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 💡 2 insights
Problem
A 10-pound nut mix that is 50% peanuts, 20% cashews, 30% almonds is combined with an unknown amount of a second mix that is 20% peanuts, 40% cashews, 40% almonds. The combined box turns out to be 40% peanuts. Find the total pounds of cashews in the combined box.

Pick an answer.

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

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The one thing blocking the count is the missing weight of Box 2, so Tool #4 (Introduce a Variable) names it x and lets every peanut and cashew amount be written in terms of it. Tool #8 (Analyze the Units) keeps the bookkeeping straight: a percent times a weight in pounds gives pounds of that nut. Tool #13 (Convert to Algebra) turns the lone numerical clue — the final mix is 40% peanuts — into an equation that pins down x, after which the cashews are a direct count.

1STEP 1

Name the missing amount

The only unknown is how much of Box 2 goes in — call it x pounds, so the combined box weighs 10+x pounds.

x = pounds of Box 2, combined weight = 10+x
2STEP 2

Count the peanuts in each box

Percent times weight gives pounds: Box 1 yields 0.50×10=5 pounds of peanuts and Box 2 yields 0.2x, so together 5+0.2x.

0.50×10=5, 0.20 × x=0.2x, peanuts=5+0.2x
3STEP 3

Turn the 40% clue into an equation

The blend is 40% peanuts, so 5+0.2x=0.4(10+x)=4+0.4x, leaving 1=0.2x and x=5 pounds of Box 2.

(5+0.2x)/(10+x)=0.40 → 5+0.2x=4+0.4x → 1=0.2x → x=5
4STEP 4

Add the cashews from both boxes

Count cashews the same way: 0.20×10=2 pounds from Box 1 and 0.40×5=2 from Box 2, so the box holds 4 pounds of cashews.

0.20×10+0.40×5=2+2=4 → (B)
Answer
4
Check the peanut condition: the combined box weighs 10+5=15 pounds and holds 5+0.2×5=6 pounds of peanuts, and 6/15=0.40, exactly the 40% required. The 4 pounds of cashews are 4/15≈27% of the mix, which sensibly lands between the two boxes' cashew rates of 20% and 40%. The tempting wrong pick 5 (choice D) is really the Box 2 weight, and mixing that up is the intended trap.
💡Key takeaway

When a problem hides an amount, give it a name, use the one percentage fact to pin it down, then count what the question actually asks for.

  • Name the missing amount
  • Count the peanuts in each box
  • Turn the 40% clue into an equation
  • Add the cashews from both boxes

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