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 machine starts in the morning and has finished one third of the job by a later time. It works at a steady rate. Find the clock time when it finishes.

Pick an answer.

(A)

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

(B)

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

(C)

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

(D)

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

(E)

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

How to solve

Strategy Analyze the Units

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.

1STEP 1

Measure the one-third interval

The first third took 2 hours 40 minutes.

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

2STEP 2

Scale up to the whole job

A steady rate scales that up to 8 hours.

3 × (2 h 40 min) = 3 × 160 min = 480 min = 8 h

3STEP 3

Add the total time to the start

Adding it to the start gives 4:30 PM, choice (D).

8{:}30 AM + 8 h = 4{:}30 PM

Answer

4:30 PM

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.

💡Key takeaway

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

Measure the one-third interval
Scale up to the whole job
Add the total time to the start

Measure the one-third interval

Scale up to the whole job

Add the total time to the start

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