Compound Interest

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We know that Interest is an extra money charged on a borrowed money at some rate for a given period of time. It can be either Simple or Compound.

Simple Interest 

It is the interest calculated on same Principal for every term for the given period of time.
SI = ^PRT/₁₀₀

Compound Interest

➣ In Compound Interest, the interest is calculated on the amount of the previous term.
➣ Interest is calculated as usual for first term, then amount of the first term become Principal for the second term and so on for the given period of time.
➣ Let's find CI on a sum of Rs 8000 for 2 years at 5% per annum compounded annually.
P₁= Rs 8000
R = 5% per annum
T = 2 years
Interest is compounded annually

Interest of first year, I₁= 5% of 8000 = ⁵/₁₀₀ × 8000 = Rs 400
Amount at the end of first year, A₁= P₁ + I₁ = Rs 8400
This amount is the Principal for second year, P₂ = Rs 8400
Interest of second year, I₂ = 5% of 8400 = ⁵/₁₀₀ × 8400 = Rs 420

Amount at the end of second year, A₂ = P₂ + I₂ = 