AMC 10 · 2004 · #7

Grade 7 arithmetic
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up
ratio-proportionlinear-equations-one-vardigit-sum convert-to-algebra ↑ Prerequisites: linear-equations-one-var
📏 Medium solution 💡 2 insights
Problem
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.

Pick an answer.

(A)
5
(B)
6
(C)
7
(D)
8
(E)
9

AMC 10 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Convert to Algebra

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.

1STEP 1

Turn the rate into a gain

Every 7 U.S. dollars traded returns 10 Canadian — a gain of 3 — so the whole pile grows to 107d\frac{10}{7}d Canadian dollars.

d USD ⟶ 10/7d CAD=d+3/7d
2STEP 2

Write the story as an equation

She spends 60 and lands back on dd, so the gain must be exactly 60: 107d−60=d\frac{10}{7}d-60=d collapses to 37d=60\frac{3}{7}d=60.

10/7d-60=d → 10/7d-d=60 → 3/7d=60
3STEP 3

Solve for the amount

Three sevenths of dd is 60, so one seventh is 20 and all seven sevenths give d=140d=140.

3/7d=60 → d=60·7/3=140
4STEP 4

Answer what was asked

The ask is the digit sum, not dd itself: the digits of 140 are 1, 4, 0, and 1+4+0=51+4+0=5 — choice (A).

1+4+0=5 → (A)
Answer
5
Check d=140 against the story directly. She exchanges 140 U.S. dollars at 10 Canadian dollars per 7 U.S. dollars: that is 140 ÷ 7=20 batches, giving 20 × 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.
💡Key takeaway

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.

  • Turn the rate into a gain
  • Write the story as an equation
  • Solve for the amount
  • Answer what was asked

More like this

Same archetype — closest grade level first.