AMC 10 · 2017 · #2

Grade 6 rate-ratio
Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum
rateunit-conversionmulti-digit-arithmetic identify-subproblemsdimensional-analysis ↑ Prerequisites: rate
📏 Medium solution 💡 1 insight
Problem
Sofia runs 5 laps of a 400-meter track. On every lap she covers the first 100 meters at 4 meters per second and the remaining 300 meters at 5 meters per second. Find the total time for all 5 laps and match it to a choice written as minutes and seconds.

Pick an answer.

(A)
5 minutes and 35 seconds
(B)
6 minutes and 40 seconds
(C)
7 minutes and 5 seconds
(D)
7 minutes and 25 seconds
(E)
8 minutes and 10 seconds

AMC 10 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Analyze the Units

Tool #8 (Analyze the Units): the speeds are in meters per second and the distances are in meters, so dividing meters by meters-per-second leaves seconds — exactly the time we want. Tool #7 (Identify Subproblems): one lap splits into a 100-meter piece and a 300-meter piece run at different speeds, so time them separately and add. Tool #5 (Look for a Pattern): all 5 laps are identical, so find one lap's time and multiply by 5, then convert the seconds into minutes and seconds.

1STEP 1

Time for the first 100 m

Distance over speed: the first 100 meters at 4 meters per second takes 100 ÷ 4 = 25 seconds.

(100 m)/(4 m/s) = 25 s
2STEP 2

Time for the last 300 m

The last 300 meters at 5 meters per second takes 300 ÷ 5 = 60 seconds.

(300 m)/(5 m/s) = 60 s
3STEP 3

Time for one lap

One lap is the two parts back to back: 25 + 60 = 85 seconds per lap.

25 s + 60 s = 85 s per lap
4STEP 4

Time for five laps

All 5 laps are identical at 85 seconds each, so 85 × 5 = 425 seconds total.

85 s × 5 = 425 s
5STEP 5

Convert to minutes and seconds

Split 425 by 60: 425 = 7 × 60 + 5, giving 7 minutes and 5 seconds — choice (C).

425 s = 7× 60 + 5 = 7 min 5 s → (C)
Answer
7 minutes and 5 seconds
Sanity-check the size: each lap is 85 seconds, a bit under 1.5 minutes, so 5 laps should be a bit under 7.5 minutes — and 7 minutes 5 seconds fits. The slow 100 meters (25 s) versus the fast 300 meters (60 s) also makes sense: even though the second part is three times longer, the higher speed keeps it from being three times the time. A rough average speed of 400 ÷ 85 ≈ 4.7 m/s sits between 4 and 5, as it must.
💡Key takeaway

Time is distance divided by speed, so each lap is 100÷4 + 300÷5 = 85 seconds; five identical laps give 425 seconds, which is 7 minutes and 5 seconds — choice (C).

  • Time for the first 100 m
  • Time for the last 300 m
  • Time for one lap
  • Time for five laps
  • Convert to minutes and seconds

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