See It 3: The 100-grid for chains of changes and points
A 10 × 10 grid makes every square 1%. It shows two things bars cannot: the small extra piece in a chain of changes, and the difference between points and percent.
See the basics first
The 100-square idea again, plus chains of changes and rates.
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:
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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.
6. Two changes in a row. The second percent is a percent of the value after the first change, not of the start. So two changes cannot simply be added.
7. A rate is already a percent. A pass rate of 40% means 40 of every 100 students pass. When a rate moves from 40% to 50%, you can describe the move in two ways: percentage points (subtract: 50 − 40 = 10 points) or percent (compare with the old rate: 10 out of 40 = 25%).
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: +20% then +10%
Start with 100 squares. +20% adds 20 squares. +10% then adds 10 squares for the old 100 and 2 more for the new 20.
| Piece | Squares |
|---|---|
| Start | 100 |
| +20% of 100 | 20 |
| +10% of 100 | 10 |
| +10% of the new 20 (the ab/100 piece) | 2 |
| Total | 132 |
On a 10 × 10 grid, a 20% rise and then a 10% rise. How many squares do you end with?
Show answer
Correct answer: C
The extra 2 squares are 10% of the 20 new squares: the ab/100 piece you can see.
Option A: Added the percents: The second 10% is of 120 (12 squares), not of 100.
Option B: Kept only the cross term: Only 2 squares added in step two.
Option D: Stopped one step early: Stopped after the first rise.
Hint: 10% of 120?
Watch one solved slowly
Question
A price rises 20% one year and 10% the next. What is the total rise?
Solution
- What is the question asking?
- Two changes happen one after the other: first +20%, then +10%.
- The second change is a percent of the NEW value, not of the start. We need the overall % change compared with the start.
- Make it a story
- A ₹200 school bag costs 20% more this year, and next year it will cost a further 10% more.
- We know / We need
- We know: start = ₹200 = 100%; first change +20% of the start; second change +10% of the value after step 1
- We need: the final value, and its % change from ₹200
- Step 1: first change (1% of the start)
- 1% of ₹200 = ₹2, so 20% of ₹200 = ₹40
- After step 1: 200 + 40 = ₹240
- Step 2: second change (1% of the NEW value)
- 1% of ₹240 = ₹2.4, so 10% of ₹240 = ₹24
- After step 2: 240 + 24 = ₹264
- Step 3: compare with the start
- Change = 264 − 200 = 64
- 1% of the start = ₹2, so 64 ÷ 2 = 32 → 32% increase
- See it on a bar
█████████████████████████start = 100% = ₹200 (each block = 4% of the start)█████████████████████████▓▓▓▓▓after step 1 = 120% of the start = ₹240█████████████████████████▓▓▓▓▓▓▓▓after step 2 = 132% of the start = ₹264
- Shortcut (only after you understand)
- Net % = a + b + (a × b) ÷ 100 = +20 +10 + (+20 × +10) ÷ 100 = +32%
- Faster way (optional): On the 100-grid: 20 squares + 10 squares + the corner 2 squares (10% of the 20 new ones) = 32 → +32%.
- Quick check
- Multipliers: 1.2 × 1.1 = 1.32 → 32% increase ✓
Answer: 32% increase
Watch out: Most important concept: Do not just add +20% and +10% (= +30%). The second percent is taken of a different, new value (₹240).
Test your understanding: on a 100-square grid, a 10% rise and then another 10% rise. How many squares do you end with?
Show answer
Correct answer: A
Start 100. +10% of 100 = 10 → 110. +10% of 110 = 11 → 121. The extra 1 square is 10% of the 10 new squares: the corner piece.
Option B: Added the percents: 120 adds 10 twice. The second 10% is of 110, which is 11.
Option C: Stopped early: 110 stops after the first rise.
Option D: Kept only the corner: 1 is only the small corner piece (10% of the first 10). The two 10s must be added too.
Hint: The second 10% is of 110, not 100.
Picture: points vs percent
Vote share 30% → 36%: 30 dark squares, 6 new squares (▓).
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Text version: 6 squares were added (6 percentage points). Compared with the 30 that were there, 6/30 = 20%.
A grid shows 45 of 100 squares shaded (45%). It becomes 54 shaded. Which is correct?
Show answer
Correct answer: A
Count added squares for points; compare them with the old 45 for percent.
Option B: Points read as percent: The relative rise is 20%, not 9%.
Option C: Percent read as points: Only 9 squares were added.
Option D: New value as base: 9/54 uses the new count.
Hint: 9 out of 45 is what fraction?
Read the picture
- Chain
- a + b + the small corner ab/100 +20, +10 → 132 squares
- Points
- count the added squares 30 → 36 = 6 points
- Percent
- added ÷ old squares × 100 6/30 = 20%
Five questions are attached below. For chains, picture the small corner piece.
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