Many highways have signs along the side of the road that list your mileage from some place behind you. You are driving down a highway and notice a milepost showing distance that is a two-digit number. Exactly one hour later, you notice a milepost that shoes the same two digits as the first, in opposite order.
Then, exactly one hour after seeing the second milepost, you notice a third one that shows a three-digit number. The middle digit on this sign is 0. The other two digits are the same as those on the first post and are in the same order as on the first milepost.
How fast did you trave

Question

Many highways have signs along the side of the road that list your mileage from some place behind you. You are driving down a highway and notice a milepost showing distance that is a two-digit number.  Exactly one hour later, you notice a milepost that shoes the same two digits as the first, in opposite order. 
Then, exactly one hour after seeing the second milepost, you notice a third one that shows a three-digit number. The middle digit on this sign is 0. The other two digits are the same as those on the first post and are in the same order as on the first milepost. 
How fast did you trave

mr.math · Answer

You're saying I'm driving away from the place the milepost indicates?

anonymous · Answer

Yes. So you are driving and the posts tell you mileage from a place behind you

mr.math · Answer

Okay, this is a nice one. Let's say the distance at time $t_0=0$ hours was $a_0a_1=10a_0+a_1$, then one hour later, it became $a_1a_0=10a_1+a_0$. One hour after than, the distance became $a_00a_1=100a_0+a_1.$ Assuming I was driving at speed $x$ speed unit, then we can write the following system:
$$10a_0+a_1+1\cdot x=10a_1+a_0$$
$$10a_1+a_0+1\cdot x=100a_0+a_1$$
$$10a_0+a_1+2\cdot x=100a_0+a_1$$

Note: $1\cdot x=x$. I wrote it this way to emphasis the units.

mr.math · Answer

Your job now, as you probably know, is to solve the above system for the variable $x$.

anonymous · Answer

Sorry but the numbers are confusing me. Is there a way to write them more clearly? 

If I solve the system, I will get three variables. What do the variables represent? 

Because I am going to need a speed as my answer, right?

mr.math · Answer

Well, $a_0$ represents the 10 place in the first milepost and $a_1$ represents the 1 place in the first milepost. $x$ is the speed of the car in $\text{mph}$, assuming the given distances are in miles.

mr.math · Answer

Key note: Keep in mind that $a_0$ and $a_1$ are integers!

anonymous · Answer

Ah! Let me try something soon and then get back you !

mr.math · Answer

Good luck!