Exam Sprint 3: Successive changes and percentage points
Goal: handle two changes one after the other, and never confuse percentage points with percent.
Start from the basics
First the basics again, then the new ideas for this lesson: two changes in a row, and percentage points.
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.
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
1. Warm-up from Lesson 2
Warm-up from Lesson 2: a price rises from ₹200 to ₹230. What is the percentage increase?
Show answer
Correct answer: B
The increase is ₹30, and it is measured against the OLD price ₹200. Step 1: 1% of 200 = ₹2. Step 2: 30 ÷ 2 = 15 → 15%.
Option A: New value as base: 30/230 uses the new price as base.
Option C: Rupees read as percent: 30 is rupees.
Option D: Ratio, not change: 115% is new ÷ old.
Hint: Divide by the old price.
2. Predict first
Predict: two successive discounts of 10% and 20%. Is that the same as one 30% discount?
Show answer
Correct answer: A
Take ₹100. First 10% off: 10% of 100 = 10 → ₹90. Then 20% off the NEW price: 20% of 90 = 18 → ₹72. ₹100 → ₹72 is ₹28 off, so 28%, not 30%.
Option B: Added the percents: The second discount is on a smaller price.
Option C: Sign error: Two discounts give less than 30%, not more.
Option D: Kept only the cross term: 2 is only the cross term.
Hint: Multiply 0.9 by 0.8.
3. Understand it with a story: two changes in a row
Question
A price rises 50% and then falls 20%. What is the net change?
Solution
- What is the question asking?
- Two changes happen one after the other: first +50%, then −20%.
- 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 ₹1,000 jacket: in winter the shop raises its price by 50%. Later it puts the jacket on sale at 20% off.
- We know / We need
- We know: start = ₹1,000 = 100%; first change +50% of the start; second change −20% of the value after step 1
- We need: the final value, and its % change from ₹1,000
- Step 1: first change (1% of the start)
- 1% of ₹1,000 = ₹10, so 50% of ₹1,000 = ₹500
- After step 1: 1,000 + 500 = ₹1,500
- Step 2: second change (1% of the NEW value)
- 1% of ₹1,500 = ₹15, so 20% of ₹1,500 = ₹300
- After step 2: 1,500 − 300 = ₹1,200
- Step 3: compare with the start
- Change = 1,200 − 1,000 = 200
- 1% of the start = ₹10, so 200 ÷ 10 = 20 → 20% increase
- See it on a bar
████████████████████start = 100% = ₹1,000 (each block = 5% of the start)████████████████████▓▓▓▓▓▓▓▓▓▓after step 1 = 150% of the start = ₹1,500████████████████████▓▓▓▓after step 2 = 120% of the start = ₹1,200
- Shortcut (only after you understand)
- Net % = a + b + (a × b) ÷ 100 = +50 −20 + (+50 × −20) ÷ 100 = +20%
- Quick check
- Multipliers: 1.5 × 0.8 = 1.2 → 20% increase ✓
Answer: 20% increase
Watch out: Most important concept: Do not just add +50% and −20% (= +30%). The second percent is taken of a different, new value (₹1,500).
Test your understanding: a price rises 10% and then falls 10%. What is the net change?
Show answer
Correct answer: C
Start ₹100. Step 1: 10% of ₹100 = ₹10 → ₹110. Step 2: 10% of ₹110 = ₹11 → ₹99. Compare with ₹100: ₹1 less = 1% decrease. Shortcut: +10 − 10 + (10 × −10) ÷ 100 = −1%.
Option A: Assumed it cancels: The fall is 10% of ₹110 (= ₹11), bigger than the rise of ₹10. They do not cancel.
Option B: Sign error: The final ₹99 is below the starting ₹100, so it is a decrease.
Option D: Added the percents: Only one of the two changes was a fall. Adding both as falls is wrong.
Hint: Use ₹100. The second 10% is of the new price.
4. Understand it with a story: points vs percent
Question
A pass rate goes from 40% to 50%. By how many points, and by what percent, did it rise?
Solution
- What is the question asking?
- the pass rate moved from 40% to 50%. The question wants two answers:
- how many percentage POINTS it changed by, and by what PERCENT it changed.
- Make it a story
- Last year 40 of every 100 students in a school passed. This year 50 of every 100 passed.
