AMC 10 · 2008 · #6

Grade 7 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Unit Conversion Chain

rateratio-proportionunit-conversion convert-to-algebradimensional-analysis ↑ Prerequisites: rate

📏 Medium solution 💡 2 insights

Problem

A race has three parts of equal length: swim, bike, and run. The athlete swims at 3 km/h, bikes at 20 km/h, and runs at 10 km/h. Which of the given choices is closest to the average speed for the whole race, in km/h?

Pick an answer.

(A)

$\ 3$

(B)

$\ 4$

(C)

$\ 5$

(D)

$\ 6$

(E)

$\ 7$

AMC 10 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The segment length is never given, so I name it with a variable. Because each part is the same length, that length appears in both the total distance and the total time, and it cancels. The unit relation distance = rate x time turns each rate into a travel time, and the whole race becomes total distance over total time.

1STEP 1

Name the segment length

Let $d$ be one segment's length in km; each of the three covers $d$ , so the total distance is $3d$ km.

total distance = d + d + d = 3d

2STEP 2

Turn each rate into a time

Time is distance over rate, so the segments take $\frac{d}{3}$ , $\frac{d}{20}$ , and $\frac{d}{10}$ hours.

t_swim = d/3, t_bike = d/20, t_run = d/10

3STEP 3

Add the three times

Over the common denominator 60 the three times add to $\frac{29d}{60}$ hours.

d/3 + d/20 + d/10 = (20d + 3d + 6d)/60 = 29d/60

4STEP 4

Divide distance by time

Divide $3d$ by $\frac{29d}{60}$ : multiply by the reciprocal, the $d$ cancels, leaving $\frac{180}{29}$ km/h.

avg speed = 3d/29d/60 = 3d · 60/29d = 180/29

5STEP 5

Round to the closest choice

Since $\frac{180}{29} \approx <span class="hl-ask">6</span>.21$ , the nearest choice among 3, 4, 5, 6, 7 is 6, choice (D).

180/29 ≈ 6.21 → 6

Answer

The average speed 180/29 ≈ 6.21 sits between the slowest rate (3) and the fastest (20), which any average must. It is much closer to the slow rates than to 20, which fits intuition: the swim is so slow that it dominates the time, dragging the average down toward the low end. 6 km/h is the closest choice, confirming (D).

💡Key takeaway

Average speed is total distance over total time, not the average of the speeds, so the slow swim pulls the whole race down toward 6 km/h.

Name the segment length
Turn each rate into a time
Add the three times
Divide distance by time
Round to the closest choice

Name the segment length

Turn each rate into a time

Add the three times

Divide distance by time

Round to the closest choice

More like this