AMC 10 · 2008 · #3
Easy mode Grade 5A baseball team has 21 players. Every player must be paid at least 15,000 dollars. The salaries of all 21 players added together cannot go over 700,000 dollars. What is the largest salary, in dollars, that one player could get?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A team has 21 players. Every player must be paid at least $15,000, and the salaries of all 21 players added together cannot go over $700,000. We want the largest salary one single player could receive.
Givens: Each team has 21 players.; Every player is paid at least $15,000.; The sum of all 21 salaries is at most $700,000.
Unknowns: The maximum possible salary for one single player.
Understand
Restated: A team has 21 players. Every player must be paid at least $15,000, and the salaries of all 21 players added together cannot go over $700,000. We want the largest salary one single player could receive.
Givens: Each team has 21 players.; Every player is paid at least $15,000.; The sum of all 21 salaries is at most $700,000.
Plan
Primary tool: #14 Extreme Principle
Secondary: #8 Analyze the Units, #13 Convert to Algebra
The $700,000 cap is shared among all 21 players. To hand one player as much as possible, push everyone else to the smallest amount the rules allow. Once the other 20 sit at the floor, whatever is left of the cap belongs to the last player.
Execute — Answer: C
4.OA.A.3 Step 1 Push the others to the minimum
The 21 salaries share a fixed cap of $700,000. To make one player's salary as large as possible, the other 20 players should each take the smallest salary the rules permit, which is $15,000.
💡 When a total is fixed, one part grows the most when every other part shrinks to its limit.
5.NBT.B.5 Step 2 Cost of the other 20 players
- Twenty players each at the $15,000 minimum cost 20 times $15,000.
- Counting in dollars, that floor cost is $300,000.
💡 Twenty equal groups of $15,000 is just repeated addition sped up by multiplication.
4.NBT.B.4 Step 3 Whatever is left goes to one player
- Subtract the $300,000 spent on the other 20 from the $700,000 cap.
- The remainder, $400,000, is the most that single player can be paid, so the answer is (C).
💡 The leftover budget after covering everyone else is exactly what one player can claim.
4.OA.A.3 The 21 salaries share a fixed cap of $700,000. To make one player's salary as la 5.NBT.B.5 Twenty players each at the $15,000 minimum cost 20 times $15,000. Counting in do 4.NBT.B.4 Subtract the $300,000 spent on the other 20 from the $700,000 cap. The remainder Review
Reasonableness: Check the total: one player at $400,000 plus twenty players at $15,000 gives 400,000 + 300,000 = 700,000, exactly the cap and not over it. Every salary is at least $15,000, so all rules hold. Choice (E) $700,000 fails because it leaves nothing for the other 20 players, who still each need $15,000.
Alternative: Use a variable: let the top salary be M. The other 20 need at least 20 times $15,000 = $300,000, so M + 300,000 must be at most 700,000, giving M at most 400,000. The largest M is $400,000.
CCSS standards used (min grade 5)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Turning the max-one/min-others idea into a subtraction from the shared cap.)5.NBT.B.5Fluently multiply multi-digit whole numbers (Computing 20 times $15,000 to get the floor cost of the other players.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Subtracting $300,000 from the $700,000 cap to find the leftover salary.)
⭐ When a total is capped, make one share the biggest by shrinking every other share to its minimum.
⭐ When a total is capped, make one share the biggest by shrinking every other share to its minimum.
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