AMC 8 · 2003 · #11

Grade 7 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation
percentagefraction-arithmeticmulti-digit-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: percentagefraction-decimal-conversionmulti-digit-arithmetic
📏 Short solution 💡 2 insights
Problem
Lou's Fine Shoes raises every Thursday price by 10% on Friday. Over the weekend, Lou advertises "10% off the listed price" starting Monday. A pair of shoes cost $40 on Thursday. What does it cost on Monday?

Pick an answer.

(A)
36
(B)
39.60
(C)
40
(D)
40.40
(E)
44

AMC 8 2003 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Solve Subproblems

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.

1STEP 1

Subproblem 1 — the Friday price: raise Thursday's 40by1040 by 10% (×1.10) to get <span class="mk-a"><span class="hl-ask">44.

P_Fri = 40 × 1.10 = 44
2STEP 2

Name the listed price L = 44</span></span>—thediscount′sbaseisthisFridayprice,nottheThursday44</span></span> — the discount's base is this Friday price, not the Thursday40.

L = 44
3STEP 3

Subproblem 2 — the Monday price: take 10% off L (×0.90), so 44×0.90=<spanclass="mk−a"><spanclass="hl−ask">44 × 0.90 = <span class="mk-a"><span class="hl-ask">39.60.

P_Mon = L × 0.90 = 44 × 0.90 = 39.60
4STEP 4

Match to the choices: $39.60 is choice (B).

P_Mon = $39.60 → (B)
Answer
39.60
The two 10% moves apply to different bases, so they should not cancel. Multiplying the factors gives 1.10 × 0.90 = 0.99, a 1% net drop from Thursday. From 40,thatis40, that is40 × 0.99 = 39.60—exactlymatchingchoice(B).Theanswerisalsolessthan39.60 — exactly matching choice (B). The answer is also less than40 (Thursday) and less than $44 (Friday), which both make sense.
💡Key takeaway

Up 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.

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