Quant · Challenge question · Week 01
Quant challenge, week 01: successive percents
One harder-than-average Quant question with a full worked solution. This week's theme is successive percentage change, which the GRE loves because it punishes anyone who adds and subtracts percents instead of multiplying them.
This is an original question written for this site; the skill it tests is checked against the GRE Quantitative Reasoning overview from ETS. The byline is a masthead name, explained in full on who writes this, and the sourcing rules are set out on the standards page.
A store marks up an item’s cost by 50%. It then offers a 20% discount on the marked price, and finally applies a 5% sales tax to the discounted price. The final amount the customer pays is what percent of the item’s original cost?
A. 112% B. 120% C. 125% D. 126% E. 130%
Answer: D, 126%.
Worked solution
Pick a clean number for the cost. With percents, 100 is almost always the right choice because the final answer drops straight out as a percent.
- Cost. Let the original cost be $100.
- Markup. A 50% markup raises it to $100 × 1.50 = $150.
- Discount. A 20% discount on $150 leaves $150 × 0.80 = $120. Note: 20% off the marked price, not off the cost.
- Tax. A 5% tax on $120 gives $120 × 1.05 = $126.
- Compare. $126 against a $100 cost is 126%.
The answer is 126%, choice D.
The method: multipliers, not additions
The whole question is a chain of multipliers. A 50% increase is ×1.50, a 20% decrease is ×0.80, and a 5% increase is ×1.05. Chain them in order and the order does not even matter, because multiplication commutes:
100 × 1.50 × 0.80 × 1.05 = 126
Notice that the markup and the discount do not cancel. A 50% up followed by 50% down is not back to the start; it is 0.75 of the original. The moment you catch yourself adding 50 and subtracting 20 to get 30, stop and switch to multipliers.
Why the traps exist
- 130% (E) comes from adding 50%, subtracting 20%, and adding 5% as if percents on different bases combine: 50 − 20 + 5 = 35. They do not.
- 120% (B) is the marked-then-discounted price with the 5% tax forgotten.
- 125% (C) comes from taking the 20% discount off the original cost instead of the marked price: 100 × 1.50 = 150, then 150 − 25 = 125, then ignoring tax.
- 112% (A) is the result of applying the discount and tax to the wrong bases in a single muddled step.
On any chained-percent question, write each change as a multiplier and multiply. Set the base to 100 when the question asks for a percent. That two-step habit clears the entire family of successive-percent traps the Quant section can throw.