 Hello and welcome to the session. In this session we will learn about a very important result of proportion which is Componento and Dividento. This is the method of simply envisioning mathematics and it helps in solving complex problems. Now firstly we will learn about Componento rule. Now let us take any four quantities A, B, C and D in proportion that is if A is to be as C is to D which implies A is to be is equal to C is to D. This implies A over B is equal to C over D. Now adding one on both sides, this implies A over B plus one is equal to C over D plus one. This implies on table here this will be A plus B over B is equal to C plus D over D which implies A plus B is to be is equal to C plus D is to D. Thus if A is to be is equal to C is to D then by Componento rule A plus B is to be is equal to C plus D is to D. Hence we can say that if four quantities are in proportion then by Componento rule the first together with the second is to the second as third together with the fourth is to the fourth. Now next we will learn about Dividento rule. Now let us take four quantities A, B, C and D in proportion that is A is to be as C is to D. This implies A is to be is equal to C is to D which implies A over B is equal to C over D. Now subtracting one from both sides this implies A over B minus one is equal to C over D minus one which implies A minus B over B is equal to C minus D over D which implies A minus B is to be is equal to C minus D is to D. Thus if A is to be is equal to C is to D then by Dividento rule A minus B is to be is equal to C minus D is to D. Hence we can say that if four quantities are in proportion then by Dividento rule the excess of first over the second is to the second as the excess of third over the fourth is to the fourth. Now next we will learn about a very important result of proportion which is Componento and Dividento rule. Now let us take four quantities A, B, C and D in proportion that is A is to be as C is to D. This implies C is to be is equal to C is to D which implies A over B is equal to C over D. Now let us name it as one. Now applying Componento rule on one this implies A plus B over B is equal to C plus D over D. Now let us name it as now applying Dividento rule on one this implies A minus B over B is equal to C minus D over D and let us name it as three. Now dividing two by three this implies A plus B over B whole divided by A minus B over B is equal to C plus D over D whole divided by C minus D over D. Now here this B will be cancelled with B well D will be cancelled with D so this implies A plus B over A minus B is equal to C plus D over C minus D which implies A plus B is to A minus B is equal to C plus D is to C minus D thus if A is to be is equal to C is to D then by Componento and Dividento rule A plus B is to A minus B is equal to C plus D is to C minus D. Hence we can say that if four quantities are in proportion then by Componento and Dividento rule the sum of first and the second is to their difference as the sum of third and the fourth is to their difference. Now let us discuss an example related to the topic of Componento and Dividento and the example says that if A is to B is equal to C is to D then show that 5A plus 9B is to 5A minus 9B is equal to 5C plus 9D is to 5C minus 9D. Now let us start with the solution. Now given A is to B is equal to C is to D this implies A over B is equal to C over D. Now multiplying both sides by 5 over 9 this implies 5A over 9B is equal to 5C over 9D. Now applying Componento and Dividento rule this implies A plus 9B over 5A minus 9B is equal to 5C plus 9D over 5C minus 9D this implies 5A plus 9B is to 5A minus 9B is equal to 5C plus 9D is to 5C minus 9D. Hence we can say that if four quantities are in proportion then by Componento and Dividento rule the sum of first and the second is to be a difference as the sum of third and fourth is to be a difference. That's all for this session and I hope that all of you have understood the concept of Componento and Dividento.