AMC 10 · 2005 · #1

Grade 6 arithmetic
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Percent / Fraction Recovery
percentagelinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: percentagefraction-arithmetic
📏 Short solution 💡 1 insight
Problem
While eating out, Mike and Joe each left the server a tip of 2 dollars. Mike's tip was 10% of his own bill, and Joe's tip was 20% of his own bill. What was the difference, in dollars, between their two bills?

Pick an answer.

(A)
2
(B)
4
(C)
5
(D)
10
(E)
20

AMC 10 2005 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

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 dividing2 by 0.10 must give a bill in dollars. Once both bills are known, the answer is one subtraction.

1STEP 1

Name the two unknowns

Let M be Mike's bill and J be Joe's bill, in dollars, then turn each tip sentence into an equation.

0.10 M = 2 0.20 J = 2
2STEP 2

Solve Mike's equation

Divide both sides by 0.10: if 2 dollars is a tenth of Mike's bill, his bill is ten times that, 20 dollars.

M = 2/0.10 = 20
3STEP 3

Solve Joe's equation

Do the same for Joe: divide by 0.20, so 2 dollars is a fifth of his bill and the bill is 10 dollars.

J = 2/0.20 = 10
4STEP 4

Subtract the two bills

Subtract the bills: Mike's 20 dollars minus Joe's 10 dollars leaves a difference of 10 dollars.

M - J = 20 - 10 = 10 → (D)
Answer
10
Check the tips against the found bills: 10% of 20 dollars is 2 dollars and 20% of 10 dollars is 2 dollars, 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.
💡Key takeaway

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.

  • Name the two unknowns
  • Solve Mike's equation
  • Solve Joe's equation
  • Subtract the two bills

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