Percentages • Topic 4 of 5

Reverse Percentages

A reverse-percentage question gives you the value AFTER a change and asks for the original. The rule: never take the same percent of the new number — divide instead. If a price after a 20% increase is $180, the original is 180 / 1.20 = $150 (not 180 × 0.8 = 144, the classic wrong move). In general, Original = Final / (1 ± x/100). This is exactly how the GRE tests "the sale price is $63 after a 30% discount — what was the list price?" and how percentages hide inside Data Interpretation ("this year’s figure is 12% above last year’s"). Whenever the words after, following, or now signal that a change has already happened, reach for division, not multiplication.

✅ Solved examples

1. After a 20% raise, a salary is $54,000. Find the original.
Original = 54000 / 1.20 = $45,000.
2. A jacket costs $63 after a 30% discount. Find the list price.
List = 63 / 0.70 = $90.
3. A number decreased by 15% becomes 170. Find the number.
170 / 0.85 = 200.
4. This year 3,360 students enrolled — 12% more than last year. Last year’s enrolment?
3360 / 1.12 = 3000.

✏️ Practice — try these, take hints as needed

1. After a 25% increase a value is 250. Original?
Divide, don’t multiply.
250 / 1.25.
= 250 × 4/5.
200
2. A price after a 10% discount is $270. List price?
Divide by 0.90.
270 / 0.9.
= 300.
$300
3. A number increased by 40% becomes 210. The number?
/ 1.4.
210 / 1.4.
= 150.
150
4. A town’s population fell 20% to 9,600. Original population?
/ 0.8.
9600 / 0.8.
= 12000.
12,000
5. A bill with 8% sales tax added is $54. Pre-tax amount?
/ 1.08.
54 / 1.08.
= 50.
$50

📝 Topic test — 8 questions

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