve its twin while you watch who finds the multiplier and who divides by the original.
The reverse pair is the one that matters most: a correct answer here is a mark many students lose.
Open the answers slide once most have finished their twins.
Shorter: Do the change and reverse pairs only (−3 min).
12 min 4 · Who's stuck? · students vote
Vote, then practise
One question shows who's stuck; then practice in levels, and a stretch.
Say“Choose an answer on your own. Nobody talks until everyone has chosen.”
Ask- Which amount is 100% here?
- Check your answer: take 20% off it. Do you get £48?
Watch forTaking 20% of whichever amount the question gives: ask which price was 100%.
A calculator is allowed. Write new = multiplier × original first. The stretch is optional.
The question that shows who's stuck
- In a sale, prices are cut by 20%. A jacket now costs £48. What did it cost before the sale?
In your Classroom, as they answer (an example) - A13
- B10
- C4
- D2
Each wrong answer is a different mistake. While they answer, your Classroom shows who chose which, so you know in seconds whom to help with what.
- Find 35% of £80.
- Sort each change onto its multiplier.They sort 8 items into × 1.05 and × 1.5 and × 0.95 and × 0.5 and × 0.05.
- Increase £240 by 12.5%.
- A plant grows from 40 cm to 50 cm. Drag the marker to its percentage increase.They drag a marker on a number line from 0 to 50.
- A bill went up by 15% and is now £92. Call the bill before x, so 1.15x = 92. Solve it a line at a time to find the bill before.They solve 1.15x = 92 line by line, each line checked as they write it.
- A price goes up by 10%. Then the new price goes down by 10%. Compared with the start, the price is now:
- A car bought for £12,500 is sold for £9,750. The percentage loss is ___ %.
- In a sale, everything is 40% off. Trainers cost £51 in the sale. What did they cost before the sale?
- A price of £186 includes VAT at 20%. How much of the £186 is VAT?
- Stretch: a price goes up by 25%. To get back to where it started, the new price must go down by ___ %.
Support · For reverse questions: write new = multiplier × original, put in what you know, and divide by the multiplier. Then check by going forwards.
For the teacher
Everyone chooses before anyone explains. Then every answer is checked by going forwards: take 20% off it. Only one gives £48.
Questions 1–3 build fluency (a sort of multipliers); 4 is a number line; 5 is solved line by line, each line checked; 6 asks for a reason; 7–9 are exam-style. The stretch asks what undoes a 25% rise.
Students may use a calculator. Their working goes on paper or the board; the quiz keeps their final answers.
If the votes split, ask everyone to take 20% off the price they chose and see whether it gives £48, before they practise.
Shorter: After the vote, questions 1–5 only (−5 min).
Longer: The stretch, then: why is a 25% rise undone by a 20% fall? (+10 min).
4 min 5 · Exit · students show
Find the price before
One reverse percentage, explained, shows who divides by the multiplier.
Say“A sale price is £84 after 30% off. Find the price before, and say why adding 30% of £84 is wrong.”
Watch forAnswers of £109.20: 30% of £84 added on.
Show the equation, the answer and one or two sentences.
- A sale price is £84 after 30% off. Find the price before the sale, and explain why adding 30% of £84 back on gives the wrong answer.
For the teacher
Students write the equation, 0.7 × original = 84, and solve it.
Then one or two sentences on which amount the 30% was of.
The answers show who divides by the multiplier: start the next lesson with one right and one wrong. Set the homework.
Longer: On the graph: draw new = 0.7 × original and read the original at £84 (+5 min).
One to one: for every question, ask the student first which amount is the original (the 100%), and to say the equation new = multiplier × original before any calculation.
Homework: multipliers and reverse
A calculator is allowed. For each, write new = multiplier × original first, then answer. Check reverse answers by going forwards.
- Find 15% of £340.
- Increase £72 by 35%.
- Decrease 850 g by 12%.
- A town's population grew from 24,000 to 27,600. Find the percentage increase.
- After a 15% discount, a bike costs £323. What did it cost before the discount?
- A £50 shirt is 20% off, and then the sale price has another 10% off. Priya says that is 30% off, so £35. Is she right? Show how you know.
Print the worksheet with working space and homework. Teacher answers come with your Library copy.
Teaching percentages and multipliers: common questions
Why use multipliers for percentages?
One multiplication does the whole change, and the same equation, new = multiplier × original, then answers every kind of question: the new amount, the percentage change and the original amount. It also makes repeated changes easy (× 1.1 then × 0.9) and leads straight into compound interest.
How do you teach reverse percentages so students remember?
Ask which amount is 100% before anything else. In a reverse question the amount given is not the original, so its percentage can't be taken. Write new = multiplier × original with the original missing, divide by the multiplier, and always check by going forwards. A graph of new = multiplier × original read backwards shows the same thing.
What is the multiplier for a percentage decrease?
100% take away the decrease, as a decimal: 30% off leaves 70%, so the multiplier is 0.7, and 4% off is 0.96. "Reduced by 30%" is × 0.7, but "reduced to 30%" is × 0.3.
Is this percentages lesson suitable for Foundation GCSE?
Yes. Percentages of amounts, increase and decrease, percentage change and reverse percentages are on both tiers, and the core is for both. The repeated changes and the stretch (what undoes a 25% rise) lead towards Higher's compound interest and growth.
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Curriculum reference: AQA GCSE Mathematics 8300: R9 (percentages, percentage change, original value problems) and N12 (percentages as operators, with a multiplier).
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