Before and after bars: change as a fraction
Draw the old value and the new value as two bars made of equal blocks. A 25% rise turns 4 blocks into 5; a 20% fall turns 5 blocks into 4. With both bars in front of you, going forward, going back and comparing all take one look.
Guess first
Guess first: after a 40% rise, a monthly internet plan costs ₹1,400. What did it cost before the rise?
Show answer
Correct answer: B
40% = 2/5, so the old price is 5 blocks and the new price is 7 blocks. 7 blocks = 1,400, so 1 block = 1,400 ÷ 7 = 200, and the old price = 5 × 200 = ₹1,000.
Option A: ₹840 takes 40% of ₹1,400 off. But the 40% was 2 blocks of the OLD price, which is 2 blocks in 7 of the new price.
Option C: ₹1,960 raises ₹1,400 by another 40% instead of going back.
Option D: ₹400 is the 2 blocks of the rise, not the old price.
Hint: If the old price is 5 blocks, how many blocks is the new price?
If you found ₹1,000 with blocks, you have the method of this lesson: one bar for before, one for after, and the value of one block.
An everyday picture
Before-and-after bars work like coaches on a train: all coaches are the same size.
An old train of 4 coaches 100%
each coach is 25% of the old train
Add 1 coach a 25% rise
the new train has 5 coaches
Take 1 coach away again a 20% fall
1 coach out of the 5 on the new train
The coach that was added is the same coach that is taken away. It is 25% of the short train but only 20% of the long one.
See it: before and after
A 25% rise. Before (4 blocks):
After (5 blocks):
The rise is 1 block on 4 blocks: 1 ÷ 4 × 100 = 25%. Going back, the same 1 block is taken from 5 blocks: 1 ÷ 5 × 100 = 20%.
| Change | Blocks before → after | Going back |
|---|---|---|
| up 25% (1/4) | 4 → 5 | down 1 in 5 = 20% |
| up 20% (1/5) | 5 → 6 | down 1 in 6 = 16⅔% |
| up 50% (1/2) | 2 → 3 | down 1 in 3 = 33⅓% |
| down 20% (1/5) | 5 → 4 | up 1 in 4 = 25% |
| down 25% (1/4) | 4 → 3 | up 1 in 3 = 33⅓% |
flowchart TD A["Old: 4 blocks"] -->|"add 1 block: 1 ÷ 4 = 25% up"| B["New: 5 blocks"] B -->|"take 1 block: 1 ÷ 5 = 20% down"| C["Old: 4 blocks again"]
Try it. Draw a 20% fall as bars (5 blocks, then 4). Write the price of one block for a ₹750 starting price, then find the new price.
Find the rule
An ₹840 electricity bill rises by 25%. With blocks: the old bill is 4 blocks of ₹210. What is the new bill?
Show answer
Correct answer: A
The new bill is 5 blocks: 5 × 210 = ₹1,050. Check: 840 × 1.25 = 1,050.
Option B: ₹210 is one block, the rise itself, not the new bill.
Option C: ₹630 removes a block (3 × 210 = 630). The bill went up.
Option D: ₹105 is ten times too small: the decimal point slipped one place.
Hint: A 25% rise adds one block to the 4.
After a 35% discount, a sofa costs ₹2,600. What was its marked price?
Show answer
Correct answer: A
35% = 7/20, so the marked price is 20 blocks and the sale price is 20 − 7 = 13 blocks. 13 blocks = 2,600, so 1 block = 2,600 ÷ 13 = 200, and 20 blocks = 20 × 200 = ₹4,000.
Option B: ₹3,510 adds 35% of ₹2,600 back. The 35% was of the marked price, which is bigger.
Option C: ₹1,690 takes another 35% off instead of undoing the discount.
Option D: ₹1,400 is the discount (7 blocks), not the marked price.
Hint: 35% is 7 blocks out of 20. How many blocks are left after the discount?
A price rises 50%. By what percent must the new price fall to get back to the old price?
Show answer
Correct answer: A
50% = 1/2: the old price is 2 blocks and the new price is 3 blocks. Going back takes 1 block away from 3: 1 ÷ 3 × 100 = 33⅓%.
Option B: A 50% fall takes half of the 3 blocks away, which leaves 1.5 blocks, less than the old 2 blocks.
Option C: 66⅔% is the old price as a share of the new one (2 ÷ 3 × 100), not the fall.
Option D: A 100% fall would take the price to zero. It uses the rule for undoing a fall, 50 ÷ (100 − 50); after a rise, divide by the new price's 3 blocks.
Hint: Old 2 blocks, new 3 blocks. How many blocks go, out of how many?
Name the rule
- Blocks for a change: write r% as a fraction a/b. The old value is b blocks; the new value is b + a blocks (a rise) or b − a blocks (a fall).
- One block = any known value ÷ its number of blocks.
- The percent change = blocks changed ÷ blocks BEFORE × 100.
- Going back = the same blocks ÷ blocks AFTER × 100.
