AMC 10 · 2025 · #2
Grade 8 rate-ratioA box contains 10 pounds of a nut mix that is 50 percent peanuts, 20 percent cashews, and 30 percent almonds. A second nut mix contains 20 percent peanuts, 40 percent cashews, and 40 percent almonds.They are all added together in one box resulting in a new nut mix that has 40 percent peanuts. How many pounds of cashews are now in the box?
Pick an answer.
AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: 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.
Givens: Box 1: $10$ pounds, made of $50\%$ peanuts, $20\%$ cashews, $30\%$ almonds; Box 2: an unstated weight, made of $20\%$ peanuts, $40\%$ cashews, $40\%$ almonds; After everything is poured together, the combined mix is $40\%$ peanuts; Answer choices: (A) $3.5$, (B) $4$, (C) $4.5$, (D) $5$, (E) $6$
Unknowns: The total pounds of cashews in the combined box; How many pounds of Box 2 were added (needed along the way)
Understand
Restated: 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.
Givens: Box 1: $10$ pounds, made of $50\%$ peanuts, $20\%$ cashews, $30\%$ almonds; Box 2: an unstated weight, made of $20\%$ peanuts, $40\%$ cashews, $40\%$ almonds; After everything is poured together, the combined mix is $40\%$ peanuts; Answer choices: (A) $3.5$, (B) $4$, (C) $4.5$, (D) $5$, (E) $6$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #8 Analyze the Units, #13 Convert to Algebra
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.
Execute — Answer: B
6.EE.B.6 Step 1 Name the missing amount
- The only quantity we are not told is how much of Box 2 gets added.
- Give it a name: let $x$ be the pounds of Box 2.
- Then the combined box weighs $10+x$ pounds.
- Nothing else is unknown, so writing $x$ down unlocks the whole problem.
💡 Naming the one missing quantity lets every other amount be written in terms of it.
6.RP.A.3 Step 2 Count the peanuts in each box
- A percent of a weight is just that fraction of the pounds.
- Box 1 gives $0.50\times10=5$ pounds of peanuts.
- Box 2 gives $0.20\times x=0.2x$ pounds of peanuts.
- Together the combined box holds $5+0.2x$ pounds of peanuts.
💡 Finding a percent of a quantity means multiplying the fraction by the amount.
8.EE.C.7 Step 3 Turn the 40% clue into an equation
- The combined mix is $40\%$ peanuts, so the peanut pounds divided by the total pounds equal $0.40$.
- Multiply both sides by $10+x$ and simplify: $5+0.2x=0.4(10+x)=4+0.4x$, which gives $1=0.2x$, so $x=5$.
- Five pounds of Box 2 were added.
💡 The percent that involves the unknown is the one equation that can pin the unknown down.
6.RP.A.3 Step 4 Add the cashews from both boxes
- Now that $x=5$, count cashews the same way peanuts were counted.
- Box 1 gives $0.20\times10=2$ pounds of cashews; Box 2 gives $0.40\times5=2$ pounds.
- Adding them, the combined box holds $2+2=4$ pounds of cashews, which is choice (B).
- The trap value $5$ in (D) is the Box 2 weight $x$, not the cashew total.
💡 Cashews come from both boxes, so total them only after the Box 2 amount is known.
6.EE.B.6 The only quantity we are not told is how much of Box 2 gets added. Give it a nam 6.RP.A.3 A percent of a weight is just that fraction of the pounds. Box 1 gives $0.50\tim 8.EE.C.7 The combined mix is $40\%$ peanuts, so the peanut pounds divided by the total po 6.RP.A.3 Now that $x=5$, count cashews the same way peanuts were counted. Box 1 gives $0. Review
Reasonableness: Check the peanut condition: the combined box weighs $10+5=15$ pounds and holds $5+0.2\times5=6$ pounds of peanuts, and $6/15=0.40$, exactly the $40\%$ required. The $4$ pounds of cashews are $4/15\approx27\%$ 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.
Alternative: Use alligation on the peanut percents. Box 1 is $50\%$ and Box 2 is $20\%$, blending to $40\%$. The distances to $40$ are $50-40=10$ and $40-20=20$, so the weight ratio Box 1 : Box 2 equals $20:10=2:1$. With Box 1 at $10$ pounds, Box 2 is $5$ pounds, and the cashews are again $0.20\times10+0.40\times5=4$ pounds.
CCSS standards used (min grade 8)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Letting $x$ stand for the unknown pounds of Box 2 and writing the combined weight as $10+x$.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Finding a percent of a quantity — computing peanut and cashew pounds as a percent of each box's weight.)8.EE.C.7Solve linear equations in one variable (Solving $\frac{5+0.2x}{10+x}=0.40$, expanding and collecting like terms to get $x=5$.)
⭐ 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.
⭐ 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.
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