AMC 10 · 2009 · #8

Grade 7 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

percentagefraction-multiplicationlinear-equations-one-var work-backwards ↑ Prerequisites: percentage

📏 Medium solution 💡 2 insights

Problem

A gasoline price rises 20% during January, falls 20% during February, rises 25% during March, then falls x% during April. At the end of April the price is exactly what it was at the beginning of January. Find x, rounded to the nearest whole number.

Pick an answer.

(A)

$\ 12$

(B)

$\ 17$

(C)

$\ 20$

(D)

$\ 25$

(E)

$\ 35$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Work Backwards

The end price is pinned to the start price, and April's drop is the only unknown. So compute the price through March forward, then work backwards from the fixed end value to find the drop April must have made. Picking a convenient start value of 100 turns every percent into an easy multiplication.

1STEP 1

Pick a starting price of 100

Only the percent changes matter, so start January 1 at 100 and every percent lands on a clean number.

P₀ = 100

2STEP 2

Apply January and February

Up 20% multiplies by 1.20 and down 20% by 0.80, so the price lands at 96, not back at 100.

100 × 1.20 = 120, 120 × 0.80 = 96

3STEP 3

Apply March

A 25% rise multiplies by 1.25, so February's 96 grows back to 120 by the end of March.

96 × 1.25 = 120

4STEP 4

Work backwards to find April's drop

April must bring 120 back down to 100, so the drop is 20 out of 120 — not out of 100.

x = (120 - 100)/120 × 100 = 20/120 × 100 = 16.6

5STEP 5

Round to the nearest integer

The decimal part is above 0.5, so 16.67 rounds up to 17 — choice (B).

16.6 ≈ 17

Answer

Check forward: 120 dropped by 17% is 120 - 0.17 * 120 = 120 - 20.4 = 99.6, very close to the target 100, confirming x is near 17 (choice B). The tempting wrong answer is 20 (choice C), which comes from measuring the 20-dollar drop against the original 100 instead of against April's actual starting price of 120.

💡Key takeaway

A percent change always works on the current price, so undoing a rise needs a smaller percent than the rise itself.

Pick a starting price of 100
Apply January and February
Apply March
Work backwards to find April's drop
Round to the nearest integer

Pick a starting price of 100

Apply January and February

Apply March

Work backwards to find April's drop

Round to the nearest integer

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