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2009 AMC 10B Problem 5

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2009 AMC 10B Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 1 of 5 · about 1 min · position P1-10

Twenty percent less than 60 is one-third more than what number?

A)
$16$
B)
$30$
C)
$32$
D)
$36$
E)
$48$

Answer

Key insight

Twenty percent less than 60 is 48, and 'one-third more than n' means 4n/3, so n = 36.

Solution

Twenty percent less than $60$ is $60 \cdot 0.8 = 48$ .

If the unknown number is $n$ , then one-third more than $n$ is $n + \frac{n}{3} = \frac{4n}{3}$ . Setting $\frac{4n}{3} = 48$ gives $n = 48 \cdot \frac34 = 36$ .

Check: one-third of $36$ is $12$ , and $36 + 12 = 48$ .

The answer is $\boxed{\textbf{(D)}\ 36}$ .

Why this works

Both phrases are multiplicative: " $p\%$ less" means multiply by $1 - p/100$ , and "a fraction $f$ more than" means multiply by $1 + f$ . Translate each phrase into a factor, then undo the factor with its reciprocal.

The trap

Reading 'one-third more than' as adding 1/3 or as dividing by 3, or subtracting one-third of 48 to get 32.

Common mistakes

Reading 'one-third more than' as adding 1/3 or as dividing by 3, or subtracting one-third of 48 to get 32.
Stopping at 48 (choice E), which is only the first half of the problem.

Techniques

Set up the equation/formula and compute; no special trick needed