AMC 10 · 2005 · #1
Grade 6 arithmeticWhile eating out, Mike and Joe each tipped their server $2. Mike tipped 10% of his bill and Joe tipped 20% of his bill. What was the difference, in dollars, between their bills?
Pick an answer.
AMC 10 2005 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: Mike and Joe each left a $\$2$ tip. Mike's tip was $10\%$ of his bill and Joe's tip was $20\%$ of his bill. Find how much more one bill was than the other.
Givens: Both tips are the same size: $\$2$ each; Mike's $\$2$ tip equals $10\%$ of Mike's bill; Joe's $\$2$ tip equals $20\%$ of Joe's bill; Answer choices: (A) $2$, (B) $4$, (C) $5$, (D) $10$, (E) $20$
Unknowns: Mike's bill amount; Joe's bill amount; The difference between the two bills
Understand
Restated: Mike and Joe each left a $\$2$ tip. Mike's tip was $10\%$ of his bill and Joe's tip was $20\%$ of his bill. Find how much more one bill was than the other.
Givens: Both tips are the same size: $\$2$ each; Mike's $\$2$ tip equals $10\%$ of Mike's bill; Joe's $\$2$ tip equals $20\%$ of Joe's bill; Answer choices: (A) $2$, (B) $4$, (C) $5$, (D) $10$, (E) $20$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #8 Analyze the Units
Each sentence 'the $\$2$ tip is a certain percent of the bill' is really a one-step equation waiting to be written. Tool #4 names each unknown bill and turns the two sentences into $0.10M = 2$ and $0.20J = 2$; solving each for the bill is a single division. Tool #8 keeps the dollars-and-percent bookkeeping honest: $10\%$ means $0.10$, so dividing $\$2$ by $0.10$ must give a bill in dollars. Once both bills are known, the answer is one subtraction.
Execute — Answer: D
6.EE.B.7 Step 1 - Name the two unknowns.
- Let $M$ be Mike's bill and $J$ be Joe's bill, both in dollars.
- Turn each tip sentence into an equation: a percent of the bill equals the $\$2$ tip.
💡 "$10\%$ of the bill is $\$2$" is just $0.10 \times \text{bill} = 2$ written in symbols.
6.RP.A.3 Step 2 - Solve Mike's equation.
- Undo the multiplication by $0.10$ by dividing both sides by $0.10$.
- Dividing $\$2$ by $0.10$ is the same as asking "$\$2$ is a tenth of what?"
💡 If $\$2$ is only a tenth of the bill, the whole bill is ten times as big: $\$20$.
6.RP.A.3 Step 3 - Solve Joe's equation the same way, dividing by $0.20$.
- Here $\$2$ is a fifth of the bill, so the bill is five times $\$2$.
💡 A bigger percent ($20\%$) reaching the same $\$2$ means a smaller bill — half of Mike's.
4.NBT.B.4 Step 4 Subtract the two bills to get the difference the problem asks for.
💡 Mike paid $\$20$ and Joe paid $\$10$, so Mike's bill was $\$10$ larger.
6.EE.B.7 Name the two unknowns. Let $M$ be Mike's bill and $J$ be Joe's bill, both in dol 6.RP.A.3 Solve Mike's equation. Undo the multiplication by $0.10$ by dividing both sides 6.RP.A.3 Solve Joe's equation the same way, dividing by $0.20$. Here $\$2$ is a fifth of 4.NBT.B.4 Subtract the two bills to get the difference the problem asks for. Review
Reasonableness: Check the tips against the found bills: $10\%$ of $\$20$ is $\$2$ and $20\%$ of $\$10$ is $\$2$, matching the story exactly. The equal tips force the $10\%$ bill to be double the $20\%$ bill, so Mike's must be the larger — and $\$20 - \$10 = \$10$ lands on choice (D). The tempting wrong answer $\$20$ (E) is Mike's bill alone, not the difference.
Alternative: Tool #11 (Work Backwards) without algebra: a $\$2$ tip that is $10\%$ of the bill means the bill is ten $\$2$-slices, so $\$20$; a $\$2$ tip that is $20\%$ is five $\$2$-slices, so $\$10$. Difference $\$10$. Same answer, reasoning straight from the meaning of percent.
CCSS standards used (min grade 6)
6.EE.B.7Solve real-world problems by writing and solving equations of the form px = q (Translating each tip sentence into a one-step equation, $0.10M = 2$ and $0.20J = 2$, and solving for the bill.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Reading a percent as a rate to recover the whole bill from a known part: dividing the $\$2$ tip by $0.10$ and by $0.20$.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Subtracting the two bills, $20 - 10$, to get the requested difference.)
⭐ When the same tip is a smaller percent of one bill, that bill is the bigger one — divide the tip by the percent to rebuild each whole bill, then subtract.
⭐ When the same tip is a smaller percent of one bill, that bill is the bigger one — divide the tip by the percent to rebuild each whole bill, then subtract.
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