AMC 10 · 2003 · #2

Grade 5 arithmetic
Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation
linear-equations-one-varmulti-digit-arithmetic convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmetic
📏 Medium solution 💡 2 insights
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Problem
Socks cost $4 a pair, and a T-shirt costs $5 more than a pair of socks. Every member of the league buys one pair of socks and one shirt for home games, and one more pair and one more shirt for away games. The whole league spent $2366. Find how many members bought this gear.

Pick an answer.

(A)
77
(B)
91
(C)
143
(D)
182
(E)
286

AMC 10 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The 2366 total looks scary until you split the job into small parts. Tool #7 (Identify Subproblems) breaks it into three easy steps: first find the T-shirt price, then find what ONE member spends, then see how many of those equal-size member bills fit inside2366. Tool #8 (Analyze the Units) keeps the dollars straight and stops the classic trap of buying only one pair of socks and one shirt instead of two of each. Once one member's bill is a single number, the last step is just a division.

1STEP 1

Price one T-shirt

Socks are $4 a pair and a shirt costs $5 more, so one T-shirt is 4+5 = $9.

4 + 5 = 9 dollars per T-shirt
2STEP 2

Add up one member's bill

Home and away means two pairs of socks and two shirts, so one member pays 2×4 + 2×9 = $26.

2 × 4 + 2 × 9 = 8 + 18 = 26 dollars per member
3STEP 3

Count how many members fit in $2366

Every member spends the same $26, so divide the total: 2366 ÷ 26 = 91 members, choice (B).

2366 ÷ 26 = 91 → (B)
Answer
91
Check that 91 members really cost $2366: 91 × 26 = 2366, an exact match with no money left over, so 91 is right. It is also easy to see the answer must be moderate in size — if you had wrongly used $13 per member (only one pair of socks and one shirt), you would get 182, which is choice (D). That is exactly the trap answer for forgetting the home-and-away doubling, so landing on half of it, 91, is a good sign the doubling was handled correctly.
💡Key takeaway

Find what one person spends first, then divide the grand total by it to see how many people there were.

  • Price one T-shirt
  • Add up one member's bill
  • Count how many members fit in $2366

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