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2020 AMC 10B Problem 6

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2020 AMC 10B Problem 6AMC 10 Step by Step
Name:                             Date: September 13, 2026
Bases & Digits · difficulty 2 of 5 · about 2 min · position P1-10

Driving along a highway, Megan noticed that her odometer showed 1595115951 (miles). This number is a palindrome-it reads the same forward and backward. Then 22 hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this 22 -hour period?

  1. A)

    5050

  2. B)

    5555

  3. C)

    6060

  4. D)

    6565

  5. E)

    7070

Answer

B

Key insight

The middle digit 9 cannot go up, so the next palindrome is 16061; the trip is 110 miles in 2 hours.

Solution

A five-digit palindrome has the form abcba‾\overline{abcba}, determined by its first three digits. The current reading is 159 51159\,51 with first three digits 159159. The next palindrome comes from the next three-digit prefix, 160160, giving 1606116061. (Nothing lies between: changing only the last two digits would break the mirror, and the middle digit 99 cannot increase.)

Distance traveled: 16061−15951=11016061 - 15951 = 110 miles, in 22 hours. Average speed =1102=55= \dfrac{110}{2} = 55 miles per hour.

The answer is (B) 55\boxed{\textbf{(B)}\ 55}.

Why this works

A palindrome is completely determined by its front half, so "next palindrome" means "increment the front half by one and mirror it." Once the number is found, average speed is plain distance over time.

The trap

Taking 15951 to 16161 or 15961 as the next palindrome, or forgetting to divide the distance by 2 hours.

Common mistakes

  • Taking 15951 to 16161 or 15961 as the next palindrome, or forgetting to divide the distance by 2 hours.
  • Reporting the distance 110110 or computing 16061−1595116061 - 15951 incorrectly as 100100 (answer 50).

Techniques

Set up the equation/formula and compute; no special trick needed