AMC 10 · 2003 · #2
Easy mode Grade 5A pair of socks costs \textdollar4. A T-shirt costs \textdollar5 more than a pair of socks. Every member of a soccer team buys two pairs of socks and two T-shirts — one set for home games and one set for away games. Altogether the team spent \textdollar2366. How many members are on the team?
Pick an answer.
AMC 10 2003 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: Socks cost $\$4$ a pair and a T-shirt costs $\$5$ more than a pair of socks. Every member 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.
Givens: A pair of socks costs $\$4$; A T-shirt costs $\$5$ more than a pair of socks; Each member needs socks and a shirt for home AND socks and a shirt for away; The total spent by all members is $\$2366$
Unknowns: The number of members in the league
Understand
Restated: Socks cost $\$4$ a pair and a T-shirt costs $\$5$ more than a pair of socks. Every member 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.
Givens: A pair of socks costs $\$4$; A T-shirt costs $\$5$ more than a pair of socks; Each member needs socks and a shirt for home AND socks and a shirt for away; The total spent by all members is $\$2366$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units, #13 Convert to Algebra
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 inside $\$2366$. 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.
Execute — Answer: B
2.OA.A.1 Step 1 Price one T-shirt
- A T-shirt costs $\$5$ more than a pair of socks, and a pair of socks is $\$4$.
- Add the extra $\$5$ to the sock price to get the shirt price.
💡 "$\$5$ more than" just means add $\$5$ to what the socks cost.
3.OA.D.8 Step 2 Add up one member's bill
- Each member plays home and away, so they buy two pairs of socks and two T-shirts, not one of each.
- Two pairs of socks cost $2 \times 4 = 8$ dollars, and two shirts cost $2 \times 9 = 18$ dollars.
- Add those together for what a single member spends.
💡 Home plus away means everything is bought twice, so one member's cost is two socks and two shirts.
5.NBT.B.6 Step 3 Count how many members fit in $2366
- Every member spends the same $\$26$, so the number of members is the total divided by one member's bill. Divide $\$2366$ by $\$26$. It comes out even at $91$, which confirms the whole thing was built from identical $\$26$ chunks.
- That is choice (B).
💡 If every member's bill is the same size, the count is just the total split into equal pieces.
2.OA.A.1 A T-shirt costs $\$5$ more than a pair of socks, and a pair of socks is $\$4$. A 3.OA.D.8 Each member plays home and away, so they buy two pairs of socks and two T-shirts 5.NBT.B.6 Every member spends the same $\$26$, so the number of members is the total divid Review
Reasonableness: Check that $91$ members really cost $\$2366$: $91 \times 26 = 91\times 26 = 2366$, 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.
Alternative: Set up algebra: let $n$ be the number of members. Each member spends $26$ dollars, so $26n = 2366$. Solving gives $n = 2366/26 = 91$, choice (B). Same arithmetic, written as one equation.
CCSS standards used (min grade 5)
2.OA.A.1Solve one- and two-step word problems using addition and subtraction within 100 (Turning "$\$5$ more than a pair of socks" into $4+5=9$ for the T-shirt price.)3.OA.D.8Solve two-step word problems using four operations within 100 (Combining two pairs of socks and two shirts into one member's bill, $2\times4 + 2\times9 = 26$.)5.NBT.B.6Find whole-number quotients with up to four-digit dividends and two-digit divisors (Dividing the $\$2366$ total by the $\$26$ per-member cost to get $91$ members.)
⭐ Find what one person spends first, then divide the grand total by it to see how many people there were.
⭐ Find what one person spends first, then divide the grand total by it to see how many people there were.
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