AMC 10 · 2008 · #1
Grade 4 rate-ratioPick an answer.
AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
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.
Measure the one-third interval
From 8:30 AM to 11:10 AM the machine ran 2 hours 40 minutes, and that stretch finished one third of the job.
The clock difference between two times is just how long the machine has been running.
3.MD.A.1Analyze The UnitsScale up to the whole job
Each third takes the same time, so the whole job is 3 times 160 minutes = 480 minutes, exactly 8 hours.
If one third takes a set time, the whole is just three of those equal times stacked together.
If one third takes a set time, the whole job is three of those equal times stacked together.
▸ Why?
A third is one of three equal shares, so three of them rebuild the whole.
▸ Why?
Three equal stretches of time are three equal groups, which is what multiplying by three records.
Add the total time to the start
Add 8 hours to the 8:30 AM start: 4 hours reaches 12:30 PM, and 4 more lands on 4:30 PM — choice (D).
Finishing time is the start time pushed forward by the total run length.
3.MD.A.1Identify SubproblemsIf 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