Up, down and back: change is measured on the old value
A percent change is always measured on the old value. Turn "up 20%" into "× 1.2" and "down 20%" into "× 0.8": then going forward is one multiplication, and going back is one division.
Guess first
Guess first: after a 20% discount you paid ₹960 for a pair of shoes. Your friend adds 20% of ₹960 to ₹960 and says the tag showed ₹1,152. What did the tag show?
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Correct answer: B
The 20% was taken off the tag price, not off ₹960. So you paid 80% of the tag: 80% of the tag = ₹960, 1% = 960 ÷ 80 = 12, and the tag = 12 × 100 = ₹1,200. Check: 20% of 1,200 = 240, and 1,200 − 240 = 960.
Option A: ₹1,152 adds 20% of ₹960. But the discount was 20% of the tag price, which is bigger than ₹960, so the friend adds too little.
Option C: ₹768 takes another 20% off ₹960. That applies the discount a second time instead of undoing it.
Option D: ₹240 is the discount itself (1,200 − 960 = 240), not the tag price.
Hint: Which price was the 20% taken from: the tag or the ₹960?
If you chose ₹1,200, you already know the main idea of this lesson: a percent change belongs to the value it started from.
An everyday picture
A rise of ₹100 means little until you know the old bill. The old value is the ruler that measures the change.
₹100 more on a ₹200 bill a 50% rise
measured on the old ₹200
₹100 more on a ₹1,000 bill a 10% rise
measured on the old ₹1,000
₹100 less on a ₹500 bill a 20% fall
measured on the old ₹500
The same ₹100 is a big change for a small bill and a small change for a big bill. That is why every percent change is divided by the old value.
See it: one multiplication forward, one division back
Start with ₹500. A rise of 20% adds 20 hundredths: 100% + 20% = 120%, so the new price is 120% of the old one. A fall of 20% leaves 100% − 20% = 80%.
| Change on ₹500 | Multiplier | New value |
|---|---|---|
| up 20% | × 1.2 | 500 × 1.2 = 600 |
| up 5% | × 1.05 | 500 × 1.05 = 525 |
| down 20% | × 0.8 | 500 × 0.8 = 400 |
| down 35% | × 0.65 | 500 × 0.65 = 325 |
flowchart TD A["Old price ₹500"] -->|"up 20%: × 1.2"| B["New price ₹600"] B -->|"back: ÷ 1.2"| C["Old price ₹500 again"] A -->|"down 20%: × 0.8"| D["Sale price ₹400"]
In words: ₹500 × 1.2 = ₹600, and to get back you divide: 600 ÷ 1.2 = 500. Taking 20% off ₹600 would give ₹480, not ₹500, because 20% of ₹600 is bigger than 20% of ₹500.
Try it on paper. Pick a price you know, say a ₹40 notebook. Find it after a 10% rise (× 1.1) and after a 10% fall (× 0.9). Then undo each one by dividing by the same multiplier.
Find the rule
A ₹650 cricket bat costs 20% more next season. What is the new price?
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Correct answer: A
Up 20% means × 1.2: 650 × 1.2 = ₹780. Or: 20% of 650 = 130, and 650 + 130 = 780.
Option B: ₹130 is only the rise, not the new price.
Option C: ₹520 takes 20% off. The price went up, not down.
Option D: ₹78 is ten times too small: the decimal point slipped one place.
Hint: The new price is 120% of the old price.
A tea stall's daily sales rise from ₹300 to ₹375. What is the percentage increase?
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Correct answer: B
The rise is 375 − 300 = 75, measured on the old value: 75 ÷ 300 × 100 = 25%.
Option A: 20% divides the rise by the new value: 75 ÷ 375 × 100 = 20%. The change is measured on the old value.
Option C: 75 is the rise in rupees, not a percent.
Option D: 125% is new ÷ old (375 ÷ 300 × 100 = 125%). That says the new sales are 125% of the old; the rise is 25%.
Hint: Divide the rise by the old sales.
After a 10% rise, a yearly newspaper subscription costs ₹836. What did it cost before the rise?
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Correct answer: B
New = old × 1.1, so old = 836 ÷ 1.1 = ₹760. Check: 10% of 760 = 76, and 760 + 76 = 836.
Option A: ₹752.40 takes 10% of ₹836 off. But the 10% was of the old price, which is smaller than ₹836.
Option C: ₹919.60 raises ₹836 by another 10% instead of undoing the rise.
Option D: ₹76 is the rise itself, not the old price.
Hint: Undo "× 1.1" by dividing by 1.1.
Name the rule
- Up r%: multiply by (100 + r) ÷ 100. Up 20% is × 1.2.
- Down r%: multiply by (100 − r) ÷ 100. Down 18% is × 0.82.
- Percent change: (new − old) ÷ old × 100. The old value is always the base.
