Competition · AMC preparation · step 4 of 4
AMC 8 · 2003 · #11
Grade 7 rate-ratioPick an answer.
AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The price changes happen in two stages — Thursday to Friday, then Friday to Monday — so Tool #7 (Solve Subproblems) is the natural fit: find the Friday price first, then the Monday price. Tool #4 (Introduce a Variable) keeps the bookkeeping clean: name the Friday price so it is crystal clear that the Monday 10% discount is taken off that listed price, not off $40. The common trap is to think the two 10% changes cancel; they do not, because they act on different bases.
Raise the price 10 percent
Subproblem 1 — the Friday price: raise Thursday's $40 by 10% (×1.10) to get $44.
Grade 7 percent problems: "10% more" means 100% + 10% = 110% of the original, i.e., multiply by 1.10.
7.RP.A.3Identify SubproblemsName the listed price
Name the listed price L = $44 — the discount's base is this Friday price, not the Thursday $40.
Giving the Friday price its own name prevents the classic mistake of discounting from $40.
6.EE.A.2Introduce A VariableTake 10 percent off
Subproblem 2 — the Monday price: take 10% off L (×0.90), so $44 × 0.90 = $39.60.
Grade 7 percent problems: "10% off" means 100% - 10% = 90% of the listed price, i.e., multiply by 0.90.
On Monday the shoes cost the Friday listed price of $44 taken down by ten percent, that is 44 × 0.90, because the discount comes off the raised listed price rather than off the original Thursday $40.
▸ Why?
The listed price the discount is taken from is the Friday price of $44, since the ad says "ten percent off the listed price" and the listed price is what the shoes are marked at after Friday raised Thursday's $40 by ten percent.
▸ Why?
Raising the Thursday price of $40 by ten percent gives $44, because keeping the whole $40 and adding another ten percent of it is the same as multiplying $40 by 1.10.
▸ Why?
One whole price plus ten percent of that same price can be gathered into a single factor (1 + 0.10) = 1.10 times the price, because a shared multiplier can be pulled back out of a sum of its own multiples.
▸ Why?
Taking ten percent off the listed price leaves ninety percent of it, so the Monday price is that listed price times 0.90.
▸ Why?
Ten percent off means keeping 100% - 10% = 90% of the price, because the removed part and the kept part are the price split with no gap or overlap and so add back to the whole 100%.
▸ Why?
Ninety percent of the price equals 0.90 times it, because a percent is one part out of every hundred, so ninety percent is ninety of those hundredths, 90/100 = 0.90, and taking that fraction of the price means multiplying it by 0.90.
Match against the choices
Match to the choices: $39.60 is choice (B).
The combined effect is 1.10 × 0.90 = 0.99, a 1% net decrease from Thursday — confirming the price drops slightly, not stays the same.
7.RP.A.3Identify SubproblemsUp 10% and then down 10% does not undo itself — the discount is taken off a bigger number, so the price ends a little below where it started.
- Raise the price 10 percent
- Name the listed price
- Take 10 percent off
- Match against the choices
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