 Hello and welcome to the session. In this session we will discuss the following question which says, a machine costing $48,000 has effective life of 20 years and its craft value is $3,500. What amount should the company put into a sinking fund earning 7.5% per annum so that it can replace the machine after its useful life? Assume that the new machine will cost 20% more than the present one after 20 years. Before we move on to the solution let's see what is sinking fund. It is basically a fund which is created by investing some amount annually at compound interest for a certain period and this is created to discharge a known future liability. Consider this small a to be the sum invested at the end of each year and this capital A is the liability to be discharged after n years. Let R% be the rate of interest and I is equal to R upon 100 then this capital A which is the amount of annuity small A payable for n years that is it is same as the liability to be discharged after n years or you can say this is the value of the sinking fund and this is equal to small a upon i into 1 plus i to the power of n minus 1 the whole. This is the key idea that we use in this question. Let's now move on to the solution. In the person we are given the present cost of a machine which is 48,000 dollars and the effective life of the machine is 20 years. Its craft value is given as 3500 dollars and also the rate of interest is given as 7.5% per annum. We also have to assume that the new machine will cost 20 percent more than the cost of the present one after 20 years and so we are supposed to find the amount that the company should put into the sinking fund that is we are supposed to find the small a which is the sum invested at the end of each year. So this is what we have to find out now we are given R% that is the rate of interest as 7.5% per annum. So i would be equal to R upon 100 that is 7.5 upon 100 which is equal to 0.075. As we have the effective life of the machine is 20 years so this means n would be equal to 20. Now the present cost of the machine is given as 48,000 dollars. Now find out the cost of the new machine and its given in the question that the cost of the new machine is 20% more than the present machine after 20 years so this would be equal to 48,000 plus 20% of 48,000 dollars. So we get this is further equal to 48,000 plus 9600 dollars and so this gives us the cost of the new machine as 57,600 dollars. In the question we are also given the scrap value of the present machine as 3500 dollars. Let us now find out the net amount required at the end of 20 years to purchase the new machine is equal to the cost of the new machine which is 57,600 dollars minus the scrap value of the present machine which is 3500 dollars and so this is equal to 54,100 dollars. Thus we have capital A that is the value of the sinking fund is equal to 54,100 dollars. Let us now use the formula A is equal to small a upon i into 1 plus i to the power of n minus 1 the whole. Now we will put the values so we have 54,100 is equal to small a which we have to find out upon i which is equal to 0.075 into 1 plus 0.075 this whole to the power of 20 minus 1 the whole. So we now have 54,100 into 0.075 is equal to A into 1.075 to the power of 20 minus 1 the whole. Now removing this decimal we write here 1000 these to 0 cancel with these to 0 this further gives us 40575 upon 10 is equal to A into 1.075 to the power of 20 minus 1 the whole. We will now find out the value of 1.075 to the power of 20 for this we suppose let x be equal to 1.075 to the power of 20 we take log on both sides so we have log x is equal to 20 into log of 1.075. Now further we have log x is equal to 20 into value of log 1.075 is 0.0 315 so further log x is equal to 20 multiplied by 0.0 315 63 that is 0.63 now from here we have x is equal to log of 0.63 which is 0.075 to the power of 20 is 4.266. Now putting this value here we have 40575 upon 10 is equal to A into 4.266 minus 1 the whole that is 75 upon 10 is equal to A into from here we get A is equal to 405 into 3.266 the decimal here we get 1000 here now this 0 cancels with this 0 we now have A is equal to 4057500 upon 3266 and from here we get the value of A as 42 we can say A is equal to 1242.35 dollars the company is equal to 1242.35 dollars so this is our final answer this completes the session hope you have understood the solution of this question