- Going back: divide the new value by the same multiplier.
The new value is the old value times the multiplier. To undo a multiplication, divide by the same number. Taking r% off the new value uses the wrong base, because the r% belonged to the old value.
Worked, half-done, solo
Question
After an 18% discount, a phone sells for ₹10,250. What was its marked price?
Solution
- Turn the discount into a multiplier
Down 18% leaves 100% − 18% = 82%, so the sale price is 82% of the marked price.
- Find 1% of the marked price
1% = 10,250 ÷ 82 = 125.
- Build 100%
100% = 125 × 100 = 12,500.
- Check forward
18% of 12,500 = 2,250, and 12,500 − 2,250 = 10,250.
Answer: ₹12,500
Watch out: Adding 18% of ₹10,250 back gives ₹12,095, which is wrong: the 18% was of the marked price, not of the sale price.
Now finish one that is half done.
A town used 48,000 litres of water a day. After a campaign, use fell by 12.5%. The multiplier is 100% − 12.5% = 87.5% = 7/8. What is the new daily use?
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Correct answer: C
48,000 × 7/8 = 42,000 litres. Check: 12.5% of 48,000 = 6,000, and 48,000 − 6,000 = 42,000.
Option A: 6,000 litres is the saving, not the new use.
Option B: 54,000 litres raises the use by 12.5%. Use fell, so the multiplier is below 1.
Option D: 4,200 litres is ten times too small: the decimal point slipped one place.
Hint: Multiply 48,000 by 7/8.
And one on your own.
A student's mock-test score fell from 160 to 120. By what percent did it fall?
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Correct answer: A
The fall is 160 − 120 = 40, measured on the old score: 40 ÷ 160 × 100 = 25%. So it is a 25% decrease.
Option B: 33⅓% divides the fall by the new score (40 ÷ 120 × 100 = 33⅓%). Measure on the old score, 160.
Option C: 40 is the fall in marks, not a percent.
Option D: A 25% increase has the right size but the wrong direction: the score went down.
Hint: Divide the fall by the score before the fall.
The traps
- Percent of the new value: to undo a 20% discount, do not add 20% of the sale price. Divide by 0.8.
- Change measured on the new value: the base of a percent change is the old value.
- Difference read as a percent: a rise of ₹75 is not a rise of 75%.
- New ÷ old read as the change: 125% means "new is 125% of old"; the rise is 25%.
- Wrong direction: a discount or a fall gives a multiplier below 1.
Asked "After a 25% rise the hostel fee is ₹9,500. What was it before?", Imran wrote: 25% of 9,500 = 2,375, so 9,500 − 2,375 = ₹7,125. What went wrong?
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Correct answer: B
The fee rose by 25% of the OLD fee, so new = old × 1.25 and old = 9,500 ÷ 1.25 = ₹7,600. Check: 25% of 7,600 = 1,900, and 7,600 + 1,900 = 9,500.
Option A: Adding gives a bigger number, but the old fee must be smaller than ₹9,500.
Option C: 9,500 ÷ 25 = 380 is neither the old fee nor 1% of it.
Option D: Check it: 25% of 7,125 = 1,781.25, and 7,125 + 1,781.25 = 8,906.25, not 9,500.
Hint: Was the 25% a share of the old fee or of the new fee?
Mixed check
A school's yearly fee of ₹18,000 rises by 7.5%. What is the new fee?
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Correct answer: C
Up 7.5% means × 1.075: 18,000 × 1.075 = ₹19,350. Or: 7.5% of 18,000 = 1,350, and 18,000 + 1,350 = 19,350.
Option A: ₹1,350 is only the rise, not the new fee.
Option B: ₹16,650 takes 7.5% off. The fee went up.
Option D: ₹1,935 is ten times too small: the decimal point slipped one place.
Hint: The new fee is 107.5% of the old fee.
From lesson 1:
48 students of a class passed with distinction. That is 16% of the class. How many students are in the class?
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Correct answer: C
16% is 48 students, so 1% = 48 ÷ 16 = 3, and 100% = 3 × 100 = 300 students.
Option A: 3 students is only 1% of the class.
Option B: 30 is ten times too small: the decimal point slipped one place.
Option D: 768 multiplies 48 by 16. Divide by the percent to reach 1%.
Hint: 48 is the 16% piece. Find 1% of the class first.
Quick reference
Change on the old value
- Up r%
- × (100 + r) ÷ 100 up 20%: × 1.2
- Down r%
- × (100 − r) ÷ 100 down 18%: × 0.82
- Percent change
- (new − old) ÷ old × 100 ₹300 to ₹375: 25% rise
- Go back
- new ÷ multiplier 10,250 ÷ 0.82 = 12,500
Why it works: a change is a percent of the old value, so old × multiplier = new. To undo a multiplication, divide by the same multiplier.
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