AMC 10 · 2020 · #6

Grade 4 rate-ratio

Archetype Small-Domain Constrained Search

place-valuepattern-recognitionrate pattern-recognitionidentify-subproblems ↑ Prerequisites: place-valuerate

📏 Short solution 💡 2 insights

Problem

Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$ -hour period?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

View mode:

Toolkit + CCSS Solution

Understand

Restated: Megan's odometer reads $15951$, a palindrome. After driving for $2$ hours, it shows the next larger palindrome. Find her average speed in miles per hour.

Givens: Starting odometer reading is $15951$ (a $5$-digit palindrome); Time elapsed is $2$ hours; Ending odometer reading is the next palindrome after $15951$; Choices: (A) $50$, (B) $55$, (C) $60$, (D) $65$, (E) $70$

Unknowns: Average speed in miles per hour during the $2$-hour drive

Understand

Restated: Megan's odometer reads $15951$, a palindrome. After driving for $2$ hours, it shows the next larger palindrome. Find her average speed in miles per hour.

Plan

Primary tool: #5 Look for a Pattern

Secondary: #7 Identify Subproblems, #3 Eliminate Possibilities

A $5$-digit palindrome has the shape $\overline{abcba}$ — the first three digits force the last two. Tool #5 says: read the palindrome pattern to find the smallest valid bump. Tool #7 then splits the problem into two clean parts: (1) find the next palindrome, (2) divide by $2$. Tool #3 cross-checks the speed against the five answer choices.

Execute — Answer: B

#5 Look for a Pattern 4.NBT.A.2 Step 1

Write the starting reading as $\overline{abcba}$ with $a = 1, b = 5, c = 9$.
To get the next palindrome, bump the middle digit $c$ by $1$ — that lifts the number by the smallest possible amount.

$$15951 = \overline{1\,5\,9\,5\,1}$$

💡 The middle digit is the cheapest knob to turn on a palindrome.

#5 Look for a Pattern 4.NBT.A.2 Step 2

But $c = 9$ already — bumping it triggers a carry into the thousands place.
The thousands digit $5$ becomes $6$, the middle digit resets to $0$, and the palindrome shape forces the hundreds digit to $0$ and the units digit to stay matching the ten-thousands digit.

$$\overline{1\,5\,9\,5\,1} \xrightarrow{\text{carry}} \overline{1\,6\,0\,6\,1} = 16061$$

💡 When a digit is already $9$, the bump cascades — like odometer rollover.

#7 Identify Subproblems 4.NBT.B.4 Step 3

Compute the distance driven by subtracting.

$$16061 - 15951 = 110 \text{ miles}$$

💡 Distance equals end reading minus start reading.

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

Divide distance by time to get average speed.

$$\dfrac{110 \text{ miles}}{2 \text{ hours}} = 55 \text{ mph}$$

💡 Speed is distance per unit time — divide.

#3 Eliminate Possibilities 4.NBT.A.2 Step 5

$55$ matches choice (B).
The other choices ($50, 60, 65, 70$) correspond to distances $100, 120, 130, 140$ — none of which lands on a palindrome reading from $15951$.

$$55 \Rightarrow \textbf{(B)}$$

💡 Match the computed speed against the listed choices.

[1] #5 4.NBT.A.2 Write the starting reading as $\overline{abcba}$ with $a = 1, b = 5, c = 9$. To

[2] #5 4.NBT.A.2 But $c = 9$ already — bumping it triggers a carry into the thousands place. The

[3] #7 4.NBT.B.4 Compute the distance driven by subtracting.

[4] #7 3.OA.A.3 Divide distance by time to get average speed.

[5] #3 4.NBT.A.2 $55$ matches choice (B). The other choices ($50, 60, 65, 70$) correspond to dist

Review

Reasonableness: Verify that $16061$ really is a palindrome: reading right-to-left gives $1, 6, 0, 6, 1$ — same as left-to-right. Also confirm no smaller palindrome sits between $15951$ and $16061$: any palindrome starting $15\_\_\_$ must end in $51$, and its middle digit would have to be $\geq 10$ to exceed $15951$, which is impossible. So $16061$ is genuinely the next one, distance $= 110$, speed $= 55$ mph. ✓

Alternative: Tool #3 (Eliminate) alone: each answer choice gives a candidate end-odometer ($16051, 16061, 16071, 16081, 16091$). Only $16061$ is a palindrome — instantly picks (B) without explicitly finding the next palindrome from scratch.

CCSS standards used (min grade 4)

4.NBT.A.2 Read and write multi-digit whole numbers and compare using symbols (Reading $15951$ as $\overline{1\,5\,9\,5\,1}$ to inspect place values and matching $16061$ against the digit positions to confirm a palindrome and the answer choice.)
4.NBT.B.4 Fluently add and subtract multi-digit whole numbers (Subtracting $16061 - 15951 = 110$ to find the distance driven.)
3.OA.A.3 Solve multiplication and division word problems within 100 (Dividing distance $110$ miles by time $2$ hours to find average speed $55$ mph.)

⭐ This AMC 10 problem only needs Grade 4 place value you already know — to find the next palindrome after $15951$, bump the middle $9$ which forces a carry to $16061$. The drive is $110$ miles in $2$ hours, so the speed is $55$ mph.

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: B

Review

CCSS standards used (min grade 4)

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