 Hello and welcome to the session. In this session we discuss the following question which says a man sets aside $2,000 at the end of every year for 12 years from his car. If the rate of interest is 10% per annum compound interest, what is the value of a sinking fund? First of all we will discuss about the sinking fund. It is a fund which is created by investing some amount annually at compound interest for a certain period. We can also say that it is a fund created to discharge a known future liability. Now if we suppose that the small a be the sum invested at the end of each year and this capital A be the liability to be discharged and this r% be the rate of interest and we get i r upon 100 then we can say that the amount of annuity and this is given by the formula small a upon i into 1 plus i to the power of n minus 1 the whole. So this capital A is the amount of annuity A payable for n years and it is given by this formula which is same as the liability to be discharged after n years and this is the value of the sinking fund. So this is the key idea that we use in this question. Let us now move on to the solution. I think the question we are given that the man sets aside $2000 at the end of every year for a period of 12 years to purchase a car and the rate of interest is given as 10% per annum and we have to find the value of a sinking fund. Now as the man invests certain amount of $2000 at the compound interest of 10% for a period of 12 years so this shows that a sinking fund is created and we have to find the value of the sinking fund. For this we would use this formula as from here we have the small a is the sum invested at the end of each year. So for this question the small a is equal to $2000 percent which is the rate of interest per annum is 10% per annum. Now n is equal to time for which the sum is invested that is 12 so n is equal to 12. Now i is equal to r upon 100 that is 10 upon 100 which is equal to 0.1. So we have the value of a sinking fund same as the amount of annuity which is capital A this is equal to small a upon i into 1 plus i to the power of n minus 1 the whole. So we will put the respective values here. So we get a is equal to 2000 upon 0.1 into 1 plus 0.1 to the power of 12 minus 1 the whole. Further we have a is equal to now we remove this decimal so we put a 0 here. So a is equal to 20,000 into 1.1 to the power of 12 minus 1 the whole. Let us now find out the value of 1.1 to the power of 12 using logs. So for this we take let x be equal to 1.1 to the power of 12 taking log on both the sides we have log x is equal to 12 into log of 1.1. Now further we have log x is equal to 12 into value of the log 1.1 which is 0.0414. So further we get log x equal to 12 multiplied by 0.0414 which comes out to be equal to 0.4968 and from here we get the value of x as nT log of 0.4968 which is equal to 3.139 this is the value of x that is the value of 1.1 to the power of 12. So putting this value here we get a is equal to 20,000 into 3.139 minus 1 the whole. This gives us a is equal to 20,000 into 2.139. Now we remove this decimal and we write here 1000 these 3 0 comes with these 3 0 and so we have 2139 multiplied by 20. So this gives us a is equal to 42780. Hence the value of the syncing fund is equal to 42,780 dollars. So this is our final answer. This completes the session. Hope you have understood the solution of this question.