AMC 10 · 2008 · #8

Grade 7 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

percentagelinear-equations-one-var convert-to-algebra ↑ Prerequisites: percentagelinear-equations-one-var

📏 Medium solution 💡 1 insight

Problem

A computer has one sticker price. Store A takes 15% off that price, then subtracts a $90 rebate. Store B takes 25% off the same sticker price, with no rebate. Buying at store A costs $15 less than buying at store B. Find the sticker price of the computer, in dollars.

Pick an answer.

(A)

$\ 750$

(B)

$\ 900$

(C)

$\ 1000$

(D)

$\ 1050$

(E)

$\ 1500$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The one number we do not know is the sticker price, so the natural first move is Tool #4: call it S and write both store prices in terms of S. "15% off" means you still pay 85%, so store A costs 0.85S - 90; "25% off" means you pay 75%, so store B costs 0.75S. Tool #13 then turns the sentence "store A saves $15" into a single equation, and Tool #8 keeps every term in dollars so the flat$ 90 rebate and the percent-based discounts can be added and subtracted safely.

1STEP 1

Name the sticker price

Let S be the sticker price in dollars; every other price in the problem is built from S.

S = sticker price (dollars)

2STEP 2

Turn percents into prices

Paying after 15% off means paying 85%, so store A charges 0.85S - 90; 25% off leaves 75%, so store B charges 0.75S.

Store A = 0.85S - 90, Store B = 0.75S

3STEP 3

Write the savings equation

Saving $15 at store A just means store B's price minus store A's price equals 15.

Store B - Store A = 15 → 0.75S - (0.85S - 90) = 15

4STEP 4

Solve for the sticker price

Clear the parentheses, combine the S terms into -0.10S, and isolate S: the sticker price is 750, choice (A).

0.75S - 0.85S + 90 = 15 → -0.10S + 90 = 15 → -0.10S = -75 → S = 750 → (A)

Answer

750

Plug S = 750 back in. Store A: 0.85 · 750 - 90 = 637.50 - 90 = 547.50. Store B: 0.75 · 750 = 562.50. The difference is 562.50 - 547.50 = 15, exactly the promised savings, and store A is indeed the cheaper one. So (A) 750 checks out.

💡Key takeaway

Turn "percent off" into "percent you pay," name the unknown price, and one clean equation from the savings clue hands you the sticker price.

Name the sticker price
Turn percents into prices
Write the savings equation
Solve for the sticker price

Name the sticker price

Turn percents into prices

Write the savings equation

Solve for the sticker price

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