 Hello and welcome to the session. In this session we discuss the following question that says the average due date of four bills was 10th June. Three of the bills were payable as follows. $400 on 29th April, $300 on 20th May, $500 on 3rd June. The fourth bill was due on 8th July. Find the amount of the fourth bill. For the average due date, first of all we take an arbitrary fixed date which is the zero date. Then we take P1, P2, P3 and so on. The different payments, D1, D2, D3 and so on days respectively, from the zero date. Then we have capital D which is the equated time when capital D is equal to P1, D1, P2, D2, P3, D3 plus and so on. And this one upon P1 plus P2 plus P3 plus and so on that is D is equal to submission PD upon submission P. Then we have the average due date which is the date the equated time B to the zero date. This is the key idea that we use in this question. Let's proceed with the solution now. Now in the question we are given the average due date of the fourth bill and three of the bills were payable as follows. That is we are given different payments for the different due dates. We are given the due date of the fourth bill also and we are supposed to find the amount of the fourth bill. Now our first due date is 29th April. So we take 29th April as the zero date. So we take net 29th April B to the zero date and we are given the average due date and June average due date is obtained by adding D that is the equated time of payment to the zero date. Now we are given the average due date and the zero date so we can easily find out the equated time of payment that is D. So the number of days which due date which is the zero date which is 29th April the equated time of payment which is capital D is equal to 42. This we are supposed to find the amount of the fourth bill. So we assume net of the fourth bill be X dollars four columns in which we have the due dates. Payments which is capital P number of days from zero date which is small D and P multiplied by D. Now from the question our first due date is 29th April and the payment for this due date is 400 dollars. So here we have 29th April and the payment is 400 dollars and number of days from the zero date would be zero since the zero date is 29th April itself where P multiplied by D would be zero so 400 multiplied by zero is zero. Our next due date is 20th May for which the amount of the payment is 300 dollars. Next we write in the table and the payment for this is 300 dollars and the number of days from the zero date which is full length April would be 21 multiplied by D would be 21 into 300 which would be 60 300 then next due date is 3rd June for which the payment is 500 dollars. So here we have 3rd June and the payment is 500 dollars and number of days from the zero date that is small D would be 35 and now 500 multiplied by 35 gives us 175 double zero where the next due date is 8th July and we have to find the amount for this bill and we have assumed that amount to be x dollars 8th July and the payment is x dollars. So number of days from zero date that is small D is 70 and P multiplied by D would be 70 x. Now we find out submission P would be equal to 400 plus 300 plus 500 plus x and that would be 1200 plus x then submission P D is equal to 6300 plus 175 double zero plus 70 x which would be 238 double zero plus 70 x. Now from the key idea we have equated time of payment that is capital D would be equal to submission P D upon submission P. So capital D is equal to submission P D upon submission P that is equal to 238 double zero plus 70 x and this whole upon 1200 plus x and we had already calculated the capital D as 42 days. So this means 42 is equal to 238 double zero plus 70 x and this whole upon 1200 plus x the cross multiplying we get 42 into plus x the whole is equal to 238 double zero plus 70 x. Further we have 42 into 1200 is 504 double zero plus 42 x is equal to 238 double zero plus 70 x now further 70 x minus 42 x is equal to 504 double zero minus 238 double zero. Or you can say we get 28 x is equal to 266 double zero and from where we have x is equal to 266 double zero upon 28 and 28 950 times is 266 double zero. So therefore we have x equal to 950 or you can say x is equal to 950 dollars. Therefore that we have assumed to be x total hope we have understood the solution of this question.