Puzzle First 2: Which number is the base?
Three puzzles about rising and falling prices. Watch which number you divide by: that number is the base, the thing the percent is a percent of.
Before the puzzles: the basic words
The meaning of the words again, plus what a percent change is. The puzzles will test it.
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
Puzzle 1
Puzzle: a ₹100 shirt goes up by ₹20. Another ₹500 jacket goes up by ₹50. Which rose by a bigger percent?
Show answer
Correct answer: C
A percent change compares the change with the old value.
Option A: Rupees read as percent: Bigger rupees is not a bigger percent: compare with the old price.
Option B: Guessed: 20% and 10% are not equal.
Option D: Missed the base: Old price and rise are enough.
Hint: Divide each rise by its old price.
Puzzle 2: there and back
Notice: 80 → 100 and 100 → 80. Are the two percent changes the same size?
Show answer
Correct answer: A
The same gap of 20 is a bigger share of the smaller number.
Option B: Assumed symmetry: Going up, the base is 80, not 100.
Option C: Assumed symmetry: Going down, the base is 100.
Option D: Base flipped: The two bases are swapped.
Hint: Which number is the old value each time?
What the two puzzles show
- A change is a percent of where you started.
- Going up from 80 is 20 out of 80 (25%). Coming down from 100 is 20 out of 100 (20%).
- So a rise and a fall of the same percent do not cancel.
Puzzle 3: the friend's shortcut
Twist: after a 20% rise a price is ₹600. A friend says 'take 20% off 600 to get back: ₹480'. What was the price really?
Show answer
Correct answer: D
₹600 is 120% of the old price (100% + 20%). 1% of the old price = 600 ÷ 120 = ₹5, so 100% = ₹500. The friend's 20% of 600 is a percent of the wrong (new) price.
Option A: Reversed on the wrong base: 20% of 600 is 120, more than the 100 that was added.
Option B: Percent read as units: Subtracting 20 treats 20% as ₹20.
Option C: Applied the change again: ₹720 raises 600 again.
Hint: Check the friend's answer: does ₹480 + (20% of ₹480) give ₹600?
Puzzle 3, solved slowly
Question
After a 20% rise a price is ₹600. What was the price before?
Solution
- What is the question asking?
- ₹600 is the value AFTER a 20% rise. It is not 100%.
- We must find the value BEFORE the change: the old price, which is 100%.
- Make it a story
- A cinema ticket cost a bit less last year. After a 20% rise it now costs ₹600.
- We know / We need
- The 20% is a percent of the old price (the value before).
- So the new price = 100% + 20% = 120% of the old price
- We know: 120% of the old price = ₹600
- We need: 100% of the old price = ?
- Step 1: find 1%
- 120% of the old price = ₹600, so share ₹600 among 120 pieces of 1%.
- 1% of the old price = 600 ÷ 120 = ₹5
- Step 2: find 100%
- 100% of the old price = 5 × 100 = ₹500
- See it on a bar
████████████████████before = 100% = ₹500 (each block = 5% of the old price)████████████████████▓▓▓▓after = 120% of before = ₹600- ▓ = the extra 20% of the old price = ₹100
- Shortcut (only after you understand)
- Before = After × 100 ÷ 120 = 600 × 100 ÷ 120 = 500
- (Same thing: divide by the multiplier 1.2.)
- Faster way (optional): 20% = 1/5, so old = 5 parts, new = 6 parts. 6 parts = ₹600 → 1 part = ₹100 → old = 5 × 100 = ₹500.
- Quick check
- 20% of ₹500 = ₹100
- 500 + 100 = 600 ✓
Answer: ₹500
Watch out: Most important concept: ₹600 is 120% of the old price, not 100%. Do not take 20% of ₹600 off: that gives ₹480, because it uses the wrong base.
Test your understanding: after a 25% rise, a room's rent is ₹10,000. What was the rent before?
Show answer
Correct answer: C
We know: 125% of the old rent = ₹10,000. Step 1: 1% = 10,000 ÷ 125 = ₹80. Step 2: 100% = 80 × 100 = ₹8,000. Check: 8,000 + 25% of 8,000 = 8,000 + 2,000 = 10,000 ✓.
Option A: Wrong base: ₹7,500 takes 25% of ₹10,000 off. But the 25% was a percent of the OLD rent.
Option B: Applied the change again: ₹12,500 raises the rent again. The old rent must be less than ₹10,000.
Option D: Percent read as rupees: ₹9,975 = 10,000 − 25 treats 25% as ₹25.
Hint: ₹10,000 is 125% of the old rent, not 100%.
What you discovered (the shortcuts)
- Percent change
- change ÷ old × 100 80 → 100 = 20 ÷ 80 × 100 = +25%
- Multiplier
- new = old × (100 ± r) ÷ 100 +20% → × 1.2
- Undo
- old = new ÷ multiplier 600 ÷ 1.2 = 500
- x% less → how much more
- x ÷ (100 − x) × 100 20% less → 20 ÷ 80 × 100 = 25% more
Divide by where you started. To go back, the new value is (100 ± r)% of the old: find 1% of the old, then 100%. Never just subtract the same percent.
Five questions are attached below. Try them before rereading anything.
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