AMC 10 · 2004 · #7
Grade 7 arithmeticOn a trip from the United States to Canada, Isabella took d U.S. dollars. At the border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had d Canadian dollars left. What is the sum of the digits of d?
Pick an answer.
AMC 10 2004 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: Isabella carries $d$ U.S. dollars into Canada and exchanges all of them at a rate of $10$ Canadian dollars for every $7$ U.S. dollars. She spends $60$ Canadian dollars and is then left with $d$ Canadian dollars — the same number she started with. Find the sum of the digits of $d$.
Givens: She starts with $d$ U.S. dollars and exchanges every one of them.; The exchange rate is $10$ Canadian dollars for each $7$ U.S. dollars.; After spending $60$ Canadian dollars she has $d$ Canadian dollars left.; Answer choices: (A) $5$, (B) $6$, (C) $7$, (D) $8$, (E) $9$.
Unknowns: The value of $d$, and then the sum of its digits.
Understand
Restated: Isabella carries $d$ U.S. dollars into Canada and exchanges all of them at a rate of $10$ Canadian dollars for every $7$ U.S. dollars. She spends $60$ Canadian dollars and is then left with $d$ Canadian dollars — the same number she started with. Find the sum of the digits of $d$.
Givens: She starts with $d$ U.S. dollars and exchanges every one of them.; The exchange rate is $10$ Canadian dollars for each $7$ U.S. dollars.; After spending $60$ Canadian dollars she has $d$ Canadian dollars left.; Answer choices: (A) $5$, (B) $6$, (C) $7$, (D) $8$, (E) $9$.
Plan
Primary tool: #13 Convert to Algebra
Secondary: #8 Analyze the Units, #4 Introduce a Variable, #7 Identify Subproblems
This is a word problem with one unknown $d$ tangled inside a story about money, the signature trigger for Tool #13 (Convert to Algebra): turn the sentences into one equation and let the algebra do the work. Tool #8 (Analyze the Units) handles the exchange rate — tracking how $7$ U.S. dollars turn into $10$ Canadian dollars keeps the conversion honest. Tool #4 (Introduce a Variable) lets $d$ carry the unknown from the setup straight through the solve. Finally the question has two layers, so Tool #7 (Identify Subproblems) splits it: first find $d$, then answer what is actually asked — the sum of its digits.
Execute — Answer: A
6.RP.A.3 Step 1 Turn the rate into a gain
- Exchanging $7$ U.S.
- dollars gives $10$ Canadian dollars — a gain of $3$ Canadian dollars for every $7$ U.S.
- dollars traded.
- Because she exchanges all $d$ U.S.
- dollars, the count of dollars grows by the fraction $\tfrac{3}{7}$: from $d$ she ends up with $\tfrac{10}{7}d$ Canadian dollars, which is her original $d$ plus an extra $\tfrac{3}{7}d$.
💡 A fixed rate scales every dollar the same way, so the whole pile grows by the same fraction one small batch does.
7.EE.B.4 Step 2 Write the story as an equation
- After the exchange she holds $\tfrac{10}{7}d$ Canadian dollars.
- She spends $60$ and is left with $d$ Canadian dollars.
- Writing that as an equation and moving the $d$ across, the extra Canadian dollars she gained must be exactly the $60$ she spent — otherwise she could not land back on the same number $d$.
💡 If spending $60$ brings her back to her starting number, then the exchange must have handed her exactly $60$ extra.
7.NS.A.3 Step 3 Solve for the amount
- The equation $\tfrac{3}{7}d=60$ says three-sevenths of $d$ is $60$, so one-seventh is $20$ and the whole is $7\times 20=140$.
- Equivalently, multiply both sides by $\tfrac{7}{3}$.
- So Isabella started with $d=140$.
💡 Undo a fraction of a number by splitting into equal parts: if three parts make $60$, one part is $20$ and seven parts is $140$.
2.NBT.A.1 Step 4 Answer what was asked
- The question does not want $d$ itself — it wants the sum of the digits of $d$.
- The digits of $140$ are $1$, $4$, and $0$, and their sum is $1+4+0=5$.
- That matches choice (A).
💡 Read the last small task literally — the digit sum, not the number — so the right work does not get thrown away on the wrong question.
6.RP.A.3 Exchanging $7$ U.S. dollars gives $10$ Canadian dollars — a gain of $3$ Canadian 7.EE.B.4 After the exchange she holds $\tfrac{10}{7}d$ Canadian dollars. She spends $60$ 7.NS.A.3 The equation $\tfrac{3}{7}d=60$ says three-sevenths of $d$ is $60$, so one-seven 2.NBT.A.1 The question does not want $d$ itself — it wants the sum of the digits of $d$. T Review
Reasonableness: Check $d=140$ against the story directly. She exchanges $140$ U.S. dollars at $10$ CAD per $7$ USD: that is $140\div 7=20$ batches, giving $20\times 10=200$ Canadian dollars. Spending $60$ leaves $200-60=140$ Canadian dollars — exactly $d$, as required. The number is also a clean multiple of $7$, which it must be for the exchange to come out whole. The digit sum $1+4+0=5$ then lands on offered choice (A), a good sign nothing was misread.
Alternative: Count in whole batches instead of fractions. Each $7$ U.S. dollars exchanged yields $3$ Canadian dollars more than it started with. To finish with the same number after spending $60$, the total extra must be $60$, so she needs $60\div 3=20$ batches of $7$ dollars. That is $20\times 7=140$ dollars, giving $d=140$ and digit sum $5$ — no fractions required.
CCSS standards used (min grade 7)
6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Applying the $10$ CAD per $7$ USD rate to convert all $d$ U.S. dollars into $\tfrac{10}{7}d$ Canadian dollars.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Translating the exchange-and-spend story into the equation $\tfrac{3}{7}d=60$.)7.NS.A.3Solve real-world problems involving the four operations with rational numbers (Solving $\tfrac{3}{7}d=60$ for $d=140$ by multiplying by $\tfrac{7}{3}$.)2.NBT.A.1Understand that the three digits of a three-digit number represent hundreds, tens, and ones (Reading the digits $1$, $4$, $0$ of $140$ to add them into the digit sum $5$.)
⭐ A fixed exchange rate grows your money by a fixed fraction, so turn the rate into that fraction, set the extra equal to what you spent, then answer the exact question asked — here the digit sum, not the number.
⭐ A fixed exchange rate grows your money by a fixed fraction, so turn the rate into that fraction, set the extra equal to what you spent, then answer the exact question asked — here the digit sum, not the number.
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