Exam Sprint 2: Increase, decrease and working backwards
Goal: find a new value after a rise or a fall, and go backwards from the new value to the old one, without falling into the wrong-base trap.
Start from the basics
First the basics again (they never change), then the one new idea for this lesson: percent change.
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
1. Warm-up from Lesson 1
Warm-up from Lesson 1: 30 is 25% of which number?
Show answer
Correct answer: B
We know: 25% of the number = 30. We need: 100%. Step 1: 1% = 30 ÷ 25 = 1.2. Step 2: 100% = 1.2 × 100 = 120. Check: 25% of 120 = 30 ✓.
Option A: Took percent of the part: 7.5 is 25% of 30.
Option C: Multiplied by the percent: 30 × 25 multiplies by the percent.
Option D: Wrong question: 37.5 adds 25% to 30.
Hint: 25% is one quarter.
2. Predict first
Predict: a ₹500 item gets 10% costlier, then you want the old price back. Is a 10% cut on the new price enough?
Show answer
Correct answer: A
After +10% the price is 550; 10% of 550 is 55; 550 − 55 = 495.
Option B: Assumed it cancels: 10% of 550 is 55, not 50.
Option C: Sign error: Direction is wrong: removing 55 from 550 gives 495.
Option D: Wrong base: ₹450 is 10% off the old ₹500.
Hint: Write the new price first.
3. Understand it with a story: going forward
Question
Your ₹800 monthly phone bill goes up by 15%. What is the new bill?
Solution
- What is the question asking?
- ₹800 is the starting value: 100%.
- It goes up by 15% OF ITSELF. We need the new value.
- Make it a story
- Your phone company says: 'From next month your ₹800 plan costs 15% more.'
- We know / We need
- We know: 100% of the old bill = ₹800; change = 15% of the old bill
- We need: the new value = 100% + 15% = 115% of the old bill
- Step 1: find 1%
- 1% of the old bill = 800 ÷ 100 = ₹8
- Step 2: find the change, then the new value
- 15% of the old bill = 15 × 8 = ₹120
- New value = 800 + 120 = ₹920
- See it on a bar
████████████████████before = 100% = ₹800 (each block = 5% of the old bill = ₹40)████████████████████▓▓▓after = 115% of before = ₹920 (▓ = 15% of before = ₹120)
- Shortcut (only after you understand)
- New = Old × 115 ÷ 100 = 800 × 1.15 = 920
- Faster way (optional): 15% = 10% + 5% of ₹800 = 80 + 40 = ₹120.
- Quick check
- Change = 920 − 800 = 120; and 120 ÷ 800 × 100 = 15% of the old value ✓
Answer: ₹920
Watch out: Most important concept: The 15% is always a percent of the OLD value (₹800), the value before the change.
4. Understand it with a story: going backwards
Going backwards is where most marks are lost. The given number is the value after the change, so it is not 100%.
Question
After a 15% discount, a watch costs ₹1,700. What was its marked price?
Solution
- What is the question asking?
- ₹1,700 is the value AFTER a 15% cut. It is not 100%.
- We must find the value BEFORE the change: the marked price, which is 100%.
- Make it a story
- A shop gives 15% off the price written on the tag. You pay ₹1,700. What was written on the tag?
- We know / We need
- The 15% is a percent of the marked price (the value before).
- So the sale price = 100% − 15% = 85% of the marked price
- We know: 85% of the marked price = ₹1,700
- We need: 100% of the marked price = ?
- Step 1: find 1%
- 85% of the marked price = ₹1,700, so share ₹1,700 among 85 pieces of 1%.
- 1% of the marked price = 1,700 ÷ 85 = ₹20
- Step 2: find 100%
- 100% of the marked price = 20 × 100 = ₹2,000
- See it on a bar
████████████████████before = 100% = ₹2,000 (each block = 5% of the marked price)█████████████████░░░after = 85% of before = ₹1,700- ░ = the 15% of the marked price taken away = ₹300
- Shortcut (only after you understand)
- Before = After × 100 ÷ 85 = 1,700 × 100 ÷ 85 = 2,000
- (Same thing: divide by the multiplier 0.85.)
- Quick check
- 15% of ₹2,000 = ₹300
- 2,000 − 300 = 1,700 ✓
Answer: ₹2,000
Watch out: Most important concept: ₹1,700 is 85% of the marked price, not 100%. Do not add 15% of ₹1,700 on: that gives ₹1,955, because it uses the wrong base.
Test your understanding: after a 20% discount a bag costs ₹480. What was the price before the discount?
