D-20MathsFri 21 Aug

Reverse percentages

Work backwards with ÷ — not by subtracting the %.

A normal percentage question gives you the whole and asks for a part. A reverse percentage question does the opposite: it gives you the amount after a rise or fall, and asks for the original. You cannot simply add or subtract the percentage back on — you need to divide.

After a decrease

The given amount is less than 100% — e.g. 20% off means you are looking at 80%.

Sale price

80% · given

Original

100%

After an increase

The given amount is more than 100% — e.g. a 10% rise means you are looking at 110%.

Original

100%

New amount

110% · given

Picture it: 20% off leaves £32 — find the original price

20%

Original price = 100% = 5 equal parts of 20%

20%

off

Sale price = 80% (4 parts) = £32 → 20% (1 part) was taken off

4 parts

£32

→

1 part

£8

→

5 parts

£40

£32 is 4 equal parts, so one part = £32 ÷ 4 = £8. The original price (5 parts) = 5 × £8 = £40.

Remember — the method

Step 1

What % is this?

After a 20% cut, the given amount is 80%.

Step 2

Find 1%

Divide the given amount by that percentage.

Step 3

Scale to 100%

Multiply by 100 to get the original.

Sanity check: after a decrease, the original must be bigger than the given amount; after an increase, the original must be smaller.

Exam trap

Adding the percentage straight back onto the given number, instead of working out what the given number actually represents.

✗ £32 + 20% of £32

£38.40

Sale

80% · £32

Wrong add-back

96% · £38.40

20% of the sale price isn’t 20% of the original

✓ £32 ÷ 80 × 100

£40.00

Sale

80% · £32

Original

100% · £40

Correct — £32 is 80% of the original

Question patterns you might see

Same divide-then-scale method. Work each one in steps — don’t jump to the answer in one line.

1 · Simple — Reverse a decrease
“A jacket is reduced by 25% to £45. Find the original price.”
25%
25%
25%
off
£45 is 3 of 4 equal parts (75%). One 25% part was taken off.
1. 1After a 25% cut, the sale price is 100% − 25% = 75%.
2. 2£45 is 3 equal parts, so one part = £45 ÷ 3 = £15.
3. 3The original (4 parts) = 4 × £15 = £60.
4. 4Check: original is bigger than £45 — that matches a decrease.
Original price = £60
2 · Simple — Reverse an increase
“After a 10% pay rise, Sam earns £550 a week. What did he earn before?”
Before
100%
After rise
110% · £550
1. 1After a 10% rise, the new wage is 100% + 10% = 110%.
2. 2£550 ÷ 110 = £5. That is 1%.
3. 3Original (100%) = £5 × 100 = £500.
4. 4Check: original is smaller than £550 — that matches an increase.
Original wage = £500
3 · Medium — A charge added on top
“The price including a 20% service charge is £96. Find the price before the charge.”
20%
20%
20%
20%
20%
20%
Bill with charge = 120% = 6 equal 20% parts
20%
20%
20%
20%
20%
extra
Price before the charge = 5 parts = 100%
1. 1£96 includes the extra 20%, so it represents 120%.
2. 2£96 ÷ 120 = £0.80. That is 1%.
3. 3Price before the charge (100%) = £0.80 × 100 = £80.
4. 4Check: £80 + 20% of £80 = £80 + £16 = £96.
Price before charge = £80
4 · Medium — Find the original, then a new percentage of it
“A number is decreased by 15% to give 68. What is 40% of the original?”
Asked for
40%
After −15%
85% · 68
Original
100%
1. 1After a 15% decrease, 68 is 85% of the original.
2. 2Original = 68 ÷ 85 × 100 = 80.
3. 3Now find 40% of that original: 40% of 80 = 32.
4. 4Do not find 40% of 68 — that is 40% of the reduced number, not the original.
85%
68
→
100%
80
→
40%
32
40% of the original = 32
5 · Trickier — Two changes chained together
“A price rises by 10%, then falls by 10%, ending at £198. Find the very original price.”
Original
100%
After +10%
110%
After −10%
99% · £198
1. 1Undo the last change first. The fall of 10% means £198 is 90% of the risen price.
2. 2Risen price = £198 ÷ 90 × 100 = £220.
3. 3That £220 is after a 10% rise, so it is 110% of the original.
4. 4Original = £220 ÷ 110 × 100 = £200.
End (90%)
£198
→
Risen (110%)
£220
→
Original
£200
Very original price = £200
6 · Trickiest — Up then down isn’t “net zero”
“A price rises 25% then falls 20%. Another price rises 10% then falls 10%. Which returns to the original?”
+25% then −20%
Start
100%
After +25%
125%
After −20%
100% · back
1. 1Multiply the multipliers: 1.25 × 0.8.
2. 21.25 × 0.8 = 1.0 — exactly the original.
+10% then −10%
Start
100%
After +10%
110%
After −10%
99% · not 100
1. 1Multiply the multipliers: 1.10 × 0.9.
2. 21.10 × 0.9 = 0.99 — a 1% net decrease, not zero.
Always multiply the multipliers. Do not subtract the percentages.

Coach tip: before dividing, say out loud what percentage the given number represents. That one sentence stops the add-back mistake almost every time.

Tomorrow (D-19): cube nets & opposite faces.

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