A semipro baseball league has teams with 21 players each. League rules state that a player must be paid at least 15,000andthatthetotalofallplayers′salariesforeachteamcannotexceed700,000. What is the maximum possible salary, in dollars, for a single player?
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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
#14 Extreme Principle 4.OA.A.3Step 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.
$$\text{one player} = 700{,}000 - (\text{salaries of the other } 20)$$
💡 When a total is fixed, one part grows the most when every other part shrinks to its limit.
#8 Analyze the Units 5.NBT.B.5Step 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.
$$20 \times 15{,}000 = 300{,}000$$
💡 Twenty equal groups of $15,000 is just repeated addition sped up by multiplication.
#13 Convert to Algebra 4.NBT.B.4Step 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).
$$700{,}000 - 300{,}000 = 400{,}000$$
💡 The leftover budget after covering everyone else is exactly what one player can claim.
[1]
#14 4.OA.A.3The 21 salaries share a fixed cap of $700,000. To make one player's salary as la
[2]
#8 5.NBT.B.5Twenty players each at the $15,000 minimum cost 20 times $15,000. Counting in do
[3]
#13 4.NBT.B.4Subtract 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.3 Solve 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.5 Fluently multiply multi-digit whole numbers (Computing 20 times $15,000 to get the floor cost of the other players.)
4.NBT.B.4 Fluently 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.