Changes in a row multiply; points are not percent
Two percent changes in a row multiply: up 10% and then down 10% is × 1.1 × 0.9 = × 0.99, a 1% fall, not zero. And when a rate itself moves, say from 40% to 50%, it rose by 10 percentage points, which is a 25% rise.
Guess first
Guess first: a shop raises the price of a ₹2,000 jacket by 10%. A week later it announces "10% off" the new price. What do you pay now?
Show answer
Correct answer: B
After the rise the price is 2,000 × 1.1 = 2,200. The 10% off is taken from ₹2,200: 2,200 × 0.9 = ₹1,980. The rise and the cut do not cancel, because the cut is 10% of a bigger price.
Option A: ₹2,000 assumes the two 10% changes cancel. The cut is 10% of ₹2,200, which is ₹220, more than the ₹200 added.
Option C: ₹1,800 takes 10% off the old price and forgets the rise.
Option D: ₹2,200 is the price after the rise only; the 10% off is missing.
Hint: Do the two changes one after the other, each on the price it starts from.
If you chose ₹1,980, you have found the rule of this lesson by yourself: every change is a percent of the value just before it.
An everyday picture
A photocopier's zoom works like successive percent changes: each zoom is applied to the copy, not to the original.
Zoom 110%, then 90% × 1.1 × 0.9 = × 0.99
the copy ends 1% smaller than the original
Zoom 50%, then 200% × 0.5 × 2 = × 1
back to the original size
Zoom 120%, then 120% again × 1.2 × 1.2 = × 1.44
44% bigger, not 40%
The zoom percentages multiply. Percent changes in a row do the same.
See it: a chain of two changes
₹1,000 rises by 20%, then falls by 10%. Each change uses the value just before it.
| Step | Value before | Change | Value after |
|---|---|---|---|
| start | ₹1,000 | ||
| up 20% | ₹1,000 | × 1.2 | 1,000 × 1.2 = 1,200 |
| down 10% | ₹1,200 | × 0.9 | 1,200 × 0.9 = 1,080 |
flowchart TD A["Start ₹1,000"] -->|"up 20%: × 1.2"| B["₹1,200"] B -->|"down 10%: × 0.9"| C["₹1,080"] A -.->|"in one step: × 1.08"| C
So the net change is 1,080 − 1,000 = 80, which is 8% of ₹1,000, not 20% − 10% = 10%. In one step: 1.2 × 0.9 = 1.08, so the two changes together are one rise of 8%.
The same answer comes from a shortcut for two changes a% and b%: net % = a + b + a × b ÷ 100. Here: 20 + (−10) + 20 × (−10) ÷ 100 = 20 − 10 − 2 = 8.
Rates are different. When the value is itself a percent, like a pass rate, there are two ways to describe a move from 40% to 50%:
| Description | Working | Result |
|---|---|---|
| percentage points | 50 − 40 = 10 | up 10 points |
| percent change | 10 ÷ 40 × 100 = 25% | up 25% |
Try it on paper. Take ₹100. Raise it by 20%, then lower the result by 20%. Then do it the other way round: lower first, then raise. Compare the two final amounts.
Find the rule
A price rises 20%, and then rises another 20%. What is the total change?
Show answer
Correct answer: B
× 1.2 × 1.2 = × 1.44, so the price is 144% of the start: a 44% increase. Shortcut: 20 + 20 + 20 × 20 ÷ 100 = 44.
Option A: 40% adds the two percents. The second 20% is of a bigger price, so it adds more than the first.
Option C: 4% is only the extra "percent of a percent" part (20 × 20 ÷ 100 = 4); the two 20% rises are missing.
Option D: 20% counts only one of the two rises.
Hint: Multiply the two multipliers.
A salary is raised by 30% and later cut by 30%. Overall, what happened to it?
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Correct answer: B
× 1.3 × 0.7 = × 0.91, so the salary ends at 91% of the start: a 9% decrease. Shortcut: 30 − 30 + 30 × (−30) ÷ 100 = −9.
Option A: The cut is 30% of the higher salary, so it removes more than the raise added. Equal ups and downs never cancel.
Option C: A 9% increase has the right size but the wrong direction: 91% of the start is a fall.
Option D: 30% decrease counts only the cut.
Hint: Multiply 1.3 by 0.7.
A district's literacy rate rose from 60% to 75%. Which statement is correct?
Show answer
Correct answer: B
Points: 75 − 60 = 15. Percent: the rise of 15 points measured on the old rate, 15 ÷ 60 × 100 = 25%.
Option A: 15 is the gap in percentage points. As a percent of the old rate it is 25%.
Option C: 25 is the percent change, not the number of points.
Option D: 75% is the new rate, not the change.
Hint: Points subtract the rates. Percent divides the gap by the old rate.
Name the rule
- Changes in a row: multiply the multipliers. Up 20% then down 10% is × 1.2 × 0.9 = × 1.08.
- Two changes a% and b%: net % = a + b + a × b ÷ 100 (a fall is a negative number).
