A reliable method
Write the original as a variable and the new value as original × multiplier. For successive changes multiply the factors. Distinguish profit divided by cost from profit divided by revenue.
The common trap
An increase followed by an equal percentage decrease does not cancel. Reversing a discount requires division by the remaining proportion.
Make the connection
Translate the same business situation into both money and ratios. This exposes whether the question uses original price, discounted price, cost or sales as its denominator.
Contrast the cases
A price rises from 80 to 100: the increase is 20/80 = 25%. Returning from 100 to 80 is a 20/100 = 20% decrease. The same dollar difference does not imply the same percentage. Likewise, a 20% discount on 100 leaves 80; adding 20% to 80 gives 96, not 100. Reversing the discount means solving 0.8p=80. Distinguish the old base, new amount and difference before choosing an operation.
Transfer and explain
Transfer: an item costs 120 after a 25% discount. Recover 160, then verify that a 25% discount removes 40. Explain why multiplying 120 by 1.25 is not the inverse operation.
After a 20% discount, a reference book costs $112. What was its original price in dollars?
- After a 20% discount, 80% of the original price remains.
- Write 0.8p = 112.
- Divide by 0.8 to get p = 140.
Compare the other answer choices
Treats the sale price as the original price.
Treats the sale price as the discount amount.
Applies a second discount instead of reversing it.
Adds the discount percentage to the sale price, using the wrong base.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference