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2009 AMC 10A Problem 7

2009 AMC 10A Problem 7AMC 10 Step by Step

Name: Date: September 13, 2026

Ratios, Percents & Averages · difficulty 2 of 5 · about 2 min · position P1-10

A carton contains milk that is $2$ % fat, an amount that is $40$ % less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk?

A)
$\frac{12}{5}$
B)
$3$
C)
$\frac{10}{3}$
D)
$38$
E)
$42$

Answer

Key insight

Forty percent less than the whole-milk fat means 2 is 60% of it, so the whole-milk percentage is 2 divided by 0.6.

Solution

Let $w$ be the fat percentage of whole milk. " $40\%$ less than $w$ " means $w$ reduced by $0.4w$ , leaving $0.6w$ . So
$0.6\,w = 2 \quad\Longrightarrow\quad w = \frac{2}{0.6} = \frac{20}{6} = \frac{10}{3}.$

Check: $40\%$ of $\frac{10}{3}$ is $\frac{4}{3}$ , and $\frac{10}{3} - \frac{4}{3} = 2$ .

The answer is $\boxed{\textbf{(C)}\ \frac{10}{3}}$ .

Why this works

A percent decrease is a multiplication: " $p\%$ less than $x$ " is $(1 - p)x$ . The comparison is always relative to the original quantity (whole milk), so to undo it you divide by $0.6$ , not multiply by $1.4$ . The two "percent" words in the problem play different roles: $2\%$ is a concentration, $40\%$ is a relative change.

The trap

Reading '40% less' as 40 percentage points less (answering 42) or adding 40% back onto 2 instead of dividing by 0.6.

Common mistakes

Reading '40% less' as 40 percentage points less (answering 42) or adding 40% back onto 2 instead of dividing by 0.6.
Treating $2$ as $40\%$ of the whole-milk fat (giving $5$ ), which confuses " $40\%$ less" with " $40\%$ of."

Techniques

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