AMC 10 · 2008 · #8
Grade 7 algebraHeather compares the price of a new computer at two different stores. Store A offers 15% off the sticker price followed by a 90$ rebate, and store $B$ offers $25\%$ off the same sticker price with no rebate. Heather saves15bybuyingthecomputeratstoreAinsteadofstoreB$. What is the sticker price of the computer, in dollars?
Pick an answer.
AMC 10 2008 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.
Toolkit + CCSS Solution
Understand
Restated: A computer has one sticker price. Store $A$ takes $15\%$ off, then subtracts a $\$90$ rebate. Store $B$ takes $25\%$ off with no rebate. Buying at store $A$ costs $\$15$ less than buying at store $B$. Find the sticker price.
Givens: Store $A$ price: sticker minus $15\%$, then minus a $\$90$ rebate; Store $B$ price: sticker minus $25\%$; Store $A$ ends up $\$15$ cheaper than store $B$; Answer choices: (A) $750$, (B) $900$, (C) $1000$, (D) $1050$, (E) $1500$
Unknowns: The sticker price of the computer, in dollars
Understand
Restated: A computer has one sticker price. Store $A$ takes $15\%$ off, then subtracts a $\$90$ rebate. Store $B$ takes $25\%$ off with no rebate. Buying at store $A$ costs $\$15$ less than buying at store $B$. Find the sticker price.
Givens: Store $A$ price: sticker minus $15\%$, then minus a $\$90$ rebate; Store $B$ price: sticker minus $25\%$; Store $A$ ends up $\$15$ cheaper than store $B$; Answer choices: (A) $750$, (B) $900$, (C) $1000$, (D) $1050$, (E) $1500$
Plan
Primary tool: #4 Introduce a Variable
Secondary: #13 Convert to Algebra, #8 Analyze the Units
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.
Execute — Answer: A
6.EE.B.6 Step 1 Name the sticker price
- Let $S$ be the sticker price in dollars.
- Everything else in the problem will be built from $S$.
💡 The one thing we are asked for is the unknown, so give it a letter and let the rest of the problem describe it.
6.RP.A.3 Step 2 Turn percents into prices
Taking $15\%$ off means you pay the other $85\%$, so store $A$'s discounted price is $0.85S$; then subtract the $\$90$ rebate. Taking $25\%$ off means you pay $75\%$, so store $B$'s price is $0.75S$.
💡 "Percent off" is easier as "percent you still pay": $100\% - 15\% = 85\%$, so multiply by $0.85$, not $0.15$.
7.EE.B.4 Step 3 Write the savings equation
- Saving $\$15$ by choosing store $A$ means store $B$ costs $\$15$ more than store $A$.
- Set the difference of the two prices equal to $15$.
💡 "Saves $\$15$ at the cheaper store" is just: dearer price minus cheaper price equals $15$.
7.EE.B.4 Step 4 Solve for the sticker price
- Remove the parentheses (the minus flips both terms), combine the $S$ terms, and isolate $S$.
- The answer is $\$750$, which is choice (A).
💡 Once the two prices are lined up, the percent parts collapse into a single $-0.10S$ and the $\$90$ carries the equation home.
6.EE.B.6 Let $S$ be the sticker price in dollars. Everything else in the problem will be 6.RP.A.3 Taking $15\%$ off means you pay the other $85\%$, so store $A$'s discounted pric 7.EE.B.4 Saving $\$15$ by choosing store $A$ means store $B$ costs $\$15$ more than store 7.EE.B.4 Remove the parentheses (the minus flips both terms), combine the $S$ terms, and Review
Reasonableness: Plug $S = 750$ back in. Store $A$: $0.85 \cdot 750 - 90 = 637.50 - 90 = 547.50$. Store $B$: $0.75 \cdot 750 = 562.50$. The difference is $562.50 - 547.50 = 15$, exactly the promised savings, and store $A$ is indeed the cheaper one. So $\textbf{(A)}\ 750$ checks out.
Alternative: Tool #3 (Eliminate Possibilities): test the choices. The two discounts differ by $10\%$ of $S$, and store $A$'s extra $\$90$ rebate must make it exactly $\$15$ cheaper, so $10\%$ of $S$ must be $\$90 - \$15 = \$75$; that gives $S = 750$ immediately, and no other choice makes $10\%$ of $S$ equal $75$.
CCSS standards used (min grade 7)
6.EE.B.6Use variables to represent numbers and write expressions to solve problems (Letting $S$ stand for the unknown sticker price so both store prices can be written in terms of it.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Converting "$15\%$ off" and "$25\%$ off" into the paid fractions $0.85S$ and $0.75S$.)7.EE.B.4Use variables to represent quantities and construct simple equations and inequalities (Building the equation $0.75S - (0.85S - 90) = 15$ from the "saves $\$15$" clue and solving it for $S = 750$.)
⭐ Turn "percent off" into "percent you pay," name the unknown price, and one clean equation from the savings clue hands you the sticker price.
⭐ Turn "percent off" into "percent you pay," name the unknown price, and one clean equation from the savings clue hands you the sticker price.
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