Question

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?

50

55

60

65

70

Solution:

We want to find the smallest palindrome larger than $15951,$ and to do so, we want to increase the value of the middle digit. This is because if we increased the value of any other digit with less place value (say, the ones place), we would also need to increase the value of the digit with greater place value (i.e. ten-thousands place).

Therefore, we must replace the number $9$ to the next greatest integer: $10.$ As all digits must be between $0$ and $9$ (inclusive), we write a zero in place of the nine, and increase the thousands digit (and the tens — to preserve the palindrome).

As such, the next smallest palindrome larger than $15951$ is $16061.$

As Megan drives $16061-15951$ $=110$ miles in $2$ hours, she drives at an average of $55$ mph.

Thus, B is the correct answer.

2020 AMC 10B Problem 6

Solution:

Problem 6 in Other Years