 Hello and welcome to the session. In this session we discussed the following question which says what number must be added to each of the numbers 6, 15, 20 and 43 to make them in proportion. We know that four quantities say A, B, C, D are in proportion implies that A is to B is equal to C is to D or you can say that A upon B is equal to C upon D. This further means that the product of extremes that is A, D is equal to the product of the means B, C. This is the key idea that we use for this question. Let's proceed with the solution now. In the question we are required to find out a number which must be added to each of these numbers so that they are in proportion. So we take let the required number be X then 6 plus X, 15 plus X, 20 plus X, 43 plus X are in proportion. Now using the key idea we get that this means 6 plus X is to 15 plus X is equal to 20 plus X is to 43 plus X. This means 6 plus X upon 15 plus X is equal to 20 plus X upon 43 plus X. Then we can say that 6 plus X the whole multiplied by 43 plus X is equal to 20 plus X the whole multiplied by 15 plus X. This gives us 43 multiplied by 6 is 258 plus 6 multiplied by X is 6X plus 43 multiplied by X is 43X plus X multiplied by X is X square is equal to 20 multiplied by 15 which is 300 plus 20 multiplied by X is 20X plus 15 multiplied by X is 15X plus X multiplied by X is X square. Now X square cancels with X square. So further we have 43X plus 6X is 49X plus 258 is equal to 20X plus 15X is 35X plus 300. Now we keep the terms containing the variable X on the left hand side and the constant terms on the right hand side. So we get 49X minus 35X is equal to 300 minus 258 which means we have 14X is equal to 42. To get the value of X we divide both sides by 14. So we have 14X upon 14 is equal to 42 upon 14. Here 14 cancels with 14 and 14 three times is 42. So therefore we get the value of X as 3 and in the starting of the solution we had assumed the required number to be X. So we can say thus 3 must be added to each of the numbers 6, 15, 20 and 43 to make them in proportion. So this is our final answer that is 3 is the final answer for this question. This completes the solution hope you have understood the solution of this question.