Which number is the base? More than and less than
"A is 20% more than B" and "B is 16⅔% less than A" describe the same two numbers. The percent changes because the base changes: the number after "than" is always the base, and the gap is divided by it.
Guess first
Guess first: an AC train ticket costs 60% more than a sleeper ticket. By what percent is the sleeper ticket cheaper than the AC ticket?
Show answer
Correct answer: B
Take the sleeper fare as ₹100. The AC fare is then ₹160 and the gap is ₹60. Now the base is the AC fare: 60 ÷ 160 × 100 = 37.5%.
Option A: 60% measures the gap on the sleeper fare. "Cheaper than the AC ticket" measures the same ₹60 on the AC fare.
Option C: 40% is 100% − 60%. Taking a percent away from 100 does not change the base.
Option D: 62.5% is the sleeper fare as a percent of the AC fare (100 ÷ 160 × 100 = 62.5%), not the gap.
Hint: Give the sleeper fare ₹100. What is the gap, measured on the AC fare?
If you chose 37.5%, you saw it: the gap is the same ₹60, but "cheaper than the AC ticket" divides it by the AC fare.
An everyday picture
A percent comparison is a measurement, and the number after "than" is the ruler.
"A is 20% more than B" B is the ruler
measure the gap on B
"B is ?% less than A" A is the ruler
measure the same gap on A
The same gap same rupees
a different percent, because the ruler changed
A ruler of 5 blocks and a ruler of 6 blocks give different readings for the same 1-block gap.
See it: one gap, two bases
Draw B as 5 equal blocks. "A is 20% more than B" adds one block (20% of 5 blocks), so A is 6 blocks.
| Statement | Base, after "than" | Gap ÷ base | Percent |
|---|---|---|---|
| A is 20% more than B | B = 5 blocks | 1 ÷ 5 × 100 = 20% | 20% |
| B is ?% less than A | A = 6 blocks | 1 ÷ 6 × 100 = 16⅔% | 16⅔% |
flowchart TD B["B: 5 blocks"] -->|"add 1 block: 1 ÷ 5 = 20% more"| A["A: 6 blocks"] A -->|"remove 1 block: 1 ÷ 6 = 16⅔% less"| B2["B: 5 blocks again"]
In words: going up from B, the 1-block gap is measured on B's 5 blocks, 20%. Coming down from A, the same gap is measured on A's 6 blocks, 16⅔%.
Try it on paper. Draw B as 4 blocks and make A 25% more. Count A's blocks, then work out by what percent B is less than A.
Find the rule
A is 20% more than B. By what percent is B less than A?
Show answer
Correct answer: B
B = 5 blocks and A = 6 blocks. The gap is 1 block, measured on A: 1 ÷ 6 × 100 = 16⅔%.
Option A: 20% measures the gap on B, but "less than A" makes A the base.
Option C: 80% is 100% − 20%; the base has not changed.
Option D: 83⅓% is B as a percent of A (5 ÷ 6 × 100), not the gap.
Hint: Draw B as 5 blocks. How many blocks is A?
A shop's selling price is 40% less than the printed price. By what percent is the printed price more than the selling price?
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Correct answer: B
Printed price = 5 blocks; 40% less removes 2 blocks, so the selling price = 3 blocks. The gap of 2 blocks measured on the selling price: 2 ÷ 3 × 100 = 66⅔%.
Option A: 40% measures the gap on the printed price, but "more than the selling price" makes the selling price the base.
Option C: 60% is 100% − 40%, the selling price as a share of the printed price.
Option D: 166⅔% is the printed price as a percent of the selling price (5 ÷ 3 × 100), not the gap.
Hint: Make the printed price 5 blocks. How many blocks is the selling price?
A share price falls by 50%. By what percent must it now rise to get back to where it was?
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Correct answer: B
After a 50% fall the price is half. To get back it must double, which is a rise of 100% of the new price.
Option A: A 50% rise on the halved price only gets back to 75% of the start, because the rise is measured on the smaller price.
Option C: 33⅓% divides the 50 by 150 instead of by the new price, 50.
Option D: 200% is the old price as a percent of the new one (100 ÷ 50 × 100), not the rise.
Hint: After the fall the price is half. What must happen to get back?
Name the rule
- The number after "than" is the base. The gap is divided by it.
- If A is r% more than B, then B is r ÷ (100 + r) × 100 percent less than A.
- If A is r% less than B, then B is r ÷ (100 − r) × 100 percent more than A.
- Undoing a change is a change on the new value: after a fall of r%, the rise back is r ÷ (100 − r) × 100 percent.
- Fastest way: turn the percent into blocks (20% = 1 block in 5, 37.5% = 3 blocks in 8), then gap ÷ base.
A percent is gap ÷ base × 100. "More than B" divides by B; "less than A" divides by A. The gap is the same, the base is not, so the percent is not.
Worked, half-done, solo
Question
A scooter costs 12.5% more than a motorbike. By what percent is the motorbike cheaper than the scooter?
