AMC 10 · 2025 · #2

Grade 8 rate-ratio

Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery

percentageweighted-averagelinear-equations-one-var convert-to-algebra ↑ Prerequisites: percentage

📏 Medium solution 💡 1 insight

Problem

Box 1 holds 10 pounds of a nut mix that is 50% peanuts, 20% cashews, and 30% almonds. Box 2 is a different mix — 20% peanuts, 40% cashews, 40% almonds — but its weight is not given. When everything is poured together, the combined mix is 40% peanuts. Find the total pounds of cashews in the combined box.

Pick an answer.

(A)

3.5

(B)

(C)

4.5

(D)

(E)

How to solve

Strategy Introduce a Variable

The one thing blocking a direct count is that Box 2's weight is missing, so Tool #4 (Introduce a Variable) names it x and turns the single fact we are given — the final mix is 40% peanuts — into an equation that fixes x. Tool #8 (Analyze the Units) keeps every percent honest: a percent times a box's pounds gives pounds of that nut, so amounts add across boxes while percents do not. Tool #7 (Identify Subproblems) splits the work into two clean stages — first solve for x using peanuts, then use that x to total the cashews — because the cashew count is impossible until the weight is known.

1STEP 1

Turn percents into pounds

A percent is a weight only after multiplying by the box's pounds: Box 1 peanuts=0.50×10=5 lb, Box 2 peanuts=0.20 of its unknown weight.

Box 1 peanuts=0.50×10=5 lb, Box 2 peanuts=0.20×(Box 2 weight)

2STEP 2

Name the weight and balance the peanuts

Let x be Box 2's weight, so the combined box is 10+x lb; the 5+0.20x lb of peanuts must be 40% of it, giving one equation in x.

(5+0.20x)/(10+x)=0.40

3STEP 3

Solve for the missing weight

Clear the fraction: 5+0.20x=4+0.40x, so 1=0.20x and x=5 — Box 2 weighs 5 pounds.

5+0.20x=4+0.40x → 1=0.20x → x=5

4STEP 4

Total the cashews

With x=5, cashews are Box 1's 0.20×10=2 lb plus Box 2's 0.40×5=2 lb, so 2+2=4 lb — choice (B); check: peanuts $\frac{6}{15}$ =40%.

0.20×10+0.40×5=2+2=4 lb → (B)

Answer

Cashews sit between the two boxes' cashew rates (20% and 40%), so the combined cashew percentage lands strictly between them; with 15 total pounds, 4 pounds of cashews is 26.7%, comfortably inside that band. The answer must also be less than the almond total (3+2=5 lb) yet more than the raw 2 lb from Box 1 alone, and 4 lb fits. Choice (D) 5 is the trap of reporting Box 2's weight instead of the cashews, and (E) 6 is the peanut total — both are numbers that appear on the way but answer the wrong question.

💡Key takeaway

When a mix's total weight is missing, let the one given percentage of the blend become an equation to find that weight — then every ingredient is just its percent times its pounds, added up.

Turn percents into pounds
Name the weight and balance the peanuts
Solve for the missing weight
Total the cashews

Turn percents into pounds

Name the weight and balance the peanuts

Solve for the missing weight

Total the cashews

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