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2025 AMC 10A Problem 2

2025 AMC 10A Problem 2AMC 10 Step by Step

Name: Date: October 28, 2026

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

A box contains $10$ pounds of a nut mix that is $50$ percent peanuts, $20$ percent cashews, and $30$ percent almonds. A second nut mix containing $20$ percent peanuts, $40$ percent cashews, and $40$ percent almonds is added to the box resulting in a new nut mix that is $40$ percent peanuts. How many pounds of cashews are now in the box?

A) $3.5$
B) $4$
C) $4.5$
D) $5$
E) $6$

Answer

Key insight

Follow only the peanuts. The peanut pounds must be 40 percent of the new total weight; that tells how much mix was added, and then count the cashews.

Solution

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.)

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 $x = 5$ .

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

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

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

Why this works

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.

Alternative approach

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.)

The trap

Finding that 5 pounds of the second mix were added and answering 5, choice (D), instead of going on to count the cashews.

Concise solution

The $10$ pounds already in the box consist of $5$ pounds of peanuts, $2$ pounds of cashews and $3$ pounds of almonds.

Let $x$ be the number of pounds of the second mix added. It brings $0.2x$ pounds of peanuts and $0.4x$ pounds of cashews, and the box now holds $10 + x$ pounds in total. The peanut condition says
$\frac{5 + 0.2x}{10 + x} = 0.4 \implies 5 + 0.2x = 4 + 0.4x \implies 1 = 0.2x \implies x = 5.$

Now return to the cashews: the box holds
$2 + 0.4(5) = 2 + 2 = 4$
pounds of them.

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

Review in the AoPS books

Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.1, §7.2, §7.3)
Prealgebra, Chapter 13: Data and Statistics (§13.1)

Techniques

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