AMC 10 · 2009 · #20

Grade 6 rate-ratio
Archetype Meeting Time with Piecewise Rates └ Sub-archetype Piecewise-Leg Time Balance
rateratio-proportionsystems-of-equations identify-subproblems ↑ Prerequisites: rate
📏 Medium solution 💡 3 insights
Problem
Andrea and Lauren start 20 kilometers apart and bike toward each other. Andrea's speed is three times Lauren's, and while both ride the gap between them shrinks by 1 kilometer every minute. After 5 minutes Andrea gets a flat tire and stops, waiting for Lauren, who keeps riding. Find the total time, counted from the start, until Lauren reaches Andrea.

Pick an answer.

(A)
$\ 20$
(B)
$\ 30$
(C)
$\ 55$
(D)
$\ 65$
(E)
$\ 80$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Analyze the Units

This is a rate problem, so I track speeds and how the closing rate changes. The given 1 km per minute is the two riders' speeds added together. I split the trip into two phases, both biking and then Lauren alone, because the closing rate is different in each, and units of km and minutes keep every step honest.

1STEP 1

Split the closing rate

That 1 km/min closing rate is both speeds added, and a 3-to-1 split gives Lauren 14\frac{1}{4} km/min, Andrea 34\frac{3}{4} km/min.

Lauren=1/4 km/min, Andrea=3/4 km/min
2STEP 2

Phase 1: both bike 5 minutes

Both ride for the first 5 minutes, closing 5 km, so 15 km of the original 20 remain when Andrea stops.

20-(1)(5)=15 km
3STEP 3

Phase 2: Lauren alone

With Andrea parked, only Lauren closes the gap: 15 divided by 14\frac{1}{4} is 15 times 4, or 60 minutes.

15÷1/4=15 × 4=60 min
4STEP 4

Add the two phases

Add the phases: 5 minutes together plus 60 minutes alone gives 65 minutes, choice (D).

5+60=65 min
Answer
65
Check with distances instead. Andrea rides only 5 minutes at 3/4 km/min, covering 3.75 km, so Lauren must cover the remaining 20 minus 3.75, which is 16.25 km. Lauren rides at 1/4 km/min, needing 16.25 times 4, which is 65 minutes. Same answer, so 65 is consistent. It is also sensibly larger than the 20 minutes it would take if both kept riding at 1 km/min, since Andrea quitting slows the closing.
💡Key takeaway

When one mover stops, the gap only closes as fast as whoever is still moving, so find the rate again for each phase.

  • Split the closing rate
  • Phase 1: both bike 5 minutes
  • Phase 2: Lauren alone
  • Add the two phases

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