2025 AMC 10A Problem 2

1. Start with what is already in the box. $10$ pounds that are $50\%$ peanuts means $5$ pounds of peanuts. $20\%$ cashews means $2$ pounds of cashews. (The almonds, $3$ pounds, will never matter.)

1. We don't know how much of the second mix is poured in, so call it $x$ pounds. That $x$ pounds brings $0.2x$ pounds of peanuts and $0.4x$ pounds of cashews.

3. The only fact we are given about the final mix is about peanuts, so use peanuts to find $x$. After mixing:
- peanuts: $5 + 0.2x$ pounds,
- total weight: $10 + x$ pounds.

4. Peanuts are $40\%$ of the total, so
$$5 + 0.2x = 0.4(10 + x) = 4 + 0.4x.$$
Subtract $4$ and $0.2x$ from both sides: $1 = 0.2x$, so $x = 5$.

1. Quick check: the box now has $15$ pounds with $5 + 1 = 6$ pounds of peanuts, and $6$ is $40\%$ of $15$. Good.

1. Now answer the actual question, about cashews: the box has $2 + 0.4 \times 5 = 2 + 2 = 4$ pounds of cashews.

The answer is $\boxed{\textbf{(B)}\ 4}$.

In a mixing problem, the pounds of one ingredient always equal its percent times the total weight. Pick the ingredient the problem tells you about (peanuts), write that one equation, and solve for the unknown amount. Then use the amount to answer whatever was really asked. Ingredients that are never mentioned in a condition, like the almonds here, can be ignored.

Try the answer choices backwards. The cashews in the box are $2 + 0.4x$. For choice (B), $2 + 0.4x = 4$ gives $x = 5$. Then the peanuts are $5 + 0.2 \times 5 = 6$ pounds out of $15$, which is exactly $40\%$. So (B) works. (For (D), $2 + 0.4x = 5$ gives $x = 7.5$, and peanuts would be $6.5$ out of $17.5$, about $37\%$, which is wrong.)