 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that a bike travels 180 miles in the same time that a bus travels 120 miles. If the bike's speed is 30 miles per hour more than the bus's speed, find the bike's speed and the bus's speed. Let us start with the solution of the given question. Now in this question we are given that time taken by bike is equal to time taken by the bus. Also we are given that distance travelled by the bike is given as 180 miles and distance travelled by the bus is given as 120 miles and it is given that bike's speed is 30 miles more than the bus's speed. Let bus's speed be x miles per hour. So according to the given condition, bike's speed will be x plus 30 miles per hour. Now we know that time taken is given by distance travelled upon speed. Now using this formula in this question, it is given that time taken by bike is equal to time taken by bus which implies that distance travelled by bike upon speed of bike will be equal to distance travelled by bus upon speed of bus. This implies distance travelled by bike that is 180 upon speed of bike that is x plus 30 is equal to distance travelled by bus that is 120 upon speed of bus that is x. Now this is a rational equation. Here least common denominator of the rational expressions will be x into x plus 30 the whole. To simplify this further, now we multiply both sides of the rational equation by least common denominator that is x into x plus 30 the whole. We get x into x plus 30 the whole into 180 upon x plus 30 the whole and this is equal to x into x plus 30 the whole into 120 upon x the whole. Now this implies that x plus 30 cancels with x plus 30 and we get x into 180 that is 180 x and this is equal to now here x cancels with x and we get 120 into x plus 30 the whole. This implies that now 180 x is equal to 120 into x that is 120 x plus 120 into 30 that is 3600. This implies that 180 x minus 120 x is equal to 3600 which implies 180 x minus 120 x is 60 x is equal to 3600. Now dividing both sides by 60 we get 60 x upon 60 is equal to 3600 upon 60 which implies that x is equal to 60. So we have got the value of x as 60 therefore speed of the bus that is equal to x is given by 60 miles per hour and speed of the bike which is given as x plus 30 will be equal to 60 plus 30 that is equal to 90 miles per hour. Thus we have got the speed of the bus as 60 miles per hour and speed of the bike as 90 miles per hour which is the required answer. This completes our session hope you enjoyed this session.