 Hello and welcome to the session. In this session we discuss the following question which says the ratio between the number of sides of two regular polygons is 3 is to 4 and the ratio between the sum of the interior angles is 2 is to 3. Find the number of sides in each polygon. Here we are discussed from result the interior angles of a convex polygon in sides is equal to 2 and minus 4 is the key idea that we use in this question. Let us now move on to the solution. So in the question we are given the ratio between the number of sides of two regular polygons and also we are given the ratio between the sum of the interior angles and we are supposed to find the number of sides in each polygon. So first of all we suppose let the two regular polygons into the question that is to n2 is equal to 3 is to 4. Since we have that the ratio between the number of sides of two regular polygons is 3 is to 4 and we have assumed n1 n2 is the number of sides of the two regular polygons. So we have n1 is to n2 is equal to 3 is to 4. n1 upon n2 is equal to 3 upon 4. Let this be result 1. See in the question we are given the ratio between the sum of the interior angles of the two polygons. The interior angles in the polygon are supposed to n1 minus 4 into 90 degrees. Then next the interior angles to regular polygon equal to 2 n2 minus 4 into 90 degrees. Next we have according to the question the ratio 2 n1 minus 4 90 degrees 2 n2 minus 4 and this whole n2 90 degrees is equal to this is equal to 2 is to this means 2 n1 minus 4 and this whole n2 90 degrees upon 2 n2 minus 4 and this whole n2 90 degrees is equal to 2 upon 3 that is 90 degrees 90 degrees cancel. Now the cross multiplying we get 3 n2 2 n1 minus 4 the whole is equal to 2 n2 2 n2 minus 4 d whole. This means 6 n1 minus 12 is equal to 4 n2 minus 8. Further we have 6 n1 minus 4 n2 is equal to 12 minus 8 that is 6 n1 minus 4 n2 is equal to 4. Now for result 1 we have n1 upon n2 is equal to 3 upon 4 so we have since n1 upon n2 is equal to 3 upon 4 this means n2 is equal to 4 n1 upon 3. So here we have further n minus 4 n2 4 n1 upon 3 is equal to so that is the place of n2 we put 4 n1 upon 3 so here we have 6 n1 minus 16 n1 upon 3 is equal to this means 18 n1 minus 16 n1 and this whole upon 3 is equal to 4 or you can say 2 n1 is equal to 12 we have n1 is equal to 12 upon 2 6 times is 12 therefore we get n1 is equal to and from here we have n2 is equal to 4 n1 upon 3 so substituting the value for n1 we get n2 is equal to 4 into 6 upon 3 3 2 times is 6 and 4 into 2 is therefore we have n2 is equal to