AMC 8 · 2008 · #5

Grade 6 rate-ratio

Archetype Arithmetic Expression Decomposition └ Sub-archetype Linear Rate Accumulation

ratemulti-digit-arithmetic identify-subproblems ↑ Prerequisites: multi-digit-arithmeticrate

📏 Short solution 💡 2 insights

Problem

Barney's bike odometer goes from 1441 to 1661 while he rides for 4 hours one day and 6 hours the next. Find his average speed in miles per hour.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Write an Equation

Average speed is defined by a single equation: speed = distance ÷ time, so Tool #3 (Write an Equation) is the direct route. The two pieces of that equation — total distance and total time — are not handed to us, so Tool #7 (Identify Subproblems) helps us break the work into two small steps: first find the distance from the odometer change, then add the two riding times.

1STEP 1

The odometer is a running total, so subtract the readings: the trip covers 220 miles.

distance = 1661 - 1441 = 220 miles

2STEP 2

Barney rode 4 hours then 6 hours, so the total riding time is 10 hours.

time = 4 + 6 = 10 hours

3STEP 3

Divide total distance by total time: 220 ÷ 10 gives an average speed of 22 mph → (E).

average speed =

\frac{220 miles}{10 hours}

= 22 mph → (E)

Answer

Check by going backward: at 22 mph for 10 hours, Barney would cover 22 × 10 = 220 miles, which matches the odometer change from 1441 to 1661. The answer also passes a quick smell test — 22 mph is a normal cruising speed for a road bike, not absurdly fast or slow.

💡Key takeaway

Average speed is just total distance divided by total time. Get each piece in its own step, then do one division.

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