Longer chains, and percentage points in the news
A chain of three or four percent changes is still one multiplication: multiply all the multipliers, in any order. And when the news reports a rate going up or down, check whether it means percentage points or percent.
Guess first
Guess first: an app's monthly users rose 20% in January, then fell 10% in February and another 10% in March. Overall, from the start of January to the end of March, what happened?
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Correct answer: B
1.2 × 0.9 × 0.9 = 0.972, so the users end at 97.2% of the start: a 2.8% decrease. The +20, −10 and −10 add to zero, but the changes multiply, they do not add.
Option A: No change adds the percents (20 − 10 − 10 = 0). Each fall is 10% of a different number of users.
Option C: A 2.8% increase has the right size but the wrong direction: 97.2% of the start is a fall.
Option D: A 10% decrease uses only March's change.
Hint: Multiply 1.2, 0.9 and 0.9.
If you chose a 2.8% decrease, you used the chain rule from lesson 3 on three steps. This lesson makes chains longer, runs them backwards, and reads rates in the news.
An everyday picture
A chain of percent changes is a relay race: each runner starts where the last one stopped.
Runner 1 the first change
runs from the starting value
Runner 2 the second change
starts from the value after change 1
The finish the final value
start × every multiplier, in order
Swap the runners and the finish line does not move: multiplication gives the same product in any order.
See it: one chain, two orders
₹10,000 goes through +20%, −10% and −10%. The table runs the chain in two orders.
| Step | Order 1 | Order 2 |
|---|---|---|
| start | ₹10,000 | ₹10,000 |
| step 1 | +20%: 10,000 × 1.2 = 12,000 | −10%: 10,000 × 0.9 = 9,000 |
| step 2 | −10%: 12,000 × 0.9 = 10,800 | −10%: 9,000 × 0.9 = 8,100 |
| step 3 | −10%: 10,800 × 0.9 = 9,720 | +20%: 8,100 × 1.2 = 9,720 |
flowchart TD S["Start ₹10,000"] -->|"× 1.2"| A["₹12,000"] A -->|"× 0.9"| B["₹10,800"] B -->|"× 0.9"| C["₹9,720"] S -.->|"one step: × 0.972"| C
Both orders end at ₹9,720, because 1.2 × 0.9 × 0.9 = 0.972 whatever the order. The whole chain is one change: 9,720 − 10,000 = −280, a 2.8% fall.
Rates in the news. When the number that changes is itself a percent, give both readings:
| News line | Points | Percent change |
|---|---|---|
| Unemployment 8% to 6% | 6 − 8 = −2 points | −2 ÷ 8 × 100 = −25% |
| Vote share 32% to 40% | 40 − 32 = 8 points | 8 ÷ 32 × 100 = 25% |
| GST slab 18% to 12% | 12 − 18 = −6 points | −6 ÷ 18 × 100 = −33⅓% |
Try it. Take ₹100 through +50% and then −50%. Then through −50% and then +50%. Both orders give the same answer, and it is not ₹100.
Find the rule
A price rises 10% every year for 3 years. What is the total rise?
Show answer
Correct answer: B
1.1 × 1.1 × 1.1 = 1.331, so the price is 133.1% of the start: a 33.1% increase.
Option A: A 30% increase adds the three percents. Each year's 10% is of a bigger price.
Option C: A 10% increase counts only one year.
Option D: A 33.1% decrease has the right size but the wrong direction: every year was a rise.
Hint: Multiply 1.1 by itself three times.
A price falls 40% and then rises 66⅔%. What is the overall change?
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Correct answer: A
0.6 × 5/3 = 1, so the price is back at 100% of the start: no change. In blocks: 5 blocks fall to 3, and a rise of 2 blocks on 3 (66⅔%) brings back 5.
Option B: A 26⅔% increase adds the percents (66⅔ − 40 = 26⅔). The 66⅔% rise is of the lower price.
Option C: A 66⅔% increase uses only the rise.
Option D: A 13⅓% increase averages the two changes ((−40 + 66⅔) ÷ 2 = 13⅓).
Hint: What is 0.6 × 5/3?
The GST rate on an item was cut from 18% to 12%. Which statement is correct?
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Correct answer: C
Points: 12 − 18 = −6, a fall of 6 points. Percent: 6 ÷ 18 × 100 = 33⅓%, so the rate was cut by a third.
Option A: 6 is the fall in percentage points; as a percent of the old rate it is 33⅓%.
Option B: 33⅓ is the percent change, not the number of points.
Option D: 12% is the new rate, not the fall.
Hint: Points subtract the rates; percent divides that gap by the old rate.
Name the rule
- Any chain: multiply every multiplier. The order does not matter.
- One step for the whole chain: (product − 1) × 100 is the single equivalent change.
- Equal up and down: up r% and down r% always lose r × r ÷ 100 percent. Up 10%, down 10% loses 1%.
- Running a chain backwards: divide the final value by the product of the multipliers.
