 Hello and welcome to the session. In this session first we will discuss about direct proportion. Consider two quantities x and y. We say that x and y are in direct proportion if the increase or decrease together in such a manner that the ratio of their corresponding values remains constant. That is we say that x and y are in direct proportion if we have x upon y is equal to k where this k is a positive number or we can say that x is equal to ky then we say that x and y are in direct proportion. And if we have that x and y are in direct proportion then we can also say that x1 upon y1 is equal to x2 upon y2 where we have y1, y2 are the values of y corresponding to the values x1, x2 of x respectively. We also have that the variables increasing or decreasing together need not always be in direct proportion. There is no direct relationship between the height of a tree and the number of leaves growing on its branches so they are not in direct proportion. Let's consider one example suppose that a machine in a soft drink factory fills 840 bottles in 6 hours. We need to find out how many bottles will it fill in 5 hours. Let us suppose that the number of bottles filled in 5 hours be equal to x. Consider this table in this we have the number of bottles filled in 6 hours is 840 and the number of bottles filled in 5 hours is x. Now more the number of hours more bottles will be filled and lesser the number of hours less number of bottles will be filled. So we say that the number of bottles and the number of hours are in direct proportion so we get that 840 upon 6 is equal to x upon 5. Since we know that if x and y are the two quantities which are in direct proportion then x1 upon y1 is equal to x2 upon y2 where y1, y2 are the values of y corresponding to the values of x that is x1 and x2. From here we get x is equal to 700 that is the number of bottles filled in 5 hours is 700. Next we shall discuss inverse proportion. Consider two quantities x and y. x and y are said to be in inverse proportion if an increase in one quantity produces degrees in the other quantity and vice versa and such that their product that is xy remains constant that is equal to k where this k is a constant. We can also say that x and y are an inverse proportion then we have x1, y1 is equal to x2, y2 or this could be written as x1 upon x2 is equal to y1 upon y2 where y1, y2 are the values of y corresponding to values x1, x2 of x respectively. Like for example if we increase the number of workers for a particular job then obviously the time taken to complete that job would decrease. So the number of workers for the job and the time to complete that job are in inverse proportion. Let's consider one example in which we have that a car takes two hours to complete the journey by travelling at a speed of 60 kilometers per hour. How long will it take when the car travels at the speed of 80 kilometers per hour? So this is our question. Let the required time that is time taken to complete the journey when the car is travelling at 80 kilometers per hour be taken as x hours. Consider this table in which we have given the time and the speed that is when the car travels at 60 kilometers per hour it completes the journey in two hours and when the car travels at 80 kilometers per hour it completes the journey in x hours and we need to find this x. So obviously as the speed of the car increases the time to complete the journey decreases. So time and speed are in inverse proportion and so we have 2 into 60 is equal to x into 80 since we know that if two quantities x and y are in inverse proportion so we have x1, y1 is equal to x2, y2 where y is equal to y1, y2 are the values of y corresponding to the values of x that is x1, x2. From here we get value of x equal to 3 upon 2 or we can say 1.5 and thus we say that the time taken to complete the journey when the car travels at the speed of 80 kilometers per hour is 1.5 hours. So this is how we solve the problems of inverse proportion. This completes the session. Hope you understood the concept of direct and inverse proportions.