Solution
- Turn the percent into blocks
12.5% = 1/8, so the motorbike is 8 blocks and the scooter is 8 + 1 = 9 blocks.
- Find the gap and the new base
The gap is 1 block; "cheaper than the scooter" makes the scooter's 9 blocks the base.
- Divide
1 ÷ 9 × 100 = 11 1/9%.
- Check with prices
Motorbike ₹80,000, scooter ₹90,000; the gap ₹10,000 ÷ ₹90,000 × 100 = 11 1/9%.
Answer: 11 1/9%
Watch out: The answer is not 12.5%: that measures the gap on the motorbike's price.
Now finish one that is half done.
Mohan's marks are 37.5% less than Rahul's. 37.5% = 3/8, so if Rahul has 8 blocks, Mohan has 8 − 3 = 5 blocks. By what percent are Rahul's marks more than Mohan's?
Show answer
Correct answer: B
The gap is 3 blocks and the base is Mohan's 5 blocks: 3 ÷ 5 × 100 = 60%.
Option A: 37.5% measures the gap on Rahul's marks; "more than Mohan's" makes Mohan's marks the base.
Option C: 62.5% is Mohan's share of Rahul's marks (5 ÷ 8 × 100 = 62.5%), not the gap.
Option D: 160% is Rahul's marks as a percent of Mohan's (8 ÷ 5 × 100 = 160%), not how much more.
Hint: Divide the 3-block gap by Mohan's 5 blocks.
And one on your own.
Asha's income is 33⅓% more than Babita's. By what percent is Babita's income less than Asha's?
Show answer
Correct answer: B
33⅓% = 1/3, so Babita = 3 blocks and Asha = 4 blocks. The gap of 1 block measured on Asha: 1 ÷ 4 × 100 = 25%.
Option A: 33⅓% measures the gap on Babita's income, but "less than Asha's" makes Asha's the base.
Option C: 66⅔% is 100% − 33⅓%; the base has not changed.
Option D: 75% is Babita's income as a percent of Asha's (3 ÷ 4 × 100 = 75%), not the gap.
Hint: 33⅓% is one block in three.
The traps
- Same percent both ways: 20% more one way is 16⅔% less the other way.
- Wrong base: the number after "than" is the base.
- 100 minus the percent: that changes the number, not the base.
- "Is x% of" read as "is x% more than": 160% of a number is 60% more than it.
- Undoing with the same percent: a 50% fall needs a 100% rise to get back.
A shop says: "Our new phone costs 300% more than the old model, so the old model is 300% cheaper." What is wrong?
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Correct answer: A
300% more means the new phone is 4 times the old one: old = 1 block, new = 4 blocks. The gap of 3 blocks measured on the new price: 3 ÷ 4 × 100 = 75%. A price cannot fall by more than 100%.
Option B: 400% is the new price as a percent of the old one (4 ÷ 1 × 100), not a discount; a fall of 400% is impossible.
Option C: 25% is the old price as a share of the new one (1 ÷ 4 × 100 = 25%), not how much cheaper it is.
Option D: A fall of 300% would make the price negative.
Hint: Draw the old model as 1 block. How many blocks is the new one?
Mixed check
A second-hand bike costs 75% less than a new one. By what percent is the new bike costlier than the second-hand one?
Show answer
Correct answer: B
New bike = 4 blocks; 75% less removes 3 blocks, so the second-hand bike = 1 block. The gap of 3 blocks measured on the second-hand bike: 3 ÷ 1 × 100 = 300%.
Option A: 75% measures the gap on the new bike, but "costlier than the second-hand one" makes the second-hand price the base.
Option C: 25% is 100% − 75%, the second-hand price as a share of the new one.
Option D: 400% is the new price as a percent of the second-hand one (4 ÷ 1 × 100), not how much more it costs.
Hint: Make the new bike 4 blocks. How many blocks is the second-hand bike?
From lesson 4:
What is 17.5% of 3,200? Use the blocks.
Show answer
Correct answer: A
17.5% = 10% + 5% + 2.5%. 10% of 3,200 = 320, 5% of 3,200 = 160 and 2.5% of 3,200 = 80, so 320 + 160 + 80 = 560.
Option B: 56 is ten times too small: the decimal point slipped one place.
Option C: 2,640 is what is left (3,200 − 560 = 2,640), not 17.5%.
Option D: 480 adds only the 10% and 5% blocks (320 + 160 = 480); the 2.5% block is missing.
Hint: 2.5% is half of the 5% block.
Quick reference
Which number is the base
- The base
- the number after "than" "more than B": divide the gap by B
- r% more, so the other is less by
- r ÷ (100 + r) × 100 20% more: 16⅔% less
- r% less, so the other is more by
- r ÷ (100 − r) × 100 40% less: 66⅔% more
- Undo a fall of r%
- rise by r ÷ (100 − r) × 100 fall 50%: rise 100%
- Blocks
- 20% = 1 in 5, 37.5% = 3 in 8 gap ÷ base, in blocks
Why it works: a percent is gap ÷ base × 100. The gap stays the same; the base is whichever number comes after "than", so the percent changes with it.
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