Puzzle First 3: Chains of changes, and points vs percent
What happens when one change follows another? Find out before you see a formula.
Before the puzzles: the basic words
The meaning of the words again, plus changes in a row and rates. The puzzles will test them.
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
Puzzle 1
Puzzle: a ₹1,000 phone gets 10% off, then 10% off the new price. Final price?
Show answer
Correct answer: B
Each step multiplies by 0.9: 1,000 × 0.81 = 810, a 19% total discount.
Option A: Added the percents: ₹800 is one 20% discount; the second 10% is on 900.
Option C: Stopped one step early: Only one discount applied.
Option D: Answered the change: ₹190 is the total discount.
Hint: Do one discount at a time.
Where did the extra 1% go?
- The first 10% came off ₹1,000: 10% of ₹1,000 = ₹100, leaving ₹900.
- The second 10% came off ₹900: 10% of ₹900 = only ₹90.
- Total off = ₹190 out of ₹1,000 = 19%. The 'missing' ₹10 is 10% of the first ₹100 that was already gone: that is the (a × b) ÷ 100 piece.
Puzzle 2: predict with the pattern
Notice: −10% then −10% gave −19%, not −20%. Using a + b + ab/100, what do −20% then −20% give?
Show answer
Correct answer: A
(−20)(−20)/100 = +4, so −40 + 4 = −36%.
Option B: Added the percents: Missing the +4 cross term.
Option C: Sign error: The cross term has the wrong sign: two falls give a positive ab.
Option D: Kept only the cross term: −4 is not even the cross term (+4).
Hint: The cross term is positive here.
A chain, solved slowly
Question
A coat gets 20% off, then a further 25% off the new price. What is the total discount?
Solution
- What is the question asking?
- Two changes happen one after the other: first −20%, then −25%.
- 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 coat has a 20% sale tag. At the counter you get a further 25% off the sale price.
- We know / We need
- We know: start = ₹1,000 = 100%; first change −20% of the start; second change −25% 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 20% of ₹1,000 = ₹200
- After step 1: 1,000 − 200 = ₹800
- Step 2: second change (1% of the NEW value)
- 1% of ₹800 = ₹8, so 25% of ₹800 = ₹200
- After step 2: 800 − 200 = ₹600
- Step 3: compare with the start
- Change = 600 − 1,000 = −400
- 1% of the start = ₹10, so 400 ÷ 10 = 40 → 40% decrease
- See it on a bar
████████████████████start = 100% = ₹1,000 (each block = 5% of the start)████████████████░░░░after step 1 = 80% of the start = ₹800████████████░░░░░░░░after step 2 = 60% of the start = ₹600
- Shortcut (only after you understand)
- Net % = a + b + (a × b) ÷ 100 = −20 −25 + (−20 × −25) ÷ 100 = −40%
- Quick check
- Multipliers: 0.8 × 0.75 = 0.6 → 40% decrease ✓
Answer: 40% decrease
Watch out: Most important concept: Do not just add −20% and −25% (= −45%). The second percent is taken of a different, new value (₹800).
Puzzle 3: the headline
Twist: a bank's interest rate goes from 8% to 10%. A headline says 'Rates up 2%'. What is the precise statement?
Show answer
Correct answer: C
Subtract rates for points; divide by the old rate for percent.
Option A: Points read as percent: 2 is points; the relative rise is 25%.
Option B: New value as base: 20% = 2/10 uses the new rate as base.
Option D: Percent read as points: The rate only moved 8 → 10.
Hint: Two different numbers answer two different questions.
Points vs percent, solved slowly
Question
A party's vote share rises from 20% to 25%. By how many points, and by what percent?
Solution
- What is the question asking?
- the vote share moved from 20% to 25%. The question wants two answers:
- how many percentage POINTS it changed by, and by what PERCENT it changed.
- Make it a story
- In the last election a party got 20 of every 100 votes. This time it got 25 of every 100.
- We know / We need
- We know: old = 20%, new = 25%
- We need: (a) the change in points; (b) the change as a percent of the OLD rate
- Step 1: points = subtract
- 25 − 20 = 5 percentage points
- Step 2: percent = compare with the old rate
- 1% of the old rate 20 = 20 ÷ 100 = 0.2
- 5 ÷ 0.2 = 25 → a 25% rise
- See it on a bar
████░░░░░░░░░░░░░░░░before: 20% (each block = 5 points)████▓░░░░░░░░░░░░░░░░after: 25% (▓ = the 5 new points)- 5 points added to the old 20 points = 5/20 = 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 20 = 5 points; 20 + 5 = 25 ✓
Answer: 5 percentage points; a 25% rise
Watch out: Most important concept: 'Points' = subtract the two rates (5). 'Percent' = divide that by the old rate (25%). Never call a 5-point rise a 5% rise.
Test your understanding: unemployment falls from 10% to 8%. Which statement is correct?
Show answer
Correct answer: B
Points = 10 − 8 = 2. Percent = 2 compared with the old 10: 1% of 10 = 0.1, 2 ÷ 0.1 = 20 → a 20% fall.
Option A: Points read as percent: The fall of 2 points is 20% of the old 10, not 2%.
Option C: New value as base: 2 ÷ 8 uses the new rate. Compare with the old rate 10.
Option D: Percent read as points: Only 2 points were lost (10 → 8).
Hint: Subtract for points; divide by the OLD rate for percent.
What you discovered (the shortcuts)
- Two changes
- a + b + (a × b) ÷ 100 (signs included) −10, −10 → −20 + 1 = −19%
- Order
- does not matter +10, +20 = +20, +10 = +32%
- Points
- new rate − old rate 8% → 10% = 2 points
- Percent of a rate
- points ÷ old rate × 100 8% → 10% = 2 ÷ 8 × 100 = 25%
The second change is a percent of a new value, so changes multiply, they do not add. A change in a rate has two names: points (subtract) and percent (divide by the old rate).
Five questions are attached below. Try them before rereading anything.
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