AMC 10 · 2013 · #5
Grade 4 arithmeticPick an answer.
AMC 10 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Tool #15 is the move that unlocks this: instead of tracking who hands what to whom, rewrite each person as a single ledger line — money paid plus money given minus money received — and require all three lines to equal one common fair share. Tool #7 then splits the work into three small computations: total the costs, divide by 3, and measure each person's gap from the share. Tool #8 keeps every quantity in dollars, so a gap and a transfer are directly comparable. Tool #3 matches the final number to one choice.
Total what the trip cost
The transfers only move dollars between friends, so the trip's total is already fixed: 105+125+175 = 405.
Money passed between friends never leaves the group, so the total spent is fixed before any settling up.
Money passed between friends never leaves the group, so the total spent is fixed before any settling up.
▸ Why?
What one hands over the other receives, so the two changes cancel exactly.
▸ Why?
The total is exactly what each person paid added together, whoever ends up owing whom.
Find the one fair share
Split that evenly among three: each friend must end up out of pocket 405 ÷ 3 = 135, a whole number of dollars.
"Everyone pays the same" turns the whole story into a single target number each person must hit.
4.NBT.B.6Identify SubproblemsRead Tom's and Dorothy's ledgers
Tom only gives, so 105+t = 135 forces t = 30, and Dorothy's line 125+d = 135 forces d = 10.
Someone who underpaid owes exactly the distance between what they paid and the fair share.
4.OA.A.3Organize Information In More WaysCheck Sammy's ledger closes
Both transfers came out positive, and Sammy's line closes too: 175-(30+10) = 135, the very same share.
A settlement is only real if the person receiving the money also lands on the fair share, not just the people paying.
4.OA.A.3Organize Information In More WaysSubtract the two transfers
The question wants the difference of the two amounts sent to Sammy: t-d = 30-10 = 20, choice (B).
Once both transfers are known numbers, the answer is one subtraction away.
3.NBT.A.2Eliminate PossibilitiesTo settle a shared bill, find the total, divide it by the number of people, and have everyone pay the distance between what they already paid and that share — then check the person collecting the money lands on the share too.
- Total what the trip cost
- Find the one fair share
- Read Tom's and Dorothy's ledgers
- Check Sammy's ledger closes
- Subtract the two transfers