Show answer
Correct answer: D
We know: 80% of the old price = ₹480 (100% − 20% discount). We need: 100% of the old price. Step 1: 1% = 480 ÷ 80 = ₹6. Step 2: 100% = 6 × 100 = ₹600. Check: 20% of 600 = 120; 600 − 120 = 480 ✓.
Option A: Wrong base: ₹576 = 480 + 20% of 480. But the 20% was a percent of the OLD price, not of ₹480.
Option B: Applied the change again: ₹384 takes another 20% off. The old price must be MORE than the sale price.
Option C: Percent read as rupees: ₹500 = 480 + 20 treats 20% as ₹20. A percent is a share of the old price.
Hint: ₹480 is 80% of the old price, not 100%.
5. Now the shortcut: one multiplier
You have seen that +15% makes the new value 115% of the old, and −15% makes it 85% of the old. Writing 115% as 1.15 gives one number to multiply by (a multiplier).
Multipliers
- Increase by r%
- × (100 + r) ÷ 100 +20% → × 1.2
- Decrease by r%
- × (100 − r) ÷ 100 −15% → × 0.85
- Percent change
- (new − old) ÷ old × 100 ₹200 → ₹230: 30 ÷ 200 × 100 = 15%
- Old from new
- new ÷ multiplier 360 after +20% → 360 ÷ 1.2 = 300
Every percent in a change is a percent of the old value.
6. Now you finish it (half-solved)
Question
A town's population grew 25% to 15,000. What was it before?
Solution
- What is the question asking?
- 15,000 is the value AFTER a 25% rise. It is not 100%.
- We must find the value BEFORE the change: the old population, which is 100%.
- Make it a story
- A town grew by 25% in ten years. Today 15,000 people live there.
- We know / We need
- The 25% is a percent of the old population (the value before).
- So the new population = 100% + 25% = 125% of the old population
- We know: 125% of the old population = 15,000
- We need: 100% of the old population = ?
- Step 1: find 1%
- 125% of the old population = 15,000, so share 15,000 among 125 pieces of 1%.
- 1% of the old population = 15,000 ÷ 125 = 120
- Step 2 is yours: find 100%
- 1% of the old population = 120. Multiply by 100, then check it below.
- See it on a bar
████████████████████before = 100% = ? (each block = 5% of the old population)████████████████████▓▓▓▓▓after = 125% of before = 15,000- ▓ = the extra 25% of the old population
- Shortcut (only after you understand)
- Before = After × 100 ÷ 125 = 15,000 × 100 ÷ 125
- (Same thing: divide by the multiplier 1.25.)
Answer: 12,000. Step 2: 100% = 120 × 100 = 12,000. Check: 25% of 12,000 = 3,000, and 12,000 + 3,000 = 15,000 ✓
Watch out: Most important concept: 15,000 is 125% of the old population, not 100%. Do not take 25% of 15,000 off: that gives 11,250, because it uses the wrong base.
7. Your turn, no help
Your turn: after a 30% increase, a value is 520. What was the original value?
Show answer
Correct answer: C
We know: the new value = 100% + 30% = 130% of the original = 520. Step 1: 1% of the original = 520 ÷ 130 = 4. Step 2: 100% = 400. Check: 30% of 400 = 120; 400 + 120 = 520 ✓.
Option A: Reversed on the wrong base: 520 × 0.7 removes 30% of the new value.
Option B: Applied the change again: 520 × 1.3 increases again.
Option D: Percent read as units: Subtracting 30 treats 30% as units.
Hint: Divide by the multiplier.
To undo a 20% rise you do not take 20% off the new value: that 20% would be a percent of the new value, which is bigger. Divide by 1.2 instead. A 20% rise is undone by a 16⅔% cut; a 20% cut is undone by a 25% rise.
Learn 12.5% = 1/8, 20% = 1/5, 25% = 1/4, 33⅓% = 1/3. Then +25% means 'old = 4 parts, new = 5 parts': 15,000 = 5 parts → 1 part = 3,000 → old = 4 parts = 12,000. No decimals needed.
8. Mixed check
Mixed check: 45 is 60% of which number?
Show answer
Correct answer: D
We know: 60% of the number = 45. Step 1: 1% = 45 ÷ 60 = 0.75. Step 2: 100% = 0.75 × 100 = 75. Check: 60% of 75 = 45 ✓.
Option A: Took percent of the part: 27 is 60% of 45.
Option B: Wrong question: 72 adds 60% to 45.
Option C: Percent read as units: 105 adds 60 to 45.
Hint: 60% is 3/5.
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: multipliers forwards and backwards.
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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