2009 AMC 10A Problem 7

1. Call the fat percentage of whole milk $w$.
2. "$40\%$ less than $w$" means $w$ minus $40\%$ of $w$. What is left is $60\%$ of $w$, which is $0.6w$.
3. The problem says that amount is $2$:
$$0.6w = 2.$$
4. Divide both sides by $0.6$:
$$w = \frac{2}{0.6} = \frac{20}{6} = \frac{10}{3}.$$
5. Check: $40\%$ of $\frac{10}{3}$ is $\frac{4}{10} \times \frac{10}{3} = \frac{4}{3}$, and $\frac{10}{3} - \frac{4}{3} = \frac{6}{3} = 2$. It works.

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

"$40\%$ less than something" means "multiply that something by $0.6$." The $40\%$ is measured against the whole-milk amount, not against $2$, so to go back you divide by $0.6$. Adding $40\%$ to $2$ would give $2.8$, which is the wrong answer. The problem uses "percent" in two ways: $2\%$ is how much of the milk is fat, while $40\%$ is how much smaller one amount is than another.

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