AMC 10 · 2013 · #5
Easy mode Grade 4Tom, Dorothy, and Sammy took a trip together and agreed to split the total cost evenly. Tom paid $105, Dorothy paid $125, and Sammy paid $175. To make everyone's share equal, Tom gave Sammy t dollars and Dorothy gave Sammy d dollars. What is t−d?
Pick an answer.
AMC 10 2013 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: Three friends share a trip's costs equally, but they paid unequal amounts up front: $\$105$, $\$125$, and $\$175$. Afterwards Tom hands Sammy $t$ dollars and Dorothy hands Sammy $d$ dollars so that everyone ends up having paid the same. Find $t-d$.
Givens: Tom paid $\$105$, Dorothy paid $\$125$, Sammy paid $\$175$; The three agreed to split the total cost evenly; Tom's only transaction afterwards is giving Sammy $t$ dollars; Dorothy's only transaction afterwards is giving Sammy $d$ dollars; Choices: (A) $15$, (B) $20$, (C) $25$, (D) $30$, (E) $35$
Unknowns: The value of $t-d$
Understand
Restated: Three friends share a trip's costs equally, but they paid unequal amounts up front: $\$105$, $\$125$, and $\$175$. Afterwards Tom hands Sammy $t$ dollars and Dorothy hands Sammy $d$ dollars so that everyone ends up having paid the same. Find $t-d$.
Givens: Tom paid $\$105$, Dorothy paid $\$125$, Sammy paid $\$175$; The three agreed to split the total cost evenly; Tom's only transaction afterwards is giving Sammy $t$ dollars; Dorothy's only transaction afterwards is giving Sammy $d$ dollars; Choices: (A) $15$, (B) $20$, (C) $25$, (D) $30$, (E) $35$
Plan
Primary tool: #15 Organize Information in More Ways
Secondary: #7 Identify Subproblems, #8 Analyze the Units, #3 Eliminate Possibilities
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.
Execute — Answer: B
3.NBT.A.2 Step 1 Total what the trip cost
- The transfers only move dollars from one friend to another, so they cannot change how much the trip cost in total.
- Add the three payments: $105+125=230$, then $230+175=405$.
- The trip cost $\$405$.
💡 Money passed between friends never leaves the group, so the total spent is fixed before any settling up.
4.NBT.B.6 Step 2 Find the one fair share
Splitting $\$405$ evenly among three people means each person should be out of pocket $405 \div 3 = 135$ dollars. Note $405$ is divisible by $3$ (its digits sum to $4+0+5=9$), so the fair share is a whole number of dollars and no one has to split a cent.
💡 "Everyone pays the same" turns the whole story into a single target number each person must hit.
4.OA.A.3 Step 3 Read Tom's and Dorothy's ledgers
- Tom gives money and receives none, so his final out-of-pocket amount is $105+t$, and it must equal $135$.
- That forces $t$ to be exactly the gap $135-105=30$.
- Dorothy's line is $125+d=135$, forcing $d=135-125=10$.
- Because each of them makes exactly one transaction, these values are not just possible — they are the only values that work.
💡 Someone who underpaid owes exactly the distance between what they paid and the fair share.
4.OA.A.3 Step 4 Check Sammy's ledger closes
- The two equations above only say what Tom and Dorothy must pay; they do not by themselves prove the described settlement is possible.
- Two things have to hold.
- First, $t=30$ and $d=10$ are both positive, so "gave Sammy" is literally true for each of them — neither is secretly owed money.
- Second, Sammy's own line must land on $135$ too: he paid $175$ and receives $t+d=30+10=40$, so his final out-of-pocket amount is $175-40=135$.
- It does.
- So one round of these two transfers really does settle the trip.
💡 A settlement is only real if the person receiving the money also lands on the fair share, not just the people paying.
3.NBT.A.2 Step 5 Subtract the two transfers
- The question asks for the difference of the two amounts sent to Sammy: $t-d = 30-10 = 20$.
- That is choice (B).
💡 Once both transfers are known numbers, the answer is one subtraction away.
3.NBT.A.2 The transfers only move dollars from one friend to another, so they cannot chang 4.NBT.B.6 Splitting $\$405$ evenly among three people means each person should be out of p 4.OA.A.3 Tom gives money and receives none, so his final out-of-pocket amount is $105+t$, 4.OA.A.3 The two equations above only say what Tom and Dorothy must pay; they do not by t 3.NBT.A.2 The question asks for the difference of the two amounts sent to Sammy: $t-d = 30 Review
Reasonableness: Three independent checks agree. (1) The fair share $135$ sits between the smallest payment $105$ and the largest $175$, exactly where an average must sit. (2) Money is conserved: the amounts leaving Tom and Dorothy, $30+10=40$, equal the amount $175-135=40$ that Sammy overpaid, so nothing is created or lost. (3) Every final ledger lands on the same number: $105+30=135$, $125+10=135$, $175-40=135$. The requested difference $t-d=20$ is choice (B), and it is smaller than $t=30$ alone, which matches the choice list.
Alternative: A faster, structurally different route skips the fair share entirely. Whatever the common share $S$ is, Tom's ledger gives $t = S-105$ and Dorothy's gives $d = S-125$. Subtracting, the unknown $S$ cancels: $t-d = (S-105)-(S-125) = 125-105 = 20$. So $t-d$ is just the gap between what Dorothy and Tom paid, and Sammy's $\$175$ is never used. That is elegant, but it is worth knowing what it does and does not prove: cancelling $S$ shows the difference is forced, yet it never checks that the story described is actually realizable. If Dorothy had paid more than the share, the same subtraction would still return $20$ while "Dorothy gave Sammy $d$ dollars" would require a negative $d$. Here the main solution verified both gaps are positive and that Sammy's ledger closes at $135$, so the shortcut's answer is the answer to the question that was actually asked.
CCSS standards used (min grade 4)
3.NBT.A.2Fluently add and subtract within 1000 (Adding $105+125+175=405$ and subtracting to get the gaps $30$ and $10$ and the final difference $20$.)4.NBT.B.6Find whole-number quotients and remainders with up to four-digit dividends (Dividing the total $405$ by $3$ to get the fair share $135$.)4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Turning each person into a ledger line, solving for $t$ and $d$, and checking that Sammy's line also closes at $135$.)
⭐ To 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.
⭐ To 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.
More like this
Same archetype — closest grade level first.