It is very easy to calculate compound interest by using formula. Show
We can derive general formulae for calculating compound interest in various cases, as given below. Compound Interest by Using Formula, when it is calculated annuallyCase I: When the interest is compounded annuallyLet principal = $ P, rate = R % per annum and time = n years.Then, the amount A is given by the formula A = P (1 + R/100)ⁿ1. Find the amount of $ 8000 for 3 years, compounded annually at 5% per annum. Also, find the compound interest.Solution: Here, P = $ 8000, R = 5 % per annum and n = 3 years. Using the formula A = $ P(1 + R/ 100)ⁿ amount after 3 years = $ {8000 × (1 + 5/100)³} = $ (8000 × 21/20 × 21/20 × 21/20) = $ 9261. Thus, amount after 3 years = $ 9261. And, compound interest = $ (9261 - 8000) Therefore, compound interest = $ 1261. 2. Find the compound interest on $ 6400 for 2 years, compounded annually at 7¹/₂ % per annum.Solution: Here, P = $ 6400, R % p. a. and n = 2 years. Using the formula A = P (1 + R/100)ⁿ Amount after 2 years = [6400 × {1 + 15/(2 × 100)}²] = $ (6400 × 43/40 × 43/40) =$ 7396. Thus, amount = $ 7396 and compound interest = $ (7396 - 6400) Therefore, compound interest = $ 996. Case 2: When the interest is compounded annually but rates are different for different yearsLet principal = $ P, time = 2 years, and let the rates of interest be p % p.a. during the first year and q % p.a. during the second year.Then, amount after 2 years = $ {P × (1 + P/100) × (1 + q/100)}. This formula may similarly be extended for any number of years. 1. Find the amount of $ 12000 after 2 years, compounded annually; the rate of interest being 5 % p.a. during the first year and 6 % p.a. during the second year. Also, find the compound interest.Solution: Here, P = $12000, p = 5 % p.a. and q = 6 % p.a. Using the formula A = {P × (1 + P/100) × (1 + q/100)} amount after 2 years = $ {12000 × (1 + 5/100) × (1 + 6/100)} = $ (12000 × 21/20 × 53/50) =$ 13356 Thus, amount after 2 years = $ 13356 And, compound interest = $ (13356 – 12000) Therefore, compound interest = $ 1356. Case 3: When interest is compounded annually but time is a fractionFor example suppose time is 2³/₅ years then,Amount = P × (1 + R/100)² × [1 + (3/5 × R)/100] 1. Find the compound interest on $ 31250 at 8 % per annum for 2 years. Solution Amount after 2³/₄ yearsSolution: Amount after 2³/₄ years = $ [31250 × (1 + 8/100)² × (1 + (3/4 × 8)/100)] = ${31250 × (27/25)² × (53/50)} = $ (31250 × 27/25 × 27/25 × 53/50) = $ 38637. Therefore, Amount = $ 38637, Hence, compound interest = $ (38637 - 31250) = $ 7387.
Compound Interest by Using Formula, when it is calculated half-yearlyInterest Compounded Half-YearlyLet principal = $ P, rate = R% per annum, time = a years.Suppose that the interest is compounded half- yearly. Then, rate = (R/2) % per half-year, time = (2n) half-years, and amount = P × (1 + R/(2 × 100))²ⁿ Compound interest = (amount) - (principal). 1. Find the compound interest on $ 15625 for 1¹/₂ years at 8 % per annum when compounded half-yearly.Solution: Here, principal = $ 15625, rate = 8 % per annum = 4% per half-year, time = 1¹/₂ years = 3 half-years. Amount = $ [15625 × (1 + 4/100)³] =$ (15625 × 26/25 × 26/25 × 26/25)= $ 17576. Compound interest = $ (17576 - 15625) = $ 1951. 2. Find the compound interest on $ 160000 for 2 years at 10% per annum when compounded semi-annually.Solution: Here, principal = $ 160000, rate = 10 % per annum = 5% per half-year, time = 2 years = 4 half-years. Amount = $ {160000 × (1 + 5/100)⁴} =$ (160000 × 21/20 × 21/20 × 21/20 × 21/20) compound interest = $ (194481- 160000) = $ 34481. Compound Interest by Using Formula, when it is calculated QuarterlyInterest Compounded QuarterlyLet principal = $ P. rate = R % per annum, time = n years.Suppose that the interest is compounded quarterly. Then, rate = (R/4) % Per quarter, time = (4n) quarters, and amount = P × (1 + R/(4 × 100))⁴ⁿ Compound interest = (amount) - (principal). 1. Find the compound interest on $ 125000, if Mike took loan from a bank for 9 months at 8 % per annum, compounded quarterly.Solution: Here, principal = $ 125000, rate = 8 % per annum = (8/4) % per quarter = 2 % per quarter, time = 9 months = 3 quarters. Therefore, amount = $ {125000 × ( 1 + 2/100)³} =$ (125000 × 51/50 × 51/50 × 51/50)= $ 132651 Therefore, compound interest $ (132651 - 125000) = $ 7651. ● Compound Interest Compound Interest Compound Interest with Growing Principal Compound Interest with Periodic Deductions Compound Interest by Using Formula Compound Interest when Interest is Compounded Yearly Compound Interest when Interest is Compounded Half-Yearly Compound Interest when Interest is Compounded Quarterly Problems on Compound Interest Variable Rate of Compound Interest Difference of Compound Interest and Simple Interest Practice Test on Compound Interest Uniform Rate of Growth Uniform Rate of Depreciation Uniform Rate of Growth and Depreciation ● Compound Interest - Worksheet Worksheet on Compound Interest Worksheet on Compound Interest when Interest is Compounded Half-Yearly Worksheet on Compound Interest with Growing Principal Worksheet on Compound Interest with Periodic Deductions Worksheet on Variable Rate of Compound Interest Worksheet on Difference of Compound Interest and Simple Interest Worksheet on Uniform Rate of Growth Worksheet on Uniform Rate of Depreciation Worksheet on Uniform Rate of Growth and Depreciation Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. |