AMC 10 · 2009 · #4

Grade 6 rate-ratio
Archetype Meeting Time with Piecewise Rates └ Sub-archetype Piecewise-Leg Time Balance
rateunit-conversionfraction-arithmetic identify-subproblemswork-backwards ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights
Problem
Eric does a triathlon with three legs: a 1/4-mile swim at 2 miles per hour, a 3-mile run at 6 miles per hour, and a 15-mile bike ride. He wants the whole thing to take exactly 2 hours. Find the average speed, in miles per hour, the bike ride must have.

Pick an answer.

(A)
$\ \frac{120}{11}$
(B)
$\ 11$
(C)
$\ \frac{56}{5}$
(D)
$\ \frac{45}{4}$
(E)
$\ 12$

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

How to solve
Strategy Analyze the Units

This is a distance-speed-time problem, so Tool #8 (Analyze the Units) is the spine: the relationship time = distance ÷ speed converts every leg into a number of hours and, at the end, turns a leftover time back into a speed. Tool #7 (Identify Subproblems) splits the race into its three legs so the two known legs can be measured first. Tool #11 (Work Backwards) is the finishing move: the finish time (2 hours) is fixed, so we subtract the known leg times to see how much time the bike is allowed, then read off the speed that fits that time.

1STEP 1

Set the time budget rule

Time = distance ÷ speed, and the three leg times must add to the 2-hour goal, so whatever the swim and run leave over belongs to the bike.

time=distance/speed, t_swim+t_run+t_bike=2 hours
2STEP 2

Time the swim and the run

Swim: 1/4 mile at 2 mph takes 1/4 ÷ 2 = 1/8 hour. Run: 3 miles at 6 mph takes 3 ÷ 6 = 1/2 hour.

t_swim=1/4/2=1/8 h, t_run=3/6=1/2 h
3STEP 3

Find the bike's time budget

Add the known times: 1/8 + 4/8 = 5/8 hour. Subtracting that from the 2-hour budget leaves the bike 11/8 hour.

t_swim+t_run=1/8+4/8=5/8 h, t_bike=2-5/8=11/8 h
4STEP 4

Turn the time back into a speed

The bike covers 15 miles in 11/8 hour, so its speed is 15 ÷ 11/8 = 15 × 8/11 = 120/11 miles per hour — choice (A).

speed=15/ 11/8 =15×8/11=120/11 mph (A)
Answer
120/11
Check the total time with speed 120/11: the bike takes 15÷120/11=(15 × 11)/120=165/120=11/8 hour, and 1/8+1/2+11/8=1/8+4/8+11/8=16/8=2 hours exactly, matching the goal. The value 120/11≈ 10.9 mph is a believable bike speed — faster than the run but well short of the smaller choices being too large — so answer (A) holds.
💡Key takeaway

A fixed finish time is a budget of hours: measure the legs you can, subtract to see what time is left, then divide the miles by that time to get the speed.

  • Set the time budget rule
  • Time the swim and the run
  • Find the bike's time budget
  • Turn the time back into a speed

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