AMC 10 · 2012 · #4

Grade 7 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain
percentageunit-conversionratio-proportion dimensional-analysisconvert-to-algebra ↑ Prerequisites: percentage
📏 Short solution 💡 2 insights
Problem
Two people hold money in different currencies at a known exchange rate. Find the percent by which one exceeds the other.

Pick an answer.

(A)
2
(B)
4
(C)
6.5
(D)
8
(E)
13
How to solve
Strategy Analyze the Units

The numbers 400 and 500 sit in different units, so comparing them directly is meaningless. The rate 1.3 dollars per euro is a conversion factor that turns euros into dollars, and once both people's money is written in one currency the question becomes a single percent comparison. Naming the percent first fixes which amount belongs in the denominator, which is the part of this problem that is easy to get backwards.

1STEP 1

Pin down the base

The wording fixes which amount is the base.

E = D(1 + p/100) ⟹ p = 100 · (E - D)/D
2STEP 2

Put both in one currency

Converting puts both in one currency.

400 euros · (1.3 dollars)/(1 euro) = 520 dollars
3STEP 3

Divide the excess by the base

Dividing the excess gives 4 percent.

p = 100 · (520 - 500)/500 = 100 · 20/500 = 4
4STEP 4

Split the factor to confirm

Splitting the factor confirms it, choice (E).

E/D = (400 × 1.3)/500 = 400/500 × 1.3 = 0.8 × 1.3 = 1.04
Answer
4
Run the answer forward. Four percent of 500 is 20, so a 4 percent increase takes Diana's 500 dollars to 520 dollars, and 520 dollars is exactly what 400 euros are worth at 1.3 dollars each. The given data is reproduced, so 4 is consistent. The size is also sensible: 520 sits only 20 above 500, a gap of one part in twenty-five, so the percent has to be small and positive, which rules out 8 and 13 immediately. One more check guards the base. Comparing the other direction, Diana's money is short by 20 dollars out of Etienne's 520, which is about 3.85 percent, a different number. Since the two directions genuinely disagree, the wording 'greater than the value of Diana's money' matters, and it is Diana's 500 that belongs underneath.
💡Key takeaway

Put both amounts in the same unit first, then divide the gap by whichever amount the question says the other is greater than.

  • Pin down the base
  • Put both in one currency
  • Divide the excess by the base
  • Split the factor to confirm

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