AMC 8 · 2013 · #12

Grade 6 arithmeticrate-ratio

Archetype Arithmetic Expression Decomposition

percentagefraction-arithmeticmental-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticpercentage

📏 Medium solution 💡 3 insights

Problem

At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage of the$ 150 regular price did he save?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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Toolkit + CCSS Solution

Understand

Restated: A vendor's "fair special" sells three pairs of $\$50$ sandals like this: pair 1 at full price, pair 2 at $40\%$ off, pair 3 at half price. Javier buys all three. The regular price for three pairs is $\$150$. What percent of that $\$150$ did he save?

Givens: Regular price per pair = $\$50$; Pair 1: full price ($\$50$); Pair 2: $40\%$ discount; Pair 3: half price ($50\%$ discount); Regular total for three pairs = $\$150$; Answer choices: (A) $25$, (B) $30$, (C) $33$, (D) $40$, (E) $45$ (percent saved)

Unknowns: The percent of the $\$150$ regular price that Javier saved

Understand

Plan

Primary tool: #7 Identify Subproblems

Secondary: #3 Eliminate Possibilities

The three pairs have three different discount rules, so the cleanest path is Tool #7 (Identify Subproblems): compute the price of each pair on its own, then add them up, then compare to $\$150$. No algebra is needed — each subproblem is a one-line decimal or fraction calculation. Tool #3 (Eliminate Possibilities) is the natural verification: once we get $30\%$, we cross-check against the five multiple-choice values and confirm only (B) matches the savings $\$45 / \$150$.

Execute — Answer: B

#7 Identify Subproblems 4.OA.A.3 Step 1

Subproblem 1 — price of pair 1.
The first pair is sold at the regular price with no discount.

$\text{Pair 1} = \$50$

💡 Naming the first subproblem and stating its answer is the Tool #7 move — break a multi-part question into one-line pieces.

#7 Identify Subproblems 5.NBT.B.7 Step 2

Subproblem 2 — price of pair 2.
A $40\%$ discount means Javier pays $100\% - 40\% = 60\%$ of $\$50$. Compute $60\%$ of $50$ by multiplying by $0.60$.

$\text{Pair 2} = 0.60 \times \$50 = \$30$

💡 Flipping "$40\%$ off" into "$60\%$ of the price" is the standard shortcut for percent discounts.

#7 Identify Subproblems 4.NF.B.4 Step 3

Subproblem 3 — price of pair 3.
"Half the regular price" means $\tfrac{1}{2}$ of $\$50$.

$\text{Pair 3} = \tfrac{1}{2} \times \$50 = \$25$

💡 Taking half of a quantity is a Grade 4 fraction-of-a-whole calculation.

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

Combine the three subproblem answers to get the total Javier actually paid.

$\$50 + \$30 + \$25 = \$105$

💡 After solving each subproblem separately, addition glues the pieces back into the full answer.

#7 Identify Subproblems 4.OA.A.3 Step 5

Find the savings: regular total minus what he paid.

$\$150 - \$105 = \$45 \text{ saved}$

💡 Savings = (what it would have cost) $-$ (what it did cost). A subtraction, by definition.

#3 Eliminate Possibilities 6.RP.A.3 Step 6

Convert the $\$45$ savings into a percent of the $\$150$ regular price by computing the ratio and multiplying by $100\%$.

$$\dfrac{45}{150} = \dfrac{3}{10} = 0.30 = 30\% \;\Rightarrow\; \textbf{(B)}$$

💡 Percent saved is a part-of-whole ratio expressed per $100$ — the core Grade 6 percent skill. Then Tool #3 confirms only choice (B) matches.

[1] #7 4.OA.A.3 Subproblem 1 — price of pair 1. The first pair is sold at the regular price with

[2] #7 5.NBT.B.7 Subproblem 2 — price of pair 2. A $40\%$ discount means Javier pays $100\% - 40\

[3] #7 4.NF.B.4 Subproblem 3 — price of pair 3. "Half the regular price" means $\tfrac{1}{2}$ of

[4] #7 4.OA.A.3 Combine the three subproblem answers to get the total Javier actually paid.

[5] #7 4.OA.A.3 Find the savings: regular total minus what he paid.

[6] #3 6.RP.A.3 Convert the $\$45$ savings into a percent of the $\$150$ regular price by comput

Review

Reasonableness: Sanity-check the percent another way. The three discounts off the regular price are $0\%$, $40\%$, and $50\%$, on three equally priced pairs. The average discount is $\tfrac{0 + 40 + 50}{3} = \tfrac{90}{3} = 30\%$, which is exactly the percent saved on the $\$150$ total. That matches (B), and $30\%$ sits in a reasonable middle of the answer range ($25$ to $45$).

Alternative: Tool #6 (Guess and Check) on the answer choices: each percent corresponds to a dollar savings of (percent) $\times \$150$. Choices give savings of $\$37.50$, $\$45$, $\$49.50$, $\$60$, $\$67.50$. Only (B) $\$45$ equals $\$150 - \$105$. Alternatively, Tool #13 (Convert to Algebra) gives the same answer via $p = \tfrac{150 - (50 + 0.6 \cdot 50 + 0.5 \cdot 50)}{150} \cdot 100\% = 30\%$, but plain subproblems are simpler here.

CCSS standards used (min grade 6)

4.OA.A.3 Solve multistep word problems with whole numbers using the four operations (Splitting the purchase into three single-pair subproblems and combining the three prices ($\$50 + \$30 + \$25 = \$105$) and the subtraction $\$150 - \$105 = \$45$.)
4.NF.B.4 Multiply a fraction by a whole number (Computing the third pair's price as $\tfrac{1}{2} \times \$50 = \$25$.)
5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths (Computing the second pair's price as $0.60 \times \$50 = \$30$ using a decimal-by-whole-number product.)
6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (including percent) (Expressing the $\$45$ savings as a percent of the $\$150$ regular price: $\tfrac{45}{150} = 30\%$.)

⭐ This AMC 8 problem really is just "price of each pair, add them up, then turn the savings into a percent" — the Grade 6 percent step is the only new idea, and the rest is patient subproblem work.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 6)

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