AMC 8 · 2012 · #8

Grade 7 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

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

📏 Short solution 💡 2 insights

Problem

A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?

Pick an answer.

(A)

$hspace{.05in}10$

(B)

$hspace{.05in}33$

(C)

$hspace{.05in}40$

(D)

$hspace{.05in}60$

(E)

$hspace{.05in}70$

AMC 8 2012 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.

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Toolkit + CCSS Solution

Understand

Restated: A shop marks everything at half price. A coupon then takes an additional $20\%$ off the sale price. What total percent off the original price does a shopper get by using the coupon during the sale?

Givens: Sale discount: $50\%$ off the original price ("half price"); Coupon: additional $20\%$ off the already-discounted sale price; Answer choices: (A) $10$, (B) $33$, (C) $40$, (D) $60$, (E) $70$ (percent)

Unknowns: The total percent discount the final price represents off the original price

Understand

Plan

Primary tool: #5 Introduce Variables

Secondary: #7 Identify Subproblems

The original price is never stated, so Tool #5 (Introduce Variables) — let the original price be $P$ — turns the abstract "percent of percent" question into concrete arithmetic. Tool #7 (Identify Subproblems) splits the chain of discounts into two clean steps: first apply the $50\%$ sale to get the sale price, then apply the $20\%$ coupon to that sale price. Comparing the final price to $P$ at the end gives the total percent off.

Execute — Answer: D

#5 Introduce Variables 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 ever needing a specific dollar amount.

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

💡 Naming the unknown original price is the Tool #5 move and is exactly the Grade 6 idea of writing an expression with a letter standing for a number.

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

Apply the $50\%$ sale.
"Half price" means the sale price is $100\% - 50\% = 50\%$ of $P$, i.e., multiply $P$ by $0.5$.

$$\text{sale price} = P \times (1 - 0.50) = 0.5P$$

💡 Treating a $50\%$ discount as multiplying by $0.5$ is the Grade 6 percent-of-a-quantity move.

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

Apply the $20\%$ coupon to the sale price.
The coupon takes $20\%$ off $0.5P$, so the final price is $100\% - 20\% = 80\%$ of $0.5P$, i.e., multiply by $0.80$.

$$\text{final price} = 0.5P \times (1 - 0.20) = 0.5P \times 0.8 = 0.4P$$

💡 Stacking a second percent discount on top of 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 price.
The final price is $0.4P$, which is $40\%$ of $P$, so the customer paid $40\%$ of the original and saved the remaining $60\%$.

$$\text{percent off} = 100\% - 40\% = 60\% \;\Rightarrow\; \textbf{(D)}$$

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

[1] #5 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 $50\%$ sale. "Half price" means the sale price is $100\% - 50\% = 50\%

[3] #7 7.RP.A.3 Apply the $20\%$ coupon to the sale price. The coupon takes $20\%$ off $0.5P$, s

[4] #7 7.RP.A.3 Compare the final price to the original price. The final price is $0.4P$, which

Review

Reasonableness: Try a concrete number: let $P = \$100$. Half price makes it $\$50$; a $20\%$ coupon off $\$50$ saves $\$10$, leaving $\$40$. The shopper paid $\$40$ out of $\$100$, a savings of $\$60$ — exactly $60\%$ off. A common trap is to add $50\% + 20\% = 70\%$ (choice E), but the $20\%$ only applies to the already-halved price, so the actual total is less than $70\%$.

Alternative: Tool #6 (Guess and Check) on a clean number: pick $P = \$10$. Sale price $= \$5$; coupon takes $\$1$ off, leaving $\$4$. Saved $\$6$ out of $\$10 \Rightarrow 60\%$ off, matching (D). This also rules out (E) $70\%$ (the additive trap) and (C) $40\%$ (the "percent paid" instead of "percent off" trap).

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 (Applying the first $50\%$ discount as multiplication by $0.5$ to find the sale price $0.5P$.)
7.RP.A.3 Use proportional relationships to solve multistep ratio and percent problems (Applying the second $20\%$ coupon on top of the sale price, then converting the final price $0.4P$ into a total $60\%$ 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: D

Review

CCSS standards used (min grade 7)

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