 Hello and welcome to the session. In this session we will discuss about compound interest. The interest calculated not only on the initial principle but also on the accumulated interest prior periods is called the compound interest. The compound interest arises when interest is added to the principle. This addition of interest to the principle is called compounding. If a man deposits certain amount say P, a savings account at the rate of interest or per annum then suppose he would gain interest i on his deposit at the end of one year. Unless he takes out this interest i in cash it will be added to the original amount P. So if he releases money in the account then the next year interest will be charged on principle plus the interest and this would continue for as long as the money is left in his account. And so this kind of interest is known as the compound interest. Now let us define the simple interest, simple interest the interest computed principle alone and we have a formula to find out the simple interest say S i and this is equal to P into R into P upon 100 where this P is the principle is the rate of interest and T is the time it is. So the basic difference between the simple interest and the compound interest is that simple interest is the interest which is computed on the principle alone and the compound interest is computed on principle plus the interest. Now let us try to find out the compound interest for different cases consider the principle given by P thousand dollars. Now for the problems of type one in which the time P is in whole number of years we take the time T the rate of interest say R be equal to 10 percent per annum. Now the principle P for the first year is taken as thousand dollars time is two years and the rate of interest R is 10 percent per annum. So the interest I first year would be equal to thousand into 10 into 1 upon 100 dollars. So this is equal to 600 dollars. Now is equal to the principle P for the first year plus the interest I for the first year which is equal to 6600 dollars. Now the principle P for the second year would be equal to 6600 dollars rate would be 10 percent per annum time would be 2 years. So now the interest I for the second year would be equal to 6600 into 1 into 10 upon 100 dollars and this is equal to 660 dollars this is the interest for the second year. The amount A at the end of second year is equal to the principle for the second year plus the interest for the second year and that is equal to 7,260 dollars. Now the compound interest after 2 years is equal to the amount at the end of the second year that is 7,260 dollars minus the principle for the first year that is 6,000 dollars this is equal to 1,260 dollars. So this is the compound interest after 2 years. This is how we calculate the compound interest when the time is given in whole number of years and let's discuss in which the time is not whole number of years. Suppose in this case the principle P is again given as 6,000 dollars say the time t is equal to 2,5 years the rate of interest R is equal to 10% as given in the previous case also. Now as the time is 2,5 years so we will find the interest for the first year for the second year and for the last half year. Now the amount A at the end of the first year calculated in the previous case is 6,600 dollars so we write here 6,600 dollars and the amount A at the end of the second year is calculated as 7,260 dollars so we have 7,260 dollars. Now we will find the interest for the last half year since the time is 2,5 years so the principle P for the last half year would be the amount at the end of the second year which is 7,260 dollars. Now where the time t is equal to half year and the rate of interest R would be equal to the same that is 10% per annum. Now in this case the interest for the last half year is equal to 7,260 into 1 upon 2 into 10 this whole upon 100 dollars so here we have interest for the last half year is equal to 363 dollars. The amount A at the end of the half year is equal to the principle for the last half year plus the interest for the last half year and this is equal to 7,623 dollars. Now the compound interest at the end of 2,5 years is equal to the amount at the end of the half year which is 7,623 dollars minus the initial principle that we had taken which is 6,000 dollars and so this would be equal to 1,623 dollars. So this is how we calculate the compound interest when the time is not in whole number of years. Now we discuss the type 3 problems in which the interest is different for successive years. For the principle P, 6,000 dollars time t as 2 years and the rate of interest for successive years are 5% and 10% for the first year the interest would be equal to 6,000 into 1 into 5 this whole upon 100 dollars this is equal to 300 dollars and the amount at the end of the first year is equal to the interest plus the principle and so this is equal to 6,300 dollars. Now for the second year the interest would be equal to the amount at the end of the first year would be taken as the principle for the second year that is 6,300 into 1 into 10 this whole upon 100 and this is equal to 630 dollars and the amount at the end of the first year is interest at the end of the second year plus the principle for the second year which is 6,300 dollars and so this is equal to 6,930 dollars and in this case the compound interest would be equal to the amount minus the initial principle which was 6,000 dollars so the amount at the end of the second year which is 6,930 dollars minus 6,000 dollars is equal to 930 dollars this is the compound interest now we have the type 4 problems where the interest is paid half yearly in this case we consider the principle p as 6,000 dollars the time t as 1 year and in this 1 year we have 2 half years the rate of interest r is taken as 10 percent and this interest is paid half yearly so we find the rate of interest half yearly is equal to half of 10 percent and this is equal to 5 percent now let's find out the interest for the first half year for this the principle p is 6,000 dollars time t is 1 half year the rate of interest r is equal to 5 percent per half year and in this case the interest i would be equal to 6,000 into 1 into 5 this whole upon 100 dollars and this is equal to 300 dollars now at the end of the first half year the amount a is equal to 300 dollars that is the interest plus the principle which is equal to 6,300 dollars now we find out the interest for the second half year for this we take the principle p as 6,300 dollars time t as 1 half year and the rate of interest r is equal to 5 percent per half year for this the interest i is equal to 6,300 into 1 into 5 this whole upon 100 and this is equal to 315 dollars and the amount a at the end of the second year is equal to the interest plus the principle for the second year and this is equal to 6,615 dollars and in this case the compound interest is equal to the amount minus the initial principle that we had taken which is 6,000 dollars so 6,615 dollars minus 6,000 dollars is equal to 6,615 dollars so this is how we calculate the compound interest when the interest is paid half yearly so we have discussed four types of problems to find out the compound interest first when the time is given in whole number of years second when the time is not in whole number of years third when the interest is different for the class of years and fourth when the interest is paid half yearly so this completes the session we hope you have understood the concept of compound interest