Because a known final value is a number of blocks too. Divide it by its blocks to get one block, then count the blocks you need. No formula has to be remembered.
Worked, half-done, solo
Question
After a 37.5% discount, a jacket costs ₹1,750. What was its marked price?
Solution
- Turn the discount into blocks
37.5% = 3/8, so the marked price is 8 blocks and the sale price is 8 − 3 = 5 blocks.
- Find one block
5 blocks = 1,750, so 1 block = 1,750 ÷ 5 = 350.
- Count the blocks you need
The marked price is 8 blocks: 8 × 350 = 2,800.
- Check forward
37.5% of 2,800 = 1,050, and 2,800 − 1,050 = 1,750.
Answer: ₹2,800
Watch out: Adding 37.5% of ₹1,750 gives ₹2,406.25, which is wrong: the discount was 3 blocks of the marked price, not of the sale price.
Now finish one that is half done.
A ₹600 price rises 20% and then falls 25%. With blocks of ₹120, the old price is 5 blocks and after the rise it is 6 blocks, ₹720. The fall is a quarter of these 6 blocks. What is the final price?
Show answer
Correct answer: A
A quarter of 6 blocks is 1.5 blocks, so 6 − 1.5 = 4.5 blocks remain: 4.5 × 120 = ₹540. Check: 600 × 1.2 × 0.75 = 540.
Option B: ₹570 adds the percents (20 − 25 = −5) and takes 5% off ₹600.
Option C: ₹450 takes 25% off the old ₹600 and forgets the rise.
Option D: ₹720 is the price after the rise only; the fall is missing.
Hint: What is a quarter of 6 blocks?
And one on your own.
After a 5% rise, a school fee is ₹1,449. What was it before the rise?
Show answer
Correct answer: A
5% = 1/20, so the old fee is 20 blocks and the new fee is 21 blocks. 1 block = 1,449 ÷ 21 = 69, and 20 blocks = 20 × 69 = ₹1,380.
Option B: ₹1,376.55 takes 5% of ₹1,449 off. The 5% was of the old fee, which is smaller.
Option C: ₹1,521.45 raises ₹1,449 by another 5% instead of going back.
Option D: ₹69 is one block, the rise itself, not the old fee.
Hint: 5% is 1 block in 20. How many blocks is the new fee?
The traps
- Measured on the wrong bar: a change is measured on the bar before; going back is measured on the bar after.
- Took the percent of the new value: to undo a rise, count blocks; do not take r% off the new value.
- Blocks of different sizes: every block in both bars must be worth the same.
- Going back with the same percent: up 1 block in 4 is down 1 block in 5.
- The change instead of the value: one block is the change, not the price.
Neha drew a 25% discount as 4 blocks before and 3 blocks after. She then said the price must rise by 25% to get back to where it was. What is right?
Show answer
Correct answer: A
To get back, 1 block must be added to 3 blocks: 1 ÷ 3 × 100 = 33⅓%. A 25% rise on 3 blocks adds only 0.75 of a block.
Option B: 25% measures the block on the old 4 blocks; going back starts from the 3 blocks.
Option C: 75% is what is left after the discount, not the rise needed.
Option D: 133⅓% is the old price as a percent of the new one (4 ÷ 3 × 100), not the rise.
Hint: Going back adds 1 block to how many blocks?
Mixed check
After a 60% rise in value, a painting is worth ₹12,800. What was it worth before?
Show answer
Correct answer: A
60% = 3/5: the old value is 5 blocks and the new one 8 blocks. 1 block = 12,800 ÷ 8 = 1,600, so the old value = 5 × 1,600 = ₹8,000.
Option B: ₹5,120 takes 60% of ₹12,800 off; the 60% was of the old value.
Option C: ₹20,480 raises ₹12,800 by another 60% instead of going back.
Option D: ₹4,800 is the rise (3 blocks), not the old value.
Hint: The new value is 8 blocks.
From lesson 7:
63 out of 84 is what percent? Simplify the fraction first.
Show answer
Correct answer: A
63/84 = 3/4 (divide both by 21), and 3/4 = 75%.
Option B: 25% is the part that is not 63, the other quarter.
Option C: 133⅓% divides 84 by 63; the whole is 84, so divide by 84.
Option D: 7.5% is ten times too small: the decimal point slipped one place.
Hint: Divide 63 and 84 by 21.
Quick reference
Before-and-after bars
- r% as blocks
- r% = a/b: before b blocks, after b ± a up 25%: 4 → 5 blocks
- One block
- known value ÷ its blocks 7 blocks = 1,400, so 1 block = 200
- Percent change
- blocks changed ÷ blocks before × 100 1 ÷ 4 × 100 = 25%
- Going back
- blocks changed ÷ blocks after × 100 1 ÷ 5 × 100 = 20%
- Reverse a discount
- sale value ÷ its blocks × all blocks 37.5% off: 1,750 ÷ 5 × 8 = 2,800
Why it works: every value is a count of equal blocks, so one block's worth turns any bar into rupees, forwards or backwards.
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