AMC 10 · 2008 · #1

Grade 4 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

rateratio-proportionunit-conversion dimensional-analysisidentify-subproblems ↑ Prerequisites: rate

📏 Short solution 💡 1 insight

Problem

A bakery owner turns on his doughnut machine at $\text{8:30}\ {\small\text{AM}}$ . At $\text{11:10}\ {\small\text{AM}}$ the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?

Pick an answer.

(A)

$\ \text{1:50}\ {\small\text{PM}}$

(B)

$\ \text{3:00}\ {\small\text{PM}}$

(C)

$\ \text{3:30}\ {\small\text{PM}}$

(D)

$\ \text{4:30}\ {\small\text{PM}}$

(E)

$\ \text{5:50}\ {\small\text{PM}}$

AMC 10 2008 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.

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Toolkit + CCSS Solution

Understand

Restated: A doughnut machine starts at 8:30 AM. By 11:10 AM it has finished one third of the day's job. Find the clock time when it finishes the whole job, assuming it keeps working at the same steady rate.

Givens: The machine turns on at 8:30 AM.; At 11:10 AM, one third of the job is complete.; The machine runs at a constant rate.

Unknowns: The clock time when the full job is finished.

Understand

Givens: The machine turns on at 8:30 AM.; At 11:10 AM, one third of the job is complete.; The machine runs at a constant rate.

Plan

Primary tool: #8 Analyze the Units

Secondary: #7 Identify Subproblems

The job runs at a steady rate, so a chunk of time is a unit of work. First measure how long one third took (a time interval), then scale that interval up to the whole job, then add it back onto the start time. Splitting it into these three time-pieces keeps the clock arithmetic clean.

Execute — Answer: D

#8 Analyze the Units 3.MD.A.1 Step 1

Measure the one-third interval

Find how much time passed from the 8:30 AM start to the 11:10 AM checkpoint.
From 8:30 to 11:30 is 3 hours, but 11:10 is 20 minutes earlier, so the gap is 2 hours 40 minutes.
That stretch of time produced one third of the job.

$$11{:}10 - 8{:}30 = 2\text{ h } 40\text{ min}$$

💡 The clock difference between two times is just how long the machine has been running.

#8 Analyze the Units 4.NF.B.4 Step 2

Scale up to the whole job

One third of the job took 2 hours 40 minutes.
Because the machine works at a steady rate, each third takes the same amount of time, so the whole job is three of those equal pieces.
Multiply the interval by 3: 2 hours 40 minutes is 160 minutes, and 3 times 160 is 480 minutes, which is exactly 8 hours.

$$3 \times (2\text{ h } 40\text{ min}) = 3 \times 160\text{ min} = 480\text{ min} = 8\text{ h}$$

💡 If one third takes a set time, the whole is just three of those equal times stacked together.

#7 Identify Subproblems 3.MD.A.1 Step 3

Add the total time to the start

The full job takes 8 hours starting from 8:30 AM.
Add 8 hours: 8:30 AM plus 4 hours is 12:30 PM, and 4 more hours is 4:30 PM.
So the machine finishes at 4:30 PM, which is choice (D).

$$8{:}30\ \text{AM} + 8\text{ h} = 4{:}30\ \text{PM}$$

💡 Finishing time is the start time pushed forward by the total run length.

[1] #8 3.MD.A.1 Find how much time passed from the 8:30 AM start to the 11:10 AM checkpoint. Fro

[2] #8 4.NF.B.4 One third of the job took 2 hours 40 minutes. Because the machine works at a ste

[3] #7 3.MD.A.1 The full job takes 8 hours starting from 8:30 AM. Add 8 hours: 8:30 AM plus 4 ho

Review

Reasonableness: One third took 2 hours 40 minutes, so two thirds more is another 5 hours 20 minutes past 11:10 AM. 11:10 AM plus 5 hours 20 minutes lands on 4:30 PM, matching the answer from adding 8 hours to the start. Both routes agree on (D), and 4:30 PM is a sensible full workday of 8 hours.

Alternative: Instead of scaling from the start, work forward from the 11:10 AM checkpoint. Two thirds of the job remain, and each third is 2 hours 40 minutes, so 2 times 2 hours 40 minutes is 5 hours 20 minutes. Adding that to 11:10 AM gives 4:30 PM.

CCSS standards used (min grade 4)

3.MD.A.1 Tell and write time to the nearest minute and solve elapsed time problems (Measuring the 2 h 40 min interval and adding 8 hours to the 8:30 AM start to land on 4:30 PM.)
4.NF.B.4 Apply and extend understanding of multiplication to multiply a fraction by a whole number (Turning 'one third of the job' into the whole by taking three equal time-pieces: whole = 3 x (one third).)

⭐ If one third of a steady job took some time, the whole job takes three times that, so measure the piece and multiply.

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: D

Measure the one-third interval

Scale up to the whole job

Add the total time to the start

Review

CCSS standards used (min grade 4)

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