AMC 10 · 2020 · #6

Grade 4 rate-ratio

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

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

📏 Short solution 💡 2 insights

Problem

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.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Look for a Pattern

A 5-digit palindrome has the shape 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.

1STEP 1

Write 15951 as abcba (a=1, b=5, c=9); the next palindrome comes from bumping the middle digit c.

15951 = 1 5 9 5 1

2STEP 2

Since c=9, bumping carries: thousands 5→6, middle resets, and the palindrome shape forces 16061.

1 5 9 5 1 → 1 6 0 6 1 = 16061

3STEP 3

Subtract the readings: the distance driven is 110 miles.

16061 - 15951 = 110 miles

4STEP 4

Divide distance by time: $\frac{110 miles}{2 hours}$ = 55 mph.

\frac{110 miles}{2 hours}

= 55 mph

5STEP 5

55 matches (B); the others need distances 100, 120, 130, 140 — none reaching a palindrome from 15951.

55 → (B)

Answer

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 ≥ 10 to exceed 15951, which is impossible. So 16061 is genuinely the next one, distance = 110, speed = 55 mph. ✓

💡Key takeaway

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.

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