AMC 10 · 2009 · #8

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Successive Multiplicative Steps

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

📏 Medium solution 💡 2 insights

Problem

In a certain year the price of gasoline rose by $20\%$ during January, fell by $20\%$ during February, rose by $25\%$ during March, and fell by $x\%$ during April. The price of gasoline at the end of April was the same as it had been at the beginning of January. To the nearest integer, what is $x$

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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: A gas price rises 20% in January, falls 20% in February, rises 25% in March, then falls x% in April. After April the price is back to exactly what it was on January 1. Find x, rounded to the nearest whole number.

Givens: January change: up 20%.; February change: down 20%.; March change: up 25%.; April change: down x% (unknown).; The end-of-April price equals the beginning-of-January price.

Unknowns: The April percent drop x, to the nearest integer.

Understand

Givens: January change: up 20%.; February change: down 20%.; March change: up 25%.; April change: down x% (unknown).; The end-of-April price equals the beginning-of-January price.

Plan

Primary tool: #11 Work Backwards

Secondary: #9 Solve an Easier Related Problem, #7 Identify Subproblems

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.

Execute — Answer: B

#9 Solve an Easier Related Problem 7.RP.A.3 Step 1

Pick a starting price of 100

Since only percent changes matter, the actual starting price does not change the answer.
Choose 100 dollars for January 1 so each percent becomes a clean number.

$$P_0 = 100$$

💡 Percent answers stay the same no matter the starting amount, so pick the amount that makes the arithmetic easiest.

#7 Identify Subproblems 7.RP.A.3 Step 2

Apply January and February

A 20% rise multiplies by 1.20; a 20% fall multiplies by 0.80.
Do them in order.
Notice the price does NOT return to 100 after these two, because the 20% fall is taken from the larger 120, not from 100.

$$100 \times 1.20 = 120,\quad 120 \times 0.80 = 96$$

💡 Up 20% then down 20% loses ground, because the drop is a bigger share of the raised price.

#7 Identify Subproblems 7.RP.A.3 Step 3

Apply March

A 25% rise multiplies by 1.25.
Apply it to the end-of-February price of 96 to get the end-of-March price.

$$96 \times 1.25 = 120$$

💡 Multiply by 1.25 to add a quarter of the current price on top.

#11 Work Backwards 7.RP.A.3 Step 4

Work backwards to find April's drop

April must carry the price from 120 back down to the original 100.
The amount lost is 120 - 100 = 20 dollars, and that loss is measured against April's starting price of 120.
So x percent is 20 out of 120.

$$x = \frac{120 - 100}{120} \times 100 = \frac{20}{120} \times 100 = 16.\overline{6}$$

💡 A percent drop is always measured against the price you are dropping from, which here is 120, not 100.

#11 Work Backwards 5.NBT.A.4 Step 5

Round to the nearest integer

16.67 rounds up to 17.
So to the nearest integer x is 17, which is choice (B).

$$16.\overline{6} \approx 17$$

💡 A decimal part above 0.5 rounds up to the next whole number.

[1] #9 7.RP.A.3 Since only percent changes matter, the actual starting price does not change the

[2] #7 7.RP.A.3 A 20% rise multiplies by 1.20; a 20% fall multiplies by 0.80. Do them in order.

[3] #7 7.RP.A.3 A 25% rise multiplies by 1.25. Apply it to the end-of-February price of 96 to ge

[4] #11 7.RP.A.3 April must carry the price from 120 back down to the original 100. The amount lo

[5] #11 5.NBT.A.4 16.67 rounds up to 17. So to the nearest integer x is 17, which is choice (B).

Review

Reasonableness: 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.

Alternative: Write one equation with a symbolic start P: P * 1.20 * 0.80 * 1.25 * (1 - x/100) = P. The product 1.20 * 0.80 * 1.25 = 1.20, so 1.20 * (1 - x/100) = 1, giving 1 - x/100 = 5/6 and x = 100/6 = 16.67, again 17.

CCSS standards used (min grade 7)

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems (Applying each month's percent change to the correct base and computing April's percent drop as 20 out of 120.)
5.NBT.A.4 Use place value understanding to round decimals to any place (Rounding 16.67 to the nearest integer, 17.)

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

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Pick a starting price of 100

Apply January and February

Apply March

Work backwards to find April's drop

Round to the nearest integer

Review

CCSS standards used (min grade 7)

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