AMC 8 · 2002 · #18

Grade 6 rate-ratioalgebra

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

mean-median-mode-rangeunit-conversionlinear-equations-one-var convert-to-algebraidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticunit-conversion

📏 Short solution 💡 2 insights

Problem

Gage skated 1 hr 15 min on each of 5 days and 1 hr 30 min on each of 3 days. How long must he skate on day 9 so that his average over all 9 days is exactly 85 minutes per day?

Pick an answer.

(A)

1 hr

(B)

1 hr 10 min

(C)

1 hr 20 min

(D)

1 hr 40 min

(E)

2 hr

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The unknown day-9 time is the only mystery quantity, so Tool #4 (Introduce a Variable) names it x and turns the average into a single equation (75 · 5 + 90 · 3 + x)/9 = 85. Tool #7 (Identify Subproblems) splits the bookkeeping into clean pieces — required total, known total, missing piece — so the arithmetic stays simple. Together they reduce a word problem about averages to one subtraction.

1STEP 1

Convert every time to minutes: 1 hr 15 min = 75 min, 1 hr 30 min = 90 min — one unit avoids a common slip.

1 hr 15 min = 75 min, 1 hr 30 min = 90 min

2STEP 2

Subproblem 1 — the required total: a 9-day average of 85 needs 765 minutes in all.

Required total = 9 × 85 = 765 minutes

3STEP 3

Subproblem 2 — minutes already skated: five 75s plus three 90s make 645 minutes over days 1–8.

5 × 75 + 3 × 90 = 375 + 270 = 645 minutes

4STEP 4

Name day 9's minutes x: since 645 + x = 765, subtracting gives x = 120 minutes.

645 + x = 765 → x = 765 - 645 = 120 minutes

5STEP 5

Turn 120 minutes back into hours: 120 ÷ 60 = 2 hr, which is choice (E).

120 min = 2 hr → (E)

Answer

2 hr

Plug x = 120 back into the average. Total minutes = 375 + 270 + 120 = 765, and 765 ÷ 9 = 85 — exactly the target. The answer also passes a quick sanity check: on the first 5 days Gage was 85 - 75 = 10 minutes short each day (50 minutes short total), and on the next 3 days he was 90 - 85 = 5 minutes over each day (15 minutes over total). Net deficit after 8 days is 50 - 15 = 35 minutes, so day 9 must run 85 + 35 = 120 minutes — the same 2 hours.

💡Key takeaway

A target average sets the required total: 85 × 9 = 765 minutes. Subtract the 645 minutes already skated and only 120 minutes — exactly 2 hours — are left for day 9, answer (E).

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