AMC 10 · 2021 · #3

Grade 6 arithmetic

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

percentagelinear-equations-one-varratio-proportionsystems-of-equations convert-to-algebraguess-and-check ↑ Prerequisites: percentagelinear-equations-one-var

📏 Short solution 💡 2 insights

Problem

An after-school program has 28 juniors and seniors total. The debate team has the same number of juniors as seniors. That same number is 25% of all juniors and 10% of all seniors. Find how many juniors are in the program.

Pick an answer.

(A)

(B)

(C)

(D)

~11

(E)

~20

AMC 10 2021 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Eliminate Possibilities

Tool #3 (Eliminate) is fast on multiple choice: juniors must be a multiple of 4 (so 25% is whole) AND total juniors ≤ 28. Only (C) 8 and (E) 20 are multiples of 4. Tool #6 (Guess and Check) tests each: if juniors = 8, then 25% = 2 on the team, so seniors on team also = 2, and seniors = 20 (since 10% of 20 = 2). Total = 8 + 20 = 28 ✓. If juniors = 20, then 25% = 5, seniors on team = 5, seniors = 50, total = 70 ≠ 28 ✗. Tool #7 (Subproblems) provides the structural backbone — "same team size" links the two percentages.

1STEP 1

25% of the juniors must be a whole number, so the junior count is a multiple of 4 — only 8 and 20 survive.

juniors ∈ {8, 20}

2STEP 2

Test juniors = 8: the team has $\frac{1}{4}$ × 8 = 2 juniors, so 2 seniors, making all seniors = 20.

juniors on team =

\frac{1}{4}

× 8 = 2 → seniors = 2 ÷

\frac{1}{10}

= 20

3STEP 3

Check the total: 8 + 20 = 28 matches the given count exactly, so juniors = 8 works.

8 + 20 = 28 ✓

4STEP 4

If juniors = 20, the team has 5 juniors, so seniors = 50 and the total is 70, not 28 — rejected.

20 + 50 = 70 ≠ 28 → juniors ≠ 20

5STEP 5

Only juniors = 8 survives — that is choice (C).

juniors = 8 → (C)

Answer

Verify the whole story: 8 juniors, 20 seniors, total 28 ✓. Team size: 25% × 8 = 2 juniors and 10% × 20 = 2 seniors — equal as required ✓. All counts are whole people. Every condition checks.

💡Key takeaway

This AMC 10 problem only needs Grade 6 percent reasoning — once you notice juniors must be a multiple of 4, just test 8: a team of 2 juniors equals 2 seniors, so seniors = 20 and the total is 28 exactly!

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