AMC 8 · 2013 · #12
Grade 6 arithmeticrate-ratioProblem
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,yougetasecondpairata40150 regular price did he save?
Pick an answer.
AMC 8 2013 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.
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
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)
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
4.OA.A.3 Step 1 - Subproblem 1 — price of pair 1.
- The first pair is sold at the regular price with no discount.
💡 Naming the first subproblem and stating its answer is the Tool #7 move — break a multi-part question into one-line pieces.
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 $0.60$.
💡 Flipping "$40\%$ off" into "$60\%$ of the price" is the standard shortcut for percent discounts.
4.NF.B.4 Step 3 - Subproblem 3 — price of pair 3.
- "Half the regular price" means $\tfrac{1}{2}$ of $\$50$.
💡 Taking half of a quantity is a Grade 4 fraction-of-a-whole calculation.
4.OA.A.3 Step 4 Combine the three subproblem answers to get the total Javier actually paid.
💡 After solving each subproblem separately, addition glues the pieces back into the full answer.
4.OA.A.3 Step 5 Find the savings: regular total minus what he paid.
💡 Savings = (what it would have cost) $-$ (what it did cost). A subtraction, by definition.
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 $100\%$.
💡 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.
4.OA.A.3 Subproblem 1 — price of pair 1. The first pair is sold at the regular price with 5.NBT.B.7 Subproblem 2 — price of pair 2. A $40\%$ discount means Javier pays $100\% - 40\ 4.NF.B.4 Subproblem 3 — price of pair 3. "Half the regular price" means $\tfrac{1}{2}$ of 4.OA.A.3 Combine the three subproblem answers to get the total Javier actually paid. 4.OA.A.3 Find the savings: regular total minus what he paid. 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.3Solve 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.4Multiply a fraction by a whole number (Computing the third pair's price as $\tfrac{1}{2} \times \$50 = \$25$.)5.NBT.B.7Add, 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.3Use 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.
⭐ 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.
More like this
Same archetype — closest grade level first.