See It 2: Before-and-after bars for increase and decrease
Draw two bars, one above the other: before and after. The before bar is always 100%, because every percent change is a percent of the value before.
See the basics first
The 100-square idea again, plus what a change looks like.
1. What is a percent? "Per cent" means per hundred. 35% means 35 out of every 100. Think of ₹100 as 100 coins of ₹1. If you spend 35 of the coins, you spent 35 out of 100, which is 35% of your money. On a 100-square grid each square is 1%, so 35% is 35 shaded squares:
███████████████████████████████████·································································
Text version: 35 of the 100 squares are shaded (35%); 65 are not (65%).
2. What does "of" mean? "Of" tells you which thing the percent is taken from. "35% of ₹200" means: cut ₹200 into 100 equal pieces (each piece is ₹200 ÷ 100 = ₹2) and take 35 pieces: 35 × ₹2 = ₹70.
3. Whole and part. The number after "of" is the whole: all of it, 100%. A part is a piece of the whole. The rest is what is left: whole − part, which is (100 − percent)%. In the example: whole ₹200, part ₹70 (35%), rest ₹130 (65%).
4. The one tool you will use in every question: find 1% first. 1% of anything = that thing ÷ 100. From 1% you can build any percent (multiply by the percent) or get back to the whole (multiply by 100). Try it now: 1% of ₹300 = 300 ÷ 100 = ₹3, so 35% of ₹300 = 35 × 3 = ₹105.
5. What is a percent change? When a value moves from an old value to a new value, the OLD value is the whole (100%), because the change is measured against where you started. A 20% rise on ₹500: 20% of ₹500 = 20 × ₹5 = ₹100, so the new value is ₹600 = 100% + 20% = 120% of the old value. A 20% fall: new value = 100% − 20% = 80% of the old value = ₹400.
Words to know
- Percent (%)
- out of every 100 35% = 35 out of 100
- Whole
- the number after 'of' = 100% ₹200 in '35% of ₹200'
- Part
- a piece of the whole ₹70 = 35% of ₹200
- Rest
- whole − part ₹130 = 65% of ₹200
- 1%
- whole ÷ 100 1% of ₹200 = ₹2
Picture: ₹500 rises 20%
| Bar | Picture (each █ = 10% of before) | Value |
|---|---|---|
| Before | ██████████ | ₹500 |
| After | ██████████▓▓ | ₹600 |
Text version: the before bar is 10 blocks of ₹50. The after bar adds 2 blocks (▓), so it is 12 blocks = ₹600. 2 extra blocks out of 10 = 20%.
The 'before' bar is 10 blocks = ₹500. The 'after' bar has 12 blocks. What is the new price, and the percent rise?
Show answer
Correct answer: D
Compare the extra blocks with the 10 'before' blocks: 2/10 = 20%. New price 12 × 50 = ₹600.
Option A: New value as base: 2 blocks out of 12 measures against the new bar.
Option B: Units confused: ₹520 adds ₹20, not 2 blocks of ₹50.
Option C: Wrong block size: ₹1,200 treats a block as ₹100.
Hint: What is one block worth?
Picture: why going back is a different percent
| Bar | Picture | Blocks |
|---|---|---|
| Before | ████████ | 8 |
| After −25% | ██████░░ | 6 |
Text version: a 25% cut removes 2 of 8 blocks. To climb back you add those 2 blocks to the 6 you have: 2/6 = 33⅓%.
A bar has 8 equal blocks. After a 25% cut, how many blocks remain?
Show answer
Correct answer: B
A quarter of 8 is 2, so 6 blocks remain. To get back you add 2 blocks to 6: 2/6 = 33⅓%.
Option A: Answered the change: 2 blocks are the part removed.
Option C: Guessed: 5.5 is a guess; a quarter of 8 is exactly 2.
Option D: Percent read as units: Subtracting 0.25 of a block, not a quarter of the bar.
Hint: A quarter of the bar.
Draw it in the exam (10 seconds)
- Draw the before bar and split it into easy blocks (4, 5, 8 or 10).
- Add or remove blocks for the change.
- Put the known rupee value on the bar it belongs to, then find one block.
Question
After a 25% rise a fee is ₹750. What was it before?
Solution
- What is the question asking?
- ₹750 is the value AFTER a 25% rise. It is not 100%.
- We must find the value BEFORE the change: the old fee, which is 100%.
- Make it a story
- A coaching centre raised its monthly fee by 25%. The new fee is ₹750.
- We know / We need
- The 25% is a percent of the old fee (the value before).
- So the new fee = 100% + 25% = 125% of the old fee
- We know: 125% of the old fee = ₹750
- We need: 100% of the old fee = ?
- Step 1: find 1%
- 125% of the old fee = ₹750, so share ₹750 among 125 pieces of 1%.
- 1% of the old fee = 750 ÷ 125 = ₹6
- Step 2: find 100%
- 100% of the old fee = 6 × 100 = ₹600
- See it on a bar
████████████████████before = 100% = ₹600 (each block = 5% of the old fee)████████████████████▓▓▓▓▓after = 125% of before = ₹750- ▓ = the extra 25% of the old fee = ₹150
- Shortcut (only after you understand)
- Before = After × 100 ÷ 125 = 750 × 100 ÷ 125 = 600
- (Same thing: divide by the multiplier 1.25.)
- Faster way (optional): With bars: 25% = 1/4, so the before bar has 4 blocks and the after bar 5 blocks = ₹750. 1 block = ₹150, before = 4 × 150 = ₹600.
- Quick check
- 25% of ₹600 = ₹150
- 600 + 150 = 750 ✓
Answer: ₹600
Watch out: Most important concept: ₹750 is 125% of the old fee, not 100%. Do not take 25% of ₹750 off: that gives ₹562.5, because it uses the wrong base.
Test your understanding: after a 10% cut, a bus pass costs ₹540. What did it cost before?
Show answer
Correct answer: B
Before bar = 10 blocks; after a 10% cut = 9 blocks = ₹540. 1 block (10% of the old price) = ₹60, so before = 10 × 60 = ₹600. Same with 1%: 540 ÷ 90 = 6, × 100 = 600.
Option A: Wrong base: ₹594 adds 10% of ₹540. The cut was 10% of the OLD price, so ₹540 is on the 'after' bar.
Option C: Applied the change again: ₹486 cuts again. The old price must be bigger than ₹540.
Option D: Percent read as rupees: ₹550 = 540 + 10 treats 10% as ₹10.
Hint: ₹540 sits on the 'after' bar: 9 of the 10 blocks.
Five questions are attached below. Draw before and after bars for each one.
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