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

Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $2366, how many members are in the League?

Pick an answer.

(A)

(B)

(C)

143

(D)

182

(E)

286

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.

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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

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

#7 Identify Subproblems 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.

$$4 + 5 = 9 \text{ dollars per T-shirt}$$

💡 "$\$5$ more than" just means add $\$5$ to what the socks cost.

#7 Identify Subproblems 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.

$$2\times 4 + 2\times 9 = 8 + 18 = 26 \text{ dollars per member}$$

💡 Home plus away means everything is bought twice, so one member's cost is two socks and two shirts.

#13 Convert to Algebra 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).

$$2366 \div 26 = 91 \;\Rightarrow\; \textbf{(B)}$$

💡 If every member's bill is the same size, the count is just the total split into equal pieces.

[1] #7 2.OA.A.1 A T-shirt costs $\$5$ more than a pair of socks, and a pair of socks is $\$4$. A

[2] #7 3.OA.D.8 Each member plays home and away, so they buy two pairs of socks and two T-shirts

[3] #13 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.1 Solve 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.8 Solve 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.6 Find 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.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Price one T-shirt

Add up one member's bill

Count how many members fit in $2366

Review

CCSS standards used (min grade 5)

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