AMC 10 · 2013 · #5
Grade 4 arithmeticTom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid 105,Dorothypaid125, and Sammy paid 175.Inordertosharecostsequally,TomgaveSammytdollars,andDorothygaveSammyddollars.Whatist-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 people paid different amounts on a trip: $105$, $125$, and $175$. They want to split the total cost evenly. Find how many dollars Tom gives Sammy ($t$), how many dollars Dorothy gives Sammy ($d$), and then compute $t-d$.
Givens: Tom paid $105$, Dorothy paid $125$, Sammy paid $175$; The three of them split the total cost evenly; Tom gives Sammy $t$ dollars; Dorothy gives Sammy $d$ dollars; Answer choices: (A) $15$, (B) $20$, (C) $25$, (D) $30$, (E) $35$
Unknowns: The value of $t-d$
Understand
Restated: Three people paid different amounts on a trip: $105$, $125$, and $175$. They want to split the total cost evenly. Find how many dollars Tom gives Sammy ($t$), how many dollars Dorothy gives Sammy ($d$), and then compute $t-d$.
Givens: Tom paid $105$, Dorothy paid $125$, Sammy paid $175$; The three of them split the total cost evenly; Tom gives Sammy $t$ dollars; Dorothy gives Sammy $d$ dollars; Answer choices: (A) $15$, (B) $20$, (C) $25$, (D) $30$, (E) $35$
Plan
Primary tool: #7 Identify Subproblems
Secondary: #8 Analyze the Units
The question hides three small jobs inside one sentence, so Tool #7 (Identify Subproblems) is the natural fit: first add up the total bill, then divide it into three equal shares, then see how far below the share Tom and Dorothy each landed. Those shortfalls are exactly $t$ and $d$. Tool #8 (Analyze the Units) keeps every number in dollars and confirms the fair share sits between the smallest and largest payment, which is a quick sanity guard against arithmetic slips.
Execute — Answer: B
3.NBT.A.2 Step 1 Add up the total cost
- First find the whole bill the three people paid together.
- Add the three payments: $105+125+175$.
- Line up the dollars and add: $105+125=230$, then $230+175=405$.
- The trip cost $405$ in total.
💡 Splitting a bill fairly starts with knowing the one number everyone is sharing.
4.NBT.B.6 Step 2 Find each person's fair share
- Split the $405$ evenly among the three friends by dividing by $3$: $405 \div 3 = 135$.
- So a fair share is $135$ each.
- Notice $135$ sits between the smallest payment $105$ and the largest $175$, which is what you expect for an average.
💡 An equal split means every person should end up having paid the same amount: the total shared out into three.
3.NBT.A.2 Step 3 Find how much Tom and Dorothy each still owe
- Compare what Tom and Dorothy actually paid against the fair share of $135$.
- Tom paid only $105$, so he is short by $135-105=30$; that shortfall is the money he hands to Sammy, so $t=30$.
- Dorothy paid $125$, so she is short by $135-125=10$, giving $d=10$.
- Both send their shortfall to Sammy, who overpaid.
💡 If you paid less than your fair share, you owe exactly the gap between what you paid and the share.
3.NBT.A.2 Step 4 Compute the difference $t-d$
- Subtract the two amounts the question asks about: $t-d = 30-10 = 20$.
- That matches choice (B).
- As a check, Sammy overpaid by $175-135=40$, and the money coming in is $t+d=30+10=40$, so everything balances.
💡 Once you know both gifts to Sammy, the answer is just one more subtraction.
3.NBT.A.2 First find the whole bill the three people paid together. Add the three payments 4.NBT.B.6 Split the $405$ evenly among the three friends by dividing by $3$: $405 \div 3 = 3.NBT.A.2 Compare what Tom and Dorothy actually paid against the fair share of $135$. Tom 3.NBT.A.2 Subtract the two amounts the question asks about: $t-d = 30-10 = 20$. That match Review
Reasonableness: The fair share $135$ lies between the smallest payment $105$ and the largest $175$, exactly as an average should. The two gifts, $t=30$ and $d=10$, add to $40$, which equals Sammy's overpayment $175-135=40$, so no money is created or lost. The difference $t-d=20$ is choice (B), a mid-range value that fits the size of the numbers.
Alternative: Skip finding the share entirely. Since $t=135-105$ and $d=135-125$, the shared $135$ cancels when you subtract: $t-d=(135-105)-(135-125)=125-105=20$. This shows $t-d$ is just the gap between what Dorothy and Tom paid, and lands on (B) without ever computing the average.
CCSS standards used (min grade 4)
3.NBT.A.2Fluently add and subtract within 1000 (Adding the payments to get $405$ and subtracting to find $t=30$, $d=10$, and $t-d=20$.)4.NBT.B.6Find whole-number quotients with multi-digit dividends (Dividing the total $405$ by $3$ to get each fair share of $135$.)
⭐ To split a bill fairly, find the total, divide it into equal shares, then each person pays the gap between what they owe and what they already paid.
⭐ To split a bill fairly, find the total, divide it into equal shares, then each person pays the gap between what they owe and what they already paid.
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