 Hello and welcome to this session. In this session we are going to discuss the following question and the question says that find the rebate fraction for a 45 month loan that is paid off early with 31 monthly payments. We know that rebate fraction is equal to sum of digits for number of payments remaining upon sum of digits for total number of payments. With this key idea we proceed with the solution. In this question we have that a 45 month loan which is paid off early with 31 monthly payments we have to calculate the rebate fraction and from the key idea we know that rebate fraction is equal to sum of digits for number of payments remaining upon sum of digits for total number of payments. So since the loan is repaid within 31 months this implies that number of payments made is equal to 31 which implies number of payments remaining is equal to 45 minus 31 which is equal to 14. So 14 month payments are remaining. Now total number of months for the payment was 45 this implies total number of payments is equal to 45 so that means rebate fraction is equal to sum of digits for number of payments remaining that is 1 plus 2 plus 3 plus 4 and so on plus 14 upon sum of digits for total number of payments that is 1 plus 2 plus 3 plus 4 plus so on plus 45. Now we know if n is very large then sum of digits is equal to n into n plus 1 upon 2. So here in the numerator n is equal to 14 and in the denominator n is equal to 45 so in the numerator the sum of digits will be equal to 14 into 14 plus 1 upon 2 and the sum of digits in the denominator will be equal to 45 into 45 plus 1 upon 2. This implies rebate fraction is equal to 14 into 14 plus 1 upon 2 the whole upon 45 into 45 plus 1 upon 2. 2 gets cancelled and this becomes 14 into 15 upon 45 into 46. 5 3s are 15 and 5 9s are 45. 2 7s are 14 and 2 into 23 is 46. 3 3s are 9. So now this is equal to 7 upon 69 so rebate fraction is equal to 7 upon 69 which is the required answer. This completes our session. Hope you enjoyed the session.