AMC 8 · 2007 · #6

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

percentagefraction-decimal-conversionestimation identify-subproblems ↑ Prerequisites: percentagefraction-arithmetic

📏 Short solution 💡 1 insight

Problem

A long-distance call in the USA cost 41 cents per minute in 1985 and 7 cents per minute in 2005. Find the approximate percent decrease in the cost per minute.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Identify Subproblems) breaks the question into three small steps: (a) find the amount of decrease, (b) form the ratio of decrease to the original price, and (c) convert that ratio to a percent. Each piece is a single arithmetic move. Tool #6 (Guess and Check) handles the word "approximate" — the answer choices are widely spaced (7, 17, 34, 41, 80), so a quick estimate of $\frac{34}{41}$ is enough to pick the only choice close to it. We avoid Tool #13 (Algebra) because no variable is needed.

1STEP 1

The price fell from 41 cents to 7 cents, so subtract to get the drop: 34 cents.

decrease = 41 - 7 = 34 cents

2STEP 2

Percent decrease compares the drop to the original 1985 price, so the ratio is $\frac{34}{41}$ .

percent decrease = decrease/original =

\frac{34}{41}

3STEP 3

Since 34 is nearly 41, the ratio $\frac{34}{41}$ sits just under 1 — about 83%.

\frac{34}{41}

≈ 0.83 = 83%

4STEP 4

The choices are spread far apart, so 83% rounds to the nearest one, 80% — choice (E).

83% closest to 80% → (E)

Answer

Double-check with the winning percent. If the 1985 price drops by 80%, the remaining price is 20% of 41, or 0.20 × 41 = 8.2 cents. The actual 2005 price is 7 cents — very close to 8.2, off by only about a cent. The other choices are not close: a 41% drop would leave 24.2 cents, and a 17% drop would leave 34 cents — both far above the real 7 cents. So 80% is the only sensible match.

💡Key takeaway

Percent decrease is just "how much it dropped" divided by "what it started at" — one subtraction and one division get you the answer.

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