AMC 8 · 2009 · #14

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Multi-Segment Path or Count Sum

ratefraction-arithmeticmean-median-mode-range identify-subproblemsdimensional-analysis ↑ Prerequisites: fraction-arithmeticrate

📏 Medium solution 💡 3 insights

Problem

Austin and Temple are 50 miles apart. Bonnie drives the 50 miles from Austin to Temple at an average speed of 60 mph, then rides a bus back the same 50 miles to Austin at an average speed of 40 mph. What is her average speed over the entire 100-mile round trip, in mph?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Analyze the Units

The trap here is averaging 60 and 40 to get 50 — that ignores units. Tool #8 (Analyze the Units) keeps us honest: "mph" is miles per hour, so the only correct average speed is (total miles)/(total hours). Tool #7 (Identify Subproblems) splits the trip into the two legs so we can compute each leg's time from time = distance / speed, then add to get the total time before dividing into the total distance.

1STEP 1

Outbound leg: 50 miles at 60 mph gives time = distance ÷ speed = $\frac{5}{6}$ hr.

t₁ = (50 mi)/(60 mph) =

\frac{5}{6}

2STEP 2

Return leg: 50 miles at 40 mph gives time = $\frac{5}{4}$ hr, longer because the speed is lower.

t₂ = (50 mi)/(40 mph) =

\frac{5}{4}

3STEP 3

Add the leg times using a common denominator of 12: $\frac{5}{6}$ + $\frac{5}{4}$ = $\frac{25}{12}$ hr total.

T =

\frac{5}{6}

\frac{5}{4}

\frac{10}{12}

\frac{15}{12}

\frac{25}{12}

4STEP 4

Total distance: the same 50-mile route twice, so 2 × 50 = 100 miles.

D = 2 × 50 = 100 mi

5STEP 5

Average speed = total distance ÷ total time = 100 ÷ $\frac{25}{12}$ = 100 × $\frac{12}{25}$ = 48 mph → (B).

v =

\frac{D}{T}

= (100 mi)/(

\frac{25}{12}

hr) = 100 ×

\frac{12}{25}

\frac{1200}{25}

= 48 mph → (B)

Answer

The naive average

\frac{60 + 40}{2}

= 50 mph would only be correct if Bonnie spent equal time at each speed. But she spent equal distance at each speed, so the slower 40-mph leg ate up more hours and pulled the average below 50. The answer 48 mph is just under 50, which matches that intuition. It also sits between 40 and 60, as any sensible average must.

💡Key takeaway

Average speed isn't just the average of two speeds — it's total miles divided by total hours, a Grade 6 rate idea you already use!

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