Find Speed ...

Number Theory Level 2

Assume a car is moving with constant speed .

The driver sees a 2-digit number on a milestone and an hour later he sees another milestone with the 2-digits reversed as the previous one .

Moving further an hour later he sees a milestone containing same 2 digits with a zero between them . What is the speed of the driver (in miles per hour) ?

This problem is part of the set : Easy Pies ...


The answer is 45.

Pranay Singh by Pranay Singh , 22, India

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2 solutions

Let the original speed be a b ‾ \overline{ab} mph. Then the first milestone reads 10 a + b 10a + b miles, the second 10 b + a 10b + a miles, and the third 100 a + b 100a + b miles. Now as these three markers represent an arithmetic progression, we require that

( 10 b + a ) − ( 10 a + b ) = ( 100 a + b ) − ( 10 b + a ) ⟹ 9 b − 9 a = 99 a − 9 b ⟹ 18 b = 108 a ⟹ b = 6 a (10b + a) - (10a + b) = (100a + b) - (10b + a) \Longrightarrow 9b - 9a = 99a - 9b \Longrightarrow 18b = 108a \Longrightarrow b = 6a .

Now as a , b a,b must be single digits, the only possibility is that a = 1 , b = 6 a = 1, b = 6 , and so the distance between successive markers is 61 − 16 = 45 61 - 16 = 45 miles. As this distance is covered in one hour, we can conclude that the constant speed of the car is 45 \boxed{45} mph.

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Pranay Singh
Nov 30, 2017

First he saw 16 , then 61 , and then 106

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