AMC 8 · 2004 · #10

Grade 5 arithmetic

Archetype Arithmetic Expression Decomposition

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

📏 Medium solution 💡 2 insights

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Problem

Handy Aaron helped a neighbor $1 \frac14$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\textdollar 3$ per hour. How much did he earn for the week?

Pick an answer.

(A)

extdollar 8

(B)

extdollar 9

(C)

extdollar 10

(D)

extdollar 12

(E)

extdollar 15

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Toolkit + CCSS Solution

Understand

Restated: Aaron worked four shifts: $1\tfrac14$ 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?

Givens: Monday: $1\tfrac14$ hours; Tuesday: $50$ minutes; Wednesday: from $8{:}20$ to $10{:}45$; Friday: a half-hour; Pay rate: $\$3$ per hour; Answer choices: (A) $\$8$, (B) $\$9$, (C) $\$10$, (D) $\$12$, (E) $\$15$

Unknowns: Total earnings for the week, in dollars

Understand

Plan

Primary tool: #7 Break into Subproblems

Secondary: #2 Make a List or Table

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.

Execute — Answer: E

#7 Break into Subproblems 5.NF.B.6 Step 1

Convert Monday to minutes.
$1\tfrac14$ hours $= \tfrac54$ hours, and each hour is $60$ minutes.

$$\tfrac54 \times 60 = 75 \text{ min}$$

💡 A quarter of $60$ is $15$, so five quarters is $5 \times 15 = 75$. The Grade 5 mixed-number multiplication move.

#7 Break into Subproblems 4.MD.A.1 Step 2

Tuesday is already given in minutes — nothing to convert.

$$50 \text{ min}$$

💡 Recognizing that one shift is already in the target unit is part of the Grade 4 unit-conversion habit: only convert what needs converting.

#7 Break into Subproblems 4.MD.A.2 Step 3

Wednesday: find the elapsed time from $8{:}20$ to $10{:}45$.
Two full hours pass from $8{:}20$ to $10{:}20$, then $25$ more minutes to $10{:}45$.
Convert to minutes.

$$2 \text{ h } 25 \text{ min} = 2(60) + 25 = 145 \text{ min}$$

💡 Counting up from the start time is the Grade 4 elapsed-time strategy: jump full hours first, then add the leftover minutes.

#7 Break into Subproblems 5.NF.B.6 Step 4

Friday's half-hour is $\tfrac12$ of $60$ minutes.

$$\tfrac12 \times 60 = 30 \text{ min}$$

💡 Half of an hour is the most familiar fraction-of-a-quantity in everyday talk; it lands at $30$ min.

#2 Make a List or Table 4.MD.A.2 Step 5

Tabulate the four shifts (Tool #2) and add.
Total minutes $\to$ total hours $\to$ dollars.

$\begin{array}{l|r} \text{Mon} & 75 \\ \text{Tue} & 50 \\ \text{Wed} & 145 \\ \text{Fri} & 30 \\ \hline \text{Total} & 300 \end{array}$ \;$\Rightarrow\; 300 \div 60 = 5 \text{ h} \;\Rightarrow\; 5 \times \$3 = \$15 \;\Rightarrow\; \textbf{(E)}$

💡 Stacking the four numbers in a small table makes the column-add automatic. Dividing $300$ by $60$ converts back to hours, then the Grade 4 "multiply to find total cost" step finishes the problem.

[1] #7 5.NF.B.6 Convert Monday to minutes. $1\tfrac14$ hours $= \tfrac54$ hours, and each hour i

[2] #7 4.MD.A.1 Tuesday is already given in minutes — nothing to convert.

[3] #7 4.MD.A.2 Wednesday: find the elapsed time from $8{:}20$ to $10{:}45$. Two full hours pass

[4] #7 5.NF.B.6 Friday's half-hour is $\tfrac12$ of $60$ minutes.

[5] #2 4.MD.A.2 Tabulate the four shifts (Tool #2) and add. Total minutes $\to$ total hours $\to

Review

Reasonableness: Cross-check in hours instead of minutes: $1.25 + \tfrac{50}{60} + 2\tfrac{25}{60} + 0.5 = 1.25 + 0.833\ldots + 2.416\ldots + 0.5 = 5$ hours, matching the $300/60$ result. $5 \times \$3 = \$15$ 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.

Alternative: Tool #2 (Make a List or Table) used on its own: list each day's time in hours instead of minutes — Mon $1\tfrac14$, Tue $\tfrac56$, Wed $2\tfrac{5}{12}$, Fri $\tfrac12$. A common denominator of $12$ gives $\tfrac{15}{12} + \tfrac{10}{12} + \tfrac{29}{12} + \tfrac{6}{12} = \tfrac{60}{12} = 5$ hours. Then $5 \times \$3 = \$15$, same answer (E).

CCSS standards used (min grade 5)

4.MD.A.1 Know relative sizes of measurement units and convert from a larger unit to a smaller unit (Converting hours to minutes ($1$ h $= 60$ min) so every shift is measured in the same unit before adding.)
4.MD.A.2 Use the four operations to solve word problems involving distances, intervals of time, and money (Finding the Wednesday interval from $8{:}20$ to $10{:}45$ and turning total hours worked into dollars at $\$3$ per hour.)
5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers (Computing $\tfrac54 \times 60 = 75$ for Monday and $\tfrac12 \times 60 = 30$ for Friday.)

⭐ 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.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 5)

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