Formulas
Let Principal = P, Rate = R% per annum, Time = n years.
When interest is compound Annually:
Amount = $P\left(1+\dfrac{R}{100}\right)^n$
When interest is compounded Half-yearly:
Amount = $P\left(1+\dfrac{\left(R/2\right)}{100}\right)^{2n}$When interest is compounded Quarterly:
Amount = $P\left(1+\dfrac{\left(R/4\right)}{100}\right)^{4n}$
When interest is compounded Annually but time is in fraction, say 3$\dfrac{2}{5}$ years.
Amount = $P\left(1+\dfrac{\left(R/4\right)}{100}\right)^3$ × $\left(1+\dfrac{\dfrac{2}{5}R}{100}\right)$
When Rates are different for different years, say$ R_{1} $%, $R_{2} $%,$ R_{3}$ % for 1st, 2nd and 3rd year respectively.
Then, Amount = $\left(1+\dfrac{ R_{1}}{100}\right)\left(1+\dfrac{ R_{2}}{100}\right)\left(1+\dfrac{ R_{3}}{100}\right)$
Present worth of Rs. x due n years hence is given by:
Present Worth = $\dfrac{x}{\left(1+\dfrac{ R_{1}}{100}\right)}$
Compound Interest:
Compound interest is the interest calculated on the principal and the interest accumulated over the previous period. It is unlike simple interest where interest is not added to the principal while calculating the interest during the next period.
How to calculate compound interest?
The compound interest formula is given below:
Compound Interest = Amount – Principal
Compound Interest by Using Formula, when it is calculated annually
Case 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)ⁿ
Example 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.
Example 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.
Example 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]
Example 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.
Formula:
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).
Example 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.
Example 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.
Formula:
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).
Example 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.