1. COMPOUND INTEREST

2. Introduction The difference between the amount at the end of the last period and the original principal is called the compound interest. It is denoted as C.I.

3. The time period after which interest is calculated and then added to the principal each time to form a new principal is called the conversion period. Note : If the conversion period is one year,the interest is said to be compounded annually.

4. Computation of compound interest There are different methods for calculating the amount and the compound interest. Let us understand with an example.

5. Example Ajika borrows a sum of Rs 10,000 from a bank for 1 year at the rate of 6% per annum. At the end of 1 year,He has to pay the principal amount and interest on Rs 10,000 for 1 year at the rate of 6%per annum. Solution : Total amount to be paid by him to the bank is : = 10,000+10,000 ×6 ×1/100 = 10,000+600 =Rs 10,600 Suppose, He is not in a position to pay this amount to the bank. Then bank will charge the interest on Rs 10,600 thereafter At the end of 2 year He has to pay principal 10,600 and the interest on Rs 10,600 for 1 year at the rate of 6 % per annum = 10,600 ×6 ×1/100 = Rs 636 Total amount he has to pay to the bank = 10,000+636 =Rs 11,236 Hence total interest payable to the bank = Rs 11,236 – Rs 10,000 = Rs 1,236 The interest calculated in this manner is called compound interest.

6. Finding compound interest when interest is compounded Half- yearly If the rate of interest is R% p.a, then (R/2)% per half year. Amount after 1 half year become principal for next half year and so on.

7. Time is also changed into the number of half years. When interest is compounded half – yearly, the rate R% p.a become R/2% per half year and n years become 2n semi years.

8. Example Compute the compound interest on Rs 5000 for 3/2 years at 16 % per annum, compound half yearly Solution : Principal for the first half – year =Rs 5000 Rate of interest = 16 % per annum = 8% per half yearly Time period = 3/2 years = 3 half years Interest for the first half year = Rs 5000 ×1 ×8/100 = Rs 400 Amount at the end of 1 half year= Rs(5000+400) = Rs 5400 Principal for the second half year = Rs 5400 Interest for the 2 nd half year= Rs [5400 ×1 ×8/100] = Rs 432 Amount at the end of 2 nd half year= Rs (5400+432) =Rs 5832 Principal for the third half year = Rs 5832 Interest for the 3 rd half year = Rs[5832 ×1 ×8/100] =Rs 466.56 Amount at the end of third half year = Rs (5832+466.56) = Rs 6298.56 Compound interest = Rs( 6298.56-5000) = Rs 1298.56

9. Finding compound interest when interest is calculated quarterly If the rate of interest is R% p.a, then for each quarter it becomes(R/4)%. Amount after 1 quarter become principal for the 2 quarter; the amount for the 2 quarter become the principal for the 3 quarter and so on. The time period is also converted into the number of quarters.

10. Example Find the compound interest on Rs 8000 for 6 months at 20 % per annum compounded quarterly Solution : Here, Rate of interest =20 % p.a =5% quarterly Time period = 6 months = 1 half year =2 quarters Principal for the 1 quarter = Rs 8000 Interest for the 1 quarter = Rs 8000×1×5/100 = Rs 400 Amount at the end of 1 quarter = Rs(6000+400) = Rs 8400 So New Principal for the 2 quarter = Rs 6800 Interest for the 2 quarter = Rs [8400×1×5/100 = Rs 420 So, at the amount of 2 quarter = Rs (8400+420) = Rs 8820 Hence compound interest = Rs(8820-8000) = Rs 820 Note : when conversion period is not mentioned in the problem given, It is take as 1 year

11. Formula for finding the compound interest The previous method is lengthy and tine consuming. We try to find a formula for calculating compound interest.

12. Derivation Suppose we have to find the compound interest on Rs P for n years %R per compound annually. Principal for the first year =Rs P Interest for the first year = Rs P ×R ×1 /100 = Ps PR/100 Hence, Amount at the end of year = Rs P+Rs PR/100 = Rs P(1+R/100) Which is the principal of second year Interest of the second year = Rs P(1+R/100) ×R ×1/100 = Rs PR/100(1+R/100) Hence, amount at the end of second year =Rs P(1+R/100)+Rs PR/100(1+R/100) = Rs P (1+R/100) (1+R/100) =Rs p(1+R/100) 2 which is the principal of third year. Interest for the third year = Rs P(1+R/100) 2 ×R ×1/100 =Rs PR/100(1+R/100) 2 Hence, Amount at the end of the third year = Rs P((1+R/100) 2 ) (1+R/100) =Rs P (1+R/100) 3 If we proceed in the same manner, we see that : Amount at the end of n years = Rs P(1+R/100) n Hence Compound interest of n years = CI = Amount -Principal = P(1+R/100) n –P = P{(1+R/100) n -1} Thus A= P(1+R/100) n and CI = P{(1+R/100) n -1} Where P is the principal R is the rate of interest per annum n is the number of conversion period ( years in the present case)

13. When interest is compounded half-yearly Let, Principal =P Rate = R/2% per half year Time (n) = 2n half years So, Amount = P[1+R/2/100] 2 or A =P[1+R/200] 2n

14. Example Compute the amount of Rs 65,536 for 3/2 years at 25/2 p.a, the interest being compounded semi – annually. Solution : Here, P= Rs 65,536 Time = 3/2= 2 half years Rate = 25/2 % p.a = 25/4 % per half - year Amount = Rs[65,536×{1+25/4} 2 ] = 65,536×{17/16} 3 = 65,536 ×17/16×17/16×17/16 = Rs 78,608

15. When interest is compounded quarterly Let, Principal =P Rate = R/4% per quarter Time (n) = 4n quarter years ( where n is the number of conversions periods) So, Amount = P × [1+R/4/100] 4n =P × [1+R/400] 4n CI= A-P

16. Example Calculate the compound interest on Rs 24,000 for 6 months if the interest is payable quarterly at the rate of 8 % p.a Solution : P = Rs 24000 Rate = 8% p.a = 2 per quaterly Time = 6 months = 2 quarters A=P[1+R/100] n = 24,000[1+2/100] 2 = 24,000 ×51/50 ×51/50 = Rs 24,969.60 CI= A-P = Rs (24969.60-24000) = Rs 969.60

17. Assessment Q1 Find the amount for Rs 15000 at 8 % per annum compound annually for 2 years Q2Find the compound interest onRs 11200 at 35/2 per annum for 2 years