 Hello and welcome to the session. In this session we discuss the following question which says if the numerator of a fraction is multiplied by 2 and its denominator is increased by 2, it becomes 6 upon 7. If instead we multiply the denominator by 2 and increase the numerator by 2, it reduces to 1 upon 2. Find the fraction. Let's move on to the solution now. Since we need to find the fraction, so we assume let the fraction be equal to x upon y. So as the fraction is x upon y, so here we have the numerator is x and the denominator is y. Now let's apply the conditions given in the question. So according to the question we have that if the numerator is multiplied by 2 and denominator is increased by 2, that is we have the numerator as 2x and denominator as y plus 2, then the fraction becomes 6 upon 7. So 2x upon y plus 2 is equal to 6 upon 7. So now cross multiplying we get 7 multiplied by 2x is equal to 6 multiplied by y plus 2 the whole. This further gives us 14x is equal to 6y plus 12 or 14x minus 6y is equal to 12. Now taking 2 common we have 7x minus 3y is equal to 12. So further we get 7x minus 3y is equal to 12 upon 2 that is 6. Thus we get one equation as 7x minus 3y is equal to 6. Let this be equation 1. Now we apply the second condition given to us in which we multiply the denominator by 2 and increase the numerator by 2 that is we now have x plus 2 upon 2y then the fraction becomes 1 upon 2. So this is equal to 1 upon 2. So further cross multiplying we get 2 multiplied by x plus 2 is equal to 2y. Now here this 2 cancels with this 2 and we are left with x plus 2 is equal to y or you can say we have x minus y is equal to minus 2. So this is our second equation. So now we have 2 equations 7x minus 3y equal to 6 and x minus y is equal to minus 2. Now we multiply the second equation by 3. So we have 3x minus 3y equal to minus 6. Let this be equation 3. And the first equation is 7x minus 3y equal to 6. This is equation 1. Now next subtracting equation 3 from equation 1 we get 7x minus 3y minus 3x minus 3y equal to 6 minus minus 6 that is 7x minus 3y minus 3x plus 3y equal to 6 plus 6. So further we have 3y cancels with minus 3y. 7x minus 3x is 4x is equal to 12. To get the value for x we divide both sides by 4. 4 cancels with 4 and 4. 3 times is 12. Thus we get the value for x as 3. Now substituting x equal to 3 in equation 1 we get 7 multiplied by 3 minus 3y is equal to 6 or you can say 21 minus 3y equal to 6. Further 3y is equal to 21 minus 6 or 3y is equal to 15. Now to get the value for y we divide both sides by 3. Here 3 cancels with 3 and 3. 5 times is 15. So we get the value for y as 5 and the value for x obtained is 3. And we had assumed this to be the numerator and this to be the denominator. So we have the numerator x is 3 and the denominator y is 5. Thus the required fraction is equal to 3 upon 5. So this is our final answer. This completes the session. Hope you have understood the solution of this question.