# Compound Interest by Using Formula

It is very easy to calculate compound interest by using formula.

We can derive general formulae for calculating compound interest in various cases, as given below.

Case I:

Let principal = $P, rate = R % per annum and time = n years. Then, the amount A is given by the formula ### A = P (1 + R/100)ⁿ ### 1. Find the amount of$ 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)

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 = $1356. Case 3: ### When interest is compounded annually but time is a fraction For example suppose time is 2³/₅ years then, Amount = P × (1 + R/100)² × [1 + (3/5 × R)/100] ### 1. Find the compound interest on$ 31250 at 8 % per annum for 2 years. Solution Amount after 2³/₄ years

Solution:

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) = $7387. ### Compound Interest by Using Formula, when it is calculated half-yearly ### Interest Compounded Half-Yearly Let principal =$ 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) = $34481. ### Compound Interest by Using Formula, when it is calculated Quarterly ### Interest Compounded Quarterly Let principal =$ 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.

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