AMC 8 · 2004 · #10

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

unit-conversionratefraction-arithmeticmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: multi-digit-arithmeticfraction-arithmetic

📏 Medium solution 💡 2 insights

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Problem

Aaron worked four shifts:

\frac{11}{4}

hours Monday, 50 minutes Tuesday, 8{:}20 to 10{:}45 Wednesday morning, and a half-hour Friday. He earns $3 per hour. How much did he earn that week?

Pick an answer.

(A)

$\textdollar 8$

(B)

$\textdollar 9$

(C)

$\textdollar 10$

(D)

$\textdollar 12$

(E)

$\textdollar 15$

AMC 8 2004 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Break into Subproblems

The four shifts come in four different formats — a mixed number of hours, a count of minutes, a clock-time range, and the phrase "a half-hour." Tool #7 (Break into Subproblems) handles each shift on its own: convert it to a common unit, then move on. Tool #2 (Make a List or Table) keeps the four numbers lined up so nothing gets dropped before the final sum. Once the total time is in hand, multiplying by $3 per hour is one short step.

1STEP 1

Monday: $\frac{11}{4}$ hours = $\frac{5}{4}$ × 60 = 75 min.

\frac{5}{4}

× 60 = 75 min

2STEP 2

Tuesday is already 50 min — no conversion needed.

50 min

3STEP 3

Wednesday: 8{:}20 to 10{:}45 is 2 h 25 min = 145 min.

2 h 25 min = 2(60) + 25 = 145 min

4STEP 4

Friday: half an hour = $\frac{1}{2}$ × 60 = 30 min.

\frac{1}{2}

× 60 = 30 min

5STEP 5

Add the four shifts: 75 + 50 + 145 + 30 = 300 min = 5 h, then × $3 gives (E).

Mon & 75 ; Tue & 50 ; Wed & 145 ; Fri & 30 ; Total & 300 → 300 ÷ 60 = 5 h → 5 ×

3 =

15 → (E)

Answer

textdollar 15

Cross-check in hours instead of minutes: 1.25 +

\frac{50}{60}

+ 2

\frac{25}{60}

+ 0.5 = 1.25 + 0.833… + 2.416… + 0.5 = 5 hours, matching the

\frac{300}{60}

result. 5 ×

3 =

15 confirms (E). The smaller choices (A)

8 and (B)

9 would need under 3 hours of work, but Wednesday alone is almost 2.5 hours, so they are far too low. (D) $12 would mean 4 hours total, which leaves out roughly an hour of work — also short.

💡Key takeaway

When a problem mixes hours, minutes, and clock times, convert every shift to one unit first — then a single addition and one multiplication finish the job.

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