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 Schwinn notices that the odometer on his bicycle reads $1441$ , a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$ . What was 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.

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

Restated: 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.

Givens: Starting odometer reading: $1441$ miles; Ending odometer reading: $1661$ miles; Time spent riding: $4$ hours on day one and $6$ hours on day two; Answer choices: (A) $15$, (B) $16$, (C) $18$, (D) $20$, (E) $22$

Unknowns: The average speed, in miles per hour, over the whole ride

Understand

Restated: 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.

Plan

Primary tool: #3 Write an Equation

Secondary: #7 Identify Subproblems

Average speed is defined by a single equation: speed $=$ distance $\div$ 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.

Execute — Answer: E

#7 Identify Subproblems 4.NBT.B.4 Step 1

Find the total distance.
The odometer reads how many miles the bike has ever traveled, so the miles ridden on this trip is the difference between the two readings.

$$\text{distance} = 1661 - 1441 = 220 \text{ miles}$$

💡 Subtracting multi-digit whole numbers is a Grade 4 skill. The odometer is a running total, so the difference is exactly the new miles.

#7 Identify Subproblems 4.OA.A.3 Step 2

Find the total time.
Barney rode for $4$ hours and then $6$ hours, so add the two sessions.

$$\text{time} = 4 + 6 = 10 \text{ hours}$$

💡 Combining the two riding sessions into one total is a Grade 4 multi-step word problem move.

#3 Write an Equation 6.RP.A.2 Step 3

Apply the average-speed equation.
Divide the total distance by the total time to get miles per hour.

$$\text{average speed} = \dfrac{220 \text{ miles}}{10 \text{ hours}} = 22 \text{ mph} \;\Rightarrow\; \textbf{(E)}$$

💡 Miles per hour is a unit rate: how many miles for each $1$ hour. Dividing $220$ by $10$ gives that per-hour amount.

[1] #7 4.NBT.B.4 Find the total distance. The odometer reads how many miles the bike has ever tra

[2] #7 4.OA.A.3 Find the total time. Barney rode for $4$ hours and then $6$ hours, so add the tw

[3] #3 6.RP.A.2 Apply the average-speed equation. Divide the total distance by the total time to

Review

Reasonableness: Check by going backward: at $22$ mph for $10$ hours, Barney would cover $22 \times 10 = 220$ 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.

Alternative: Tool #6 (Guess and Check) on the answer choices: try (D) $20$ mph. In $10$ hours that gives $200$ miles, but the odometer actually advanced $220$ miles, so $20$ is too small. Try (E) $22$ mph: $22 \times 10 = 220$ miles, which is exactly right. (E) is the answer.

CCSS standards used (min grade 6)

4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm (Subtracting the two odometer readings, $1661 - 1441 = 220$, to get the total distance ridden.)
4.OA.A.3 Solve multistep word problems with whole numbers using the four operations (Adding the two riding sessions, $4 + 6 = 10$ hours, to get the total time.)
6.RP.A.2 Understand the concept of a unit rate associated with a ratio (Dividing $220$ miles by $10$ hours to find the unit rate (miles per hour), i.e., the average speed.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 6)

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