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?
In order to get the smallest palindrome greater than $15951$, we need to raise the middle digit. If we were to raise any of the digits after the middle, we would be forced to also raise a digit before the middle to keep it a palindrome, making it unnecessarily larger. So we raise $9$ to the next largest value, $10$, but obviously, that's not how place value works, so we're in the $16000$s now. To keep this a palindrome, our number is now $16061$. So Megan drove $16061-15951=110$ miles. Since this happened over $2$ hours, she drove at $\frac{110}{2}=\fbox{55}$ mph.