QUANTITATIVE REASONING · GMAT-Q-PCT

Percent Fundamentals and Reverse Percent

GMAT lesson development preview · DRAFT · Original learning and assessment material

Concepts and problem-solving strategy

Model the relationship

F = O (1 + r)

Identify constraints and signs

F = O (1 + r_{1}) (1 + r_{2})

Verify with substitution

O = \frac{F}{( 1 + r _{1} ) ( 1 + r _{2} )}

Think Like a GMAT Test-Taker

Identify the base quantity before calculating a percentage. For successive changes, multiply factors; to recover the original, divide by the combined factor.

Faster Methods

Use an original value of 100 when only a relative change is requested.

For a reverse-percent problem, write final = original × multiplier before calculating.

Ask: percent of what? Label the original amount, discounted amount, cost and selling price before using any multiplier.

Worked example 1

A fee increases by 20% and then decreases by 20%. What is its net percentage change?

100 (1.20) (0.80) = 96; \frac{96 - 100}{100} = - 4%

The decrease applies to 120, not 100. The net result is a 4% decrease.

Worked example 2

After a 15% discount and 10% tax on the discounted price, a bill is $187. Find the original price.

0.85 (1.10) P = 187 \Rightarrow P = 200

Combine the two multipliers before reversing them. Check: 200 becomes 170, then 187.

Worked example 3

A team's error rate falls from 12% to 9%. What is the relative reduction?

\frac{12 - 9}{12} = \frac{1}{4} = 25%

The change is 3 percentage points. Dividing that change by the original rate gives a 25% reduction.

Worked example 4

A shop sells an item for $144 after a 20% discount. The selling price is 20% above the shop's cost. Find the marked price and the cost.

0.80 M 1.20 C = 144 = 144 M C = 180 = 120

There are two different percentage bases. The discount is based on the marked price, while profit is based on cost. Write a separate relationship for each. Check: 20% of 180 is 36, and 20% of 120 is 24; both routes lead to 144.

Common Traps

Equal percentage increases and decreases do not cancel: their bases differ.
A rise from 20% to 25% is 5 percentage points but a 25% relative increase.