AMC 8 · 2013 · #12

Grade 6 arithmeticrate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

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

📏 Medium solution 💡 3 insights

Problem

A vendor's "fair special" sells three pairs of

50</span> sandals like this: pair 1 at full price, pair 2 at <span class="mk-c">40% off</span>, pair 3 at <span class="mk-c">half price</span>. Javier buys all three. The regular price for three pairs is <span class="mk-c">

150. What percent of that $150 did he save?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2013 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

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 $150. No algebra is needed — each subproblem is a one-line decimal or fraction calculation. Tool #3 (Eliminate Possibilities) is the natural verification: once we get 30%, we cross-check against the five multiple-choice values and confirm only (B) matches the savings$ \frac{45}{150}$.

1STEP 1

Subproblem 1 — pair 1 carries no discount, so it stays at the full $50.

Pair 1 = $50

2STEP 2

Subproblem 2 — a 40% discount means paying only 60%, so pair 2 costs $30.

Pair 2 = 0.60 ×

50 =

3STEP 3

Subproblem 3 — "half price" is a 50% discount, so pair 3 costs $25.

Pair 3 =

\frac{1}{2}

50 =

4STEP 4

Add the three pair prices to get what Javier actually paid: $105.

50 +

30 +

25 =

105

5STEP 5

Savings is the regular total minus what he paid: $45.

150 -

105 = $45 saved

6STEP 6

Divide the savings by the $150 regular total:$ \frac{45}{150}$ = 30%, choice (B).

\frac{45}{150}

\frac{3}{10}

= 0.30 = 30% → (B)

Answer

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

\frac{0 + 40 + 50}{3}

\frac{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).

💡Key takeaway

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.

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