- We know / We need
- We know: old = 40%, new = 50%
- We need: (a) the change in points; (b) the change as a percent of the OLD rate
- Step 1: points = subtract
- 50 − 40 = 10 percentage points
- Step 2: percent = compare with the old rate
- 1% of the old rate 40 = 40 ÷ 100 = 0.4
- 10 ÷ 0.4 = 25 → a 25% rise
- See it on a bar
████████░░░░░░░░░░░░before: 40% (each block = 5 points)████████▓▓░░░░░░░░░░░░after: 50% (▓ = the 10 new points)- 10 points added to the old 40 points = 10/40 = 25% of the old rate
- Shortcut (only after you understand)
- Points = new rate − old rate
- Percent change = points ÷ old rate × 100
- Quick check
- 25% of 40 = 10 points; 40 + 10 = 50 ✓
Answer: 10 percentage points; a 25% rise
Watch out: Most important concept: 'Points' = subtract the two rates (10). 'Percent' = divide that by the old rate (25%). Never call a 10-point rise a 10% rise.
5. Now the shortcuts
Successive changes and points
- Two changes a% and b%
- a + b + (a × b) ÷ 100 +20, +10 → 20 + 10 + 2 = +32%
- Same, with multipliers
- (1 + a/100) × (1 + b/100) 1.2 × 1.1 = 1.32
- Equal rise and fall of x%
- net = −(x × x) ÷ 100 % +10, −10 → −1%
- Percentage points
- new rate − old rate 40% → 50% = 10 points
- Percent change of a rate
- points ÷ old rate × 100 40% → 50% = 10 ÷ 40 × 100 = 25%
Use a minus sign for a decrease: −20 then +10 → −20 + 10 + (−20 × 10) ÷ 100 = −20 + 10 − 2 = −12%.
6. Now you finish it (half-solved)
Question
Sales fall 25% one year and rise 20% the next. What is the net change?
Solution
- What is the question asking?
- Two changes happen one after the other: first −25%, then +20%.
- 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 small shop sold ₹2,000 of goods a day. One bad year cut that by 25%; the next year it grew 20%.
- We know / We need
- We know: start = ₹2,000 = 100%; first change −25% of the start; second change +20% of the value after step 1
- We need: the final value, and its % change from ₹2,000
- Step 1: first change (1% of the start)
- 1% of ₹2,000 = ₹20, so 25% of ₹2,000 = ₹500
- After step 1: 2,000 − 500 = ₹1,500
- Step 2 is yours: second change on the NEW value
- Find 1% of ₹1,500 (not of ₹2,000), then 20% of it, then the final value and its % change from ₹2,000.
- See it on a bar
████████████████████start = 100% = ₹2,000 (each block = 5% of the start)███████████████░░░░░after step 1 = 75% of the start = ₹1,500- after step 2 = ? (draw it yourself)
Answer: 10% decrease. Step 2: 1% of ₹1,500 = ₹15; 20% of it = ₹300; final = ₹1,800. Change −200 on 2,000 = −10% of the start. Check: shortcut −25 +20 + (−25 × +20) ÷ 100 = −10% ✓
Watch out: Most important concept: Do not just add −25% and +20% (= −5%). The second percent is taken of a different, new value (₹1,500).
7. Your turn, no help
Your turn: +20% then −25%. Net change?
Show answer
Correct answer: D
Take ₹100. +20%: 20% of 100 = 20 → ₹120. −25% of the NEW price: 25% of 120 = 30 → ₹90. ₹100 → ₹90 = 10% decrease.
Option A: Added the percents: 20 − 25 = −5 ignores the cross term −5.
Option B: Assumed it cancels: Changes do not cancel.
Option C: Sign error: Direction is wrong.
Hint: Use a + b + ab/100 with b = −25.
If a pass rate goes from 40% to 50%, it rose by 10 percentage points, which is a 25% rise (10 compared with the old 40). Read the question: 'by how many points' means subtract; 'by what percent' means divide by the old rate.
The extra piece is (a × b) ÷ 100. Same direction (both up or both down): it is positive. Opposite directions: it is negative. So +x then −x always ends below the start, by (x × x) ÷ 100 %.
8. Mixed check
Mixed check: a price is cut 25%. What rise brings it back?
Show answer
Correct answer: C
Take ₹100. A 25% cut leaves ₹75. To get back to ₹100 you must add ₹25 to ₹75. 1% of 75 = 0.75, and 25 ÷ 0.75 = 33⅓ → 33⅓%.
Option A: Assumed symmetry: 75 × 1.25 = 93.75, short of 100.
Option B: Ratio, not change: 75% is the reduced price as a share.
Option D: Wrong pattern: 20% is the reverse of a 25% rise, not of a 25% cut.
Hint: Start from 100.
Write one line for every miss: question → trap name → the fix in five words. Trap names to use: wrong base, answered the remainder, added the percents, points vs percent. Return to this lesson: tomorrow (redo the mixed check), day 3 (practice test cold), day 7 (only your error-log questions). Today's skill: successive changes and points.
Five exam-style questions are attached below this lesson. Every option has a reason: read why each wrong option is wrong, not only the right one.
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