AMC 8 · 2008 · #5
Grade 6 rate-ratioProblem
Barney Schwinn notices that the odometer on his bicycle reads , a palindrome, because it reads the same forward and backward. After riding more hours that day and the next, he notices that the odometer shows another palindrome, . What was his average speed in miles per hour?
Pick an answer.
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.
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) $22$
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
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.
💡 Subtracting multi-digit whole numbers is a Grade 4 skill. The odometer is a running total, so the difference is exactly the new miles.
4.OA.A.3 Step 2 - Find the total time.
- Barney rode for $4$ hours and then $6$ hours, so add the two sessions.
💡 Combining the two riding sessions into one total is a Grade 4 multi-step word problem move.
6.RP.A.2 Step 3 - Apply the average-speed equation.
- Divide the total distance by the total time to get miles per hour.
💡 Miles per hour is a unit rate: how many miles for each $1$ hour. Dividing $220$ by $10$ gives that per-hour amount.
4.NBT.B.4 Find the total distance. The odometer reads how many miles the bike has ever tra 4.OA.A.3 Find the total time. Barney rode for $4$ hours and then $6$ hours, so add the tw 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.4Fluently 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.3Solve 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.2Understand 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.
⭐ Average speed is just total distance divided by total time. Get each piece in its own step, then do one division.