 Hello and welcome to the session. In this session we discuss the following question which says find two numbers whose mean proportional is 24 and third proportional is 1536. If we have three quantities A, B and C and it's given that A, B and C are in continued proportion, then we have A is to B is equal to B is to C and here this B is called the mean proportional between A and C and C is called the third proportional to A and B. This is the key idea that we use in this question. Let's move on to the solution now. We are supposed to find two numbers whose mean proportional and third proportional are given to us. So first of all we assume let the required two numbers B, X, Y, we know that if A is to B is equal to B is to C then B is the mean proportional between A and C and we are given that the mean proportional between the two numbers that we have assumed that is X and Y is given as 24. Therefore this would mean that square root of XY is equal to 24. Since we know that A upon B is equal to B upon C when A, B and C are in continued proportion. So this gives us B square equal to AC or you can say that B is equal to square root of AC and we know that B is the mean proportional. So mean proportional is equal to square root of the numbers A and C. So in this way we get square root of XY is equal to 24. Now this means that XY is equal to 24 square. So we have XY is equal to 576. Let this be result one. Since we know that the third proportional to X and Y is 1536. Therefore X is to Y is equal to Y is to 1536 which means that X upon Y is equal to Y upon 1536 or you can say that X is equal to Y square upon 1536. Let this be result two. Now consider the result one in which we have XY equal to 576. So from here we have that X is equal to 576 upon Y. Let this be result three. Now from the results two and three we have Y square upon 1536 is equal to 576 upon Y. That is Y square is equal to 1536 multiplied by 576 upon Y or further YQ is equal to 1536 multiplied by 576. On factorizing 1536 we get two multiplied by itself nine times and this whole multiplied by three and on factorizing 576 we get this. That is we get three multiplied by itself two times and two multiplied by itself six times. Now to get the value of Y we will make triplets of these. So we get Y is equal to two multiplied by two multiplied by three multiplied by two multiplied by two which is equal to 96. Therefore we have value of Y is 96. From three we have that X is equal to 576 upon Y. Now substituting the value of Y we get X is equal to 576 upon 96 and we know 96. Six times is 576. So we get the value of X as six. Hence we have the required numbers six and 96. So this is our final answer. This completes the session. Hope you have understood the solution of this question.