# CAT Practice : Speed Time, Races

Number Theory combined with Circular Races, whowuda thunk?

## Speed Time - Races

Q.1: Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6 m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?
1. 3
2. 4
3. 7
4. 5

Choice A. 3

## Detailed Solution

Let track length be equal to T.
Time taken to meet for the first time = ${{T \over relative speed} {=} {T \over 6-b}}$ or ${T \over b-6}$

Time taken for a lap for A = ${T \over 6}$
Time taken for a lap for B = ${T \over b}$

So, time taken to meet for the first time at the starting point = LCM ${{\left({{T \over 6}, {T \over b}} \right)} {=} {T \over HCF(6, b)}}$

Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b).

So, in essence we have to find values for b such that ${6 - b \over HCF(6, b)}$ = 2 or ${b - 6 \over HCF(6, b)}$ = 2

b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.

## Our Online Course, Now on Google Playstore!

### Fully Functional Course on Mobile

All features of the online course, including the classes, discussion board, quizes and more, on a mobile platform.

### Cache Content for Offline Viewing

Download videos onto your mobile so you can learn on the fly, even when the network gets choppy!

## More questions from Speed Time, Races

"Life is a race ... if you don't run fast ... you will be like a broken undaa" - Veerusahasra Buddhi