AMC 8 · 2002 · #14

Grade 7 rate-ratio

Archetype Arithmetic Expression Decomposition

percentagefraction-multiplicationfraction-arithmetic identify-subproblems ↑ Prerequisites: percentagefraction-arithmetic

📏 Short solution 💡 2 insights

Problem

A merchant offers a large group of items at $30\%$ off. Later, the merchant takes $20\%$ off these sale prices. The total discount is

Pick an answer.

(A)

35%

(B)

44%

(C)

50%

(D)

56%

(E)

60%

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Toolkit + CCSS Solution

Understand

Restated: A store first takes $30\%$ off every item. Later it takes another $20\%$ off those already-discounted prices. What is the total percent discount compared to the original price?

Givens: First discount: $30\%$ off the original price; Second discount: $20\%$ off the already-reduced sale price; Answer choices: (A) $35\%$, (B) $44\%$, (C) $50\%$, (D) $56\%$, (E) $60\%$

Unknowns: The total percent off the original price after both discounts

Understand

Restated: A store first takes $30\%$ off every item. Later it takes another $20\%$ off those already-discounted prices. What is the total percent discount compared to the original price?

Givens: First discount: $30\%$ off the original price; Second discount: $20\%$ off the already-reduced sale price; Answer choices: (A) $35\%$, (B) $44\%$, (C) $50\%$, (D) $56\%$, (E) $60\%$

Plan

Primary tool: #4 Introduce a Variable

Secondary: #7 Identify Subproblems

The original price is never given, so Tool #4 (Introduce a Variable) — let the original price be $P$ — turns the abstract "percent of percent" question into clean arithmetic in $P$. Tool #7 (Identify Subproblems) splits the chain of discounts into two clear steps: first apply the $30\%$ discount to get the sale price, then apply the $20\%$ discount to that sale price. Comparing the final price to $P$ at the end gives the total percent off.

Execute — Answer: B

#4 Introduce a Variable 6.EE.A.2 Step 1

Let the original price be $P$.
Using a variable lets us track "what fraction of $P$" the price is at each step, without needing a specific dollar amount.

$$\text{original price} = P$$

💡 Naming the unknown original price is the Tool #4 move and matches the Grade 6 idea of writing an expression with a letter that stands for a number.

#7 Identify Subproblems 6.RP.A.3 Step 2

Apply the first discount.
A $30\%$ discount means the sale price is $100\% - 30\% = 70\%$ of $P$, i.e., multiply $P$ by $0.7$.

$$\text{sale price} = P \times (1 - 0.30) = 0.7P$$

💡 Treating a $30\%$ discount as multiplying by $0.7$ is the Grade 6 percent-of-a-quantity move.

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

Apply the second discount on top of the sale price.
A $20\%$ discount makes the final price $100\% - 20\% = 80\%$ of $0.7P$, i.e., multiply by $0.8$.

$$\text{final price} = 0.7P \times (1 - 0.20) = 0.7P \times 0.8 = 0.56P$$

💡 Stacking a second percent discount on an already-discounted price is a Grade 7 multi-step percent-change problem.

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

Compare the final price to the original.
The final price is $0.56P$, which is $56\%$ of $P$, so the customer paid $56\%$ and saved the other $44\%$.

$$\text{total discount} = 100\% - 56\% = 44\% \;\Rightarrow\; \textbf{(B)}$$

💡 Subtracting the "percent paid" from $100\%$ to get the "percent off" is the standard Grade 7 percent-change wrap-up.

[1] #4 6.EE.A.2 Let the original price be $P$. Using a variable lets us track "what fraction of

[2] #7 6.RP.A.3 Apply the first discount. A $30\%$ discount means the sale price is $100\% - 30\

[3] #7 7.RP.A.3 Apply the second discount on top of the sale price. A $20\%$ discount makes the

[4] #7 7.RP.A.3 Compare the final price to the original. The final price is $0.56P$, which is $5

Review

Reasonableness: Try a concrete number: let $P = \$100$. The $30\%$ discount drops the price to $\$70$; the $20\%$ discount takes $\$14$ off $\$70$, leaving $\$56$. The customer paid $\$56$ out of $\$100$, a savings of $\$44$ — exactly $44\%$ off. The common trap is to add $30\% + 20\% = 50\%$ (choice C), but the $20\%$ only applies to the already-reduced price, so the actual total is less than $50\%$. Also note choice (D) $56\%$ is the percent paid, not the percent off.

Alternative: Tool #16 (Transform the Problem): combine the two percent factors into a single multiplier. "$30\%$ off then $20\%$ off" means multiplying by $0.7 \times 0.8 = 0.56$, so the final price is $56\%$ of the original and the total discount is $1 - 0.56 = 0.44 = 44\%$. Same answer (B), in one multiplication.

CCSS standards used (min grade 7)

6.EE.A.2 Write, read, and evaluate expressions in which letters stand for numbers (Introducing the variable $P$ for the unspecified original price so each discounted price can be written as an expression in $P$.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems, including percent (Applying the first $30\%$ discount as multiplication by $0.7$ to get the sale price $0.7P$.)
7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems (Stacking the second $20\%$ discount on the sale price to get $0.56P$, then converting the final price into a total $44\%$ discount off the original.)

⭐ Stacked discounts multiply, not add — once you write the original price as $P$, this AMC 8 question is just Grade 7 percent reasoning.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 7)

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