AMC 8 · 2003 · #11

Grade 7 rate-ratio

Archetype Arithmetic Expression Decomposition

percentagefraction-arithmeticmulti-digit-arithmetic identify-subproblemspattern-recognition ↑ Prerequisites: percentagefraction-decimal-conversionmulti-digit-arithmetic

📏 Short solution 💡 2 insights

Problem

Business is a little slow at Lou's Fine Shoes, so Lou decides to have a
sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the
weekend, Lou advertises the sale: "Ten percent off the listed price. Sale
starts Monday." How much does a pair of shoes cost on Monday that
cost $40$ dollars on Thursday?

Pick an answer.

(A)

(B)

39.60

(C)

(D)

40.40

(E)

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Toolkit + CCSS Solution

Understand

Restated: 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?

Givens: Thursday price of the shoes is $\$40$; Friday: prices go up by $10\%$ from Thursday; Monday: prices drop by $10\%$ off the listed (Friday) price; Answer choices: (A) $\$36$, (B) $\$39.60$, (C) $\$40$, (D) $\$40.40$, (E) $\$44$

Unknowns: The Monday price of the shoes

Understand

Plan

Primary tool: #7 Solve Subproblems

Secondary: #4 Introduce a Variable

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.

Execute — Answer: B

#7 Solve Subproblems 7.RP.A.3 Step 1

Subproblem 1: find the Friday price.
A $10\%$ increase multiplies the Thursday price by $1.10$.

$$P_{\text{Fri}} = 40 \times 1.10 = 44$$

💡 Grade 7 percent problems: "$10\%$ more" means $100\% + 10\% = 110\%$ of the original, i.e., multiply by $1.10$.

#4 Introduce a Variable 6.EE.A.2 Step 2

Name the listed price.
Let $L = P_{\text{Fri}} = \$44$. The Monday sale advertises "$10\%$ off the listed price," so $L$ — not the Thursday $\$40$ — is the base for the discount.

$$L = 44$$

💡 Giving the Friday price its own name prevents the classic mistake of discounting from $\$40$.

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

Subproblem 2: find the Monday price.
A $10\%$ discount on the listed price multiplies $L$ by $0.90$.

$$P_{\text{Mon}} = L \times 0.90 = 44 \times 0.90 = 39.60$$

💡 Grade 7 percent problems: "$10\%$ off" means $100\% - 10\% = 90\%$ of the listed price, i.e., multiply by $0.90$.

#7 Solve Subproblems 7.RP.A.3 Step 4

Match the result to the answer choices.

$P_{\text{Mon}} = \$39.60 \;\Rightarrow\; \textbf{(B)}$

💡 The combined effect is $1.10 \times 0.90 = 0.99$, a $1\%$ net decrease from Thursday — confirming the price drops slightly, not stays the same.

[1] #7 7.RP.A.3 Subproblem 1: find the Friday price. A $10\%$ increase multiplies the Thursday p

[2] #4 6.EE.A.2 Name the listed price. Let $L = P_{\text{Fri}} = \$44$. The Monday sale advertis

[3] #7 7.RP.A.3 Subproblem 2: find the Monday price. A $10\%$ discount on the listed price multi

[4] #7 7.RP.A.3 Match the result to the answer choices.

Review

Reasonableness: The two $10\%$ moves apply to different bases, so they should not cancel. Multiplying the factors gives $1.10 \times 0.90 = 0.99$, a $1\%$ net drop from Thursday. From $\$40$, that is $\$40 \times 0.99 = \$39.60$ — exactly matching choice (B). The answer is also less than $\$40$ (Thursday) and less than $\$44$ (Friday), which both make sense.

Alternative: Tool #16 (Transform the Problem): instead of two separate steps, combine the two percent factors into a single multiplier. Up $10\%$ then down $10\%$ means multiplying by $1.10 \times 0.90 = 0.99$, so the Monday price is $40 \times 0.99 = \$39.60$. Same answer, in one multiplication.

CCSS standards used (min grade 7)

7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems (Modeling the $10\%$ Friday increase as multiplying by $1.10$ and the $10\%$ Monday discount as multiplying by $0.90$ to chain the two percent changes.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, including percent (Reading "$10\%$" as the rate $10$ per $100$, i.e., $0.10$, so a $10\%$ change scales the price by $1 \pm 0.10$.)
6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Naming the Friday price $L$ so the Monday discount is clearly applied to the listed price, not to the Thursday price.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 7)

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