There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same.

There is a circular path around a sports field. Sonia takes 18 minutes to drive one round of the field, while Ravi takes 12 minutes for the same. Suppose they both start at the same point and at the same time, and go in the same direction. After how many minutes will they meet again at the starting poin?
Ans. It can be observed that Ravi takes lesser time than Sonia for completing 1 round of the circular path. As they are going in the same direction, they will meet again at the same time when Ravi will have completed 1 round of that circular path with respect to Sonia. And the total time taken for completing this 1 round of circular path will be the LCM of time taken by Sonia and Ravi for completing 1 round of circular path respectively i.e., LCM of 18 minutes and 12 minutes.
18 = 2 x 3 x 3
And, 12 = 2 x 2 x 3
LCM of 12 and 18 = 2 x 2 x 3 x 3 = 36
Therefore, Ravi and Sonia will meet together at the starting point after.

Ques- Prove that √5 is irrational.
Ans. let √5 is a rational number.
Therefore, we can find two integers a, b (b≠0) such that √5 = a/b
Let a and b have a common factor other than 1. Then we can divide them by the common factor, and assume that a and b are co-prime,
a =√5 b
a² =5b²
therefore, a² is divisible by 5 and it can be said that a is divisible by 5.
Let a = 5k, where k is an integer
(5k) ² = 5b²
b²=5k² this means that b² is divisible by 5 and hence, b is divisible by 5.
This implies that a and b have 5 as a common factor.
And this is a contradiction to the fact that a and b are co-prime.
Hence, √5 cannot be expressed as p/q or it can be said that √5 is irrational.