 Hello and welcome to the session. In this session we discuss the following question which says, the sum of the areas of two squares is 640 meters square. If the difference in their perimeters be 64 meters, find the sides of the two squares. Let's move on to the solution. Let the side of first square be equal to x meters and let the side of the second square be equal to y meters. We also take x that is the side of first square to be greater than the side of the second square that is y. Now we have the area of the first square would be equal to psi square that is x square meter square then area of the second square would be equal to y square meter square. Now according to the question we have that the sum of the areas of the two squares is 640 meters square. Thus we have x square plus y square is equal to 640. Now we consider the perimeters of both the squares. So perimeters of the first square is equal to 4 into psi that is 4 x meters then perimeters of the second square is equal to 4 y meters. Now again according to the question we have that the difference in the perimeters of both the squares is 64 meters and we have taken x and y so we take 4 x minus 4 y is equal to 64. So from here we get x minus y is equal to 16. Now we have got two equations this equation and this equation. So let's write both these equations. So we have x square plus y square equal to 640. Let's take it as equation 1 and we have x minus y equal to 16. Let this be equation 2. Now we need to find the values for x and y from both these equations. So what we do is from equation 2 we have x is equal to 16 plus y. Now we substitute x equal to 16 plus y in equation 1. So we get 16 plus y the whole square plus y square is equal to 640. Now this further implies that 256 plus y square plus 32 y plus y square is equal to 640. Then this further implies 2 y square plus 32 y minus 384 is equal to 0. So this further implies y square plus 16 y minus 192 is equal to 0. From here we have y square plus 24 y minus 8 y minus 192 is equal to 0. That is here we have split this middle term plus 16 y. Now we can factorize them. So we get y into y plus 24 minus 8 into y plus 24 equal to 0. That is now we get y plus 24 into y minus 8 is equal to 0. So this gives us y plus 24 equal to 0 or y minus 8 equal to 0. That is we have y equal to minus 24 or y equal to 8. Now y equal to minus 24 is not possible as the side of a square can't be negative. So we take y equal to 8. Now we have got the value for y. Now using this value for y we will find the value for x. Now from equation 2 we have x minus y is equal to 16. That is x is equal to y plus 16. Now we put y equal to 8 in this equation. So we get x equal to 8 plus 16. This further implies that x is equal to 24. That is we get that the side of the first square that we have taken to be equal to x meters would be equal to 24 meters. And then the side of the second square equal to y meters would be equal to 8 meters. Now let's do the verification. We find the sum of the areas of two squares are equal to 24 square plus 8 square since we have found out that the side of the first square is 24 meters and the side of the second square is 8 meters. So this is equal to 576 plus 64 which comes out to be equal to 640 meters square. And we are already given in the question that sum of the areas of two squares is 640 meters square. Now the difference of the perimeters of two squares is equal to 96. That is the perimeter of the first square with side 24 meters minus 32 which is the perimeter of the second square with side 8 meters. And this is equal to 64 meters. And it is given in the question that the differences in the perimeters is 64 meters. Hence our answer is correct. So final answer is the sides of the two squares are 24 meters and 8 meters respectively. So this completes the session. Hope you have understood the solution for this question.