- Equal up and down never cancel: up r% then down r% always ends below the start.
- A rate that moves: percentage points = new − old; percent change = (new − old) ÷ old × 100.
Because each percent is of a different value. The first change is a percent of the start; the second is a percent of the value after the first change. The extra a × b ÷ 100 is the second change acting on the first change.
Worked, half-done, solo
Question
A ₹60,000 scooter loses 15% of its value every year. What is it worth after 2 years?
Solution
- Turn the yearly loss into a multiplier
Down 15% leaves 85%, so each year is × 0.85.
- Year 1
60,000 × 0.85 = 51,000.
- Year 2: the loss is 15% of ₹51,000, not of ₹60,000
51,000 × 0.85 = 43,350.
- Check with the shortcut
−15 − 15 + 15 × 15 ÷ 100 = −27.75, and 72.25% of 60,000 = 43,350.
Answer: ₹43,350
Watch out: Taking 30% off ₹60,000 gives ₹42,000. That adds the two yearly losses as if both were of the first price.
Now finish one that is half done.
A price of ₹8,000 rises by 25% and then falls by 20%. After the rise it is 8,000 × 1.25 = ₹10,000. What is it after the fall?
Show answer
Correct answer: A
The fall is 20% of ₹10,000: 10,000 × 0.8 = ₹8,000. The price is back where it started, because 1.25 × 0.8 = 1.
Option B: ₹8,400 adds the percents (25 − 20 = 5) and raises ₹8,000 by 5%. The 20% fall is of ₹10,000, not of ₹8,000.
Option C: ₹6,400 takes 20% off the old ₹8,000 and forgets the rise.
Option D: ₹10,000 is the price after the rise only; the fall is missing.
Hint: Take 20% off ₹10,000.
And one on your own.
The share of electric scooters among new two-wheelers sold goes up from 4% to 5%. By what percent did the share go up?
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Correct answer: B
The share rose by 5 − 4 = 1 percentage point. As a percent of the old share: 1 ÷ 4 × 100 = 25%.
Option A: 1% is the rise in percentage points, not in percent.
Option C: 5% is the new share, not the rise.
Option D: 20% divides the rise by the new share (1 ÷ 5 × 100). Measure on the old share, 4%.
Hint: Divide the gap in points by the old share.
The traps
- Added the percents: two 20% rises are a 44% rise, not 40%.
- Assumed they cancel: up r% and down r% always end below the start.
- Forgot a step: use every change, in order, each on the value before it.
- Points called percent: 4% to 5% is 1 point, but a 25% rise.
- New rate as the change: a rate that ends at 9% did not rise by 9%.
A news line says: "Petrol rose 10% in March and another 10% in April, so it is now 20% dearer than in February." What is wrong?
Show answer
Correct answer: A
× 1.1 × 1.1 = × 1.21, so petrol is 21% dearer. Shortcut: 10 + 10 + 10 × 10 ÷ 100 = 21.
Option B: 19% subtracts the extra part (10 + 10 − 1). Two rises make it bigger, not smaller.
Option C: Adding the percents ignores that April's rise was on a higher price.
Option D: 1% is only the extra part (10 × 10 ÷ 100 = 1), not the whole change.
Hint: Multiply 1.1 by 1.1.
Mixed check
A share price rises 10%, then rises 20%, then falls 25%. What is the overall change?
Show answer
Correct answer: B
× 1.1 × 1.2 × 0.75 = × 0.99, so the price ends at 99% of the start: a 1% decrease.
Option A: 5% adds the three percents (10 + 20 − 25 = 5). Each change is on a different value.
Option C: A 1% increase has the right size but the wrong direction: 99% of the start is a fall.
Option D: 25% decrease uses only the last change.
Hint: Multiply all three multipliers: 1.1, 1.2 and 0.75.
From lesson 2:
After a 15% fall, a share costs ₹6,630. What was it before the fall?
Show answer
Correct answer: B
After a 15% fall the price is 85% of the old one: old = 6,630 ÷ 0.85 = ₹7,800. Check: 15% of 7,800 = 1,170, and 7,800 − 1,170 = 6,630.
Option A: ₹7,624.50 adds 15% of ₹6,630. The 15% was of the old price, which is bigger than ₹6,630.
Option C: ₹5,635.50 takes another 15% off instead of undoing the fall.
Option D: ₹1,170 is the fall itself, not the old price.
Hint: Undo "× 0.85" by dividing by 0.85.
Quick reference
Changes in a row and percentage points
- Changes in a row
- multiply the multipliers up 20%, down 10%: × 1.2 × 0.9 = × 1.08
- Two changes a%, b%
- a + b + a × b ÷ 100 20, −10: 20 − 10 − 2 = 8
- Up r% then down r%
- always below the start up 30%, down 30%: × 0.91
- Points
- new rate − old rate 4% to 5%: 1 point
- Percent change of a rate
- (new − old) ÷ old × 100 4% to 5%: 25%
Why it works: each change is a percent of the value just before it, so the multipliers multiply. Points only subtract; a percent change divides by the old rate.
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