- A rate in the news: points = new − old; percent change = (new − old) ÷ old × 100. Say which one you mean.
Because a chain is one long multiplication, and 1.2 × 0.9 × 0.9 = 0.9 × 0.9 × 1.2. Each step changes the base, but the product of all the multipliers is the same.
Worked, half-done, solo
Question
A fan's price was raised 20%, and later the shop gave 25% off. The fan now sells for ₹1,800. What was the price before both changes?
Solution
- Turn the chain into one multiplier
1.2 × 0.75 = 0.9, so the two changes together are × 0.9.
- Run the chain backwards
Original = 1,800 ÷ 0.9 = 2,000.
- Check forward, step by step
2,000 × 1.2 = 2,400, then 2,400 × 0.75 = 1,800.
Answer: ₹2,000
Watch out: Adding the percents (20 − 25 = −5) and dividing by 0.95 gives about ₹1,895, which is wrong: the 25% off was of the raised price.
Now finish one that is half done.
A ₹50,000 investment grows 10%, then falls 20%, then grows 5%. After the first two steps it is 50,000 × 1.1 × 0.8 = ₹44,000. What is it after the third step?
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Correct answer: A
44,000 × 1.05 = ₹46,200. In one step: 1.1 × 0.8 × 1.05 = 0.924, and 50,000 × 0.924 = 46,200.
Option B: ₹47,500 adds the percents (10 − 20 + 5 = −5) and takes 5% off ₹50,000.
Option C: ₹52,500 uses only the last change (50,000 × 1.05 = 52,500).
Option D: ₹55,000 uses only the first change (50,000 × 1.1 = 55,000).
Hint: Grow ₹44,000 by 5%.
And one on your own.
After two price cuts, first 10% and then 16%, a TV costs ₹7,560. What was its price before the cuts?
Show answer
Correct answer: B
0.9 × 0.84 = 0.756, so the price before = 7,560 ÷ 0.756 = ₹10,000. Check: 10,000 × 0.9 = 9,000, and 9,000 × 0.84 = 7,560.
Option A: ₹9,525.60 adds 26% of ₹7,560 back (7,560 × 1.26). The cuts were of the earlier, higher prices.
Option C: ₹5,715.36 applies both cuts again to ₹7,560 instead of undoing them.
Option D: ₹2,440 is the total amount cut (10,000 − 7,560 = 2,440), not the old price.
Hint: Divide by the product of the two multipliers.
The traps
- Added the percents: +20, −10, −10 add to zero, but the chain is a 2.8% fall.
- Forgot a step: every change in the chain must be used.
- Thought the order matters: it changes the steps, never the end.
- Points called percent: 8% to 6% is 2 points, but a 25% fall.
- New rate called the change: a rate cut to 12% did not fall by 12%.
A news line says: "Unemployment fell from 8% to 6%, a 2% drop." Which is the precise statement?
Show answer
Correct answer: A
Points: 6 − 8 = −2, a fall of 2 points. Percent: 2 ÷ 8 × 100 = 25%. The rate fell by a quarter.
Option B: 2 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: 6% is the new rate, not the fall.
Hint: Divide the gap in points by the old rate.
Mixed check
A car loses 10% of its value every year. Over 3 years, what is the total fall in its value?
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Correct answer: B
0.9 × 0.9 × 0.9 = 0.729, so the car is worth 72.9% of its first value: a 27.1% decrease.
Option A: A 30% decrease adds the three yearly falls; each fall is of a smaller value.
Option C: A 10% decrease counts only one year.
Option D: A 27.1% increase has the right size but the wrong direction: the value fell.
Hint: Multiply 0.9 by itself three times.
From lesson 5:
A's salary is 16⅔% less than B's. By what percent is B's salary more than A's?
Show answer
Correct answer: B
16⅔% = 1/6, so B = 6 blocks; 16⅔% less removes 1 block, so A = 5 blocks. The gap of 1 block measured on A: 1 ÷ 5 × 100 = 20%.
Option A: 16⅔% measures the gap on B, but "more than A's" makes A the base.
Option C: 83⅓% is 100% − 16⅔%, A's salary as a share of B's.
Option D: 120% is B's salary as a percent of A's (6 ÷ 5 × 100), not how much more it is.
Hint: Make B 6 blocks. How many blocks is A?
Quick reference
Long chains and rates in the news
- Any chain
- multiply every multiplier, any order +20%, −10%, −10%: 1.2 × 0.9 × 0.9 = 0.972
- Whole chain as one change
- (product − 1) × 100 0.972: a 2.8% fall
- Up r%, then down r%
- lose r × r ÷ 100 percent up 10%, down 10%: lose 1%
- Chain backwards
- final ÷ product of multipliers 1,800 ÷ 0.9 = 2,000
- Rate in the news
- points = new − old; percent = gap ÷ old × 100 8% to 6%: 2 points, 25%
Why it works: each change multiplies the value before it, so a chain is one product. Points subtract two rates; percent divides that gap by the old rate.
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