 Hello and welcome to the session. Today I'll help you with the following question. The question says, a factory requires 42 machines to produce a given number of articles in 63 days. How many machines would be required to produce the same number of articles in 54 days? Two quantities x and y are said to vary in inverse proportion if there exists a relation of the type multiplied by y is equal to k, k being a constant. This is the key idea. Move on to the solution. Let the number of machines be x. We have the following table which says that the number of machines produced in 63 days is 42 and we are supposed to find the number of machines produced in 54 days. Let the number of days be represented by y. We see that more the number of days less machines would be required to produce the given number of articles and lesser the number of days more will be the number of machines required. So we say more the machines lesser would be the days. Hence we have the number of machines and the number of days in inverse proportion. So from the key idea we say x multiplied by y is equal to k which is the constant where x is the number of machines and y is the number of days. Using this table and this relation we say that 42 multiplied by 63 is equal to k where 42 is x that is number of machines and 63 is y that is number of days. Also x multiplied by 54 is equal to k where x is the number of machines and 54 that is y is the number of days. From these two relations we get 42 multiplied by 63 is equal to x multiplied by 54. So from here we get x is equal to 42 multiplied by 63 upon 54. 9 6 times is 54 and 9 7 times is 63 and 6 7 times is 42 so we have x is equal to 7 multiplied by 7 that is 49. Thus we have number of machines required is equal to 49. Hence our final answer is 49 machines. So hope you enjoyed the session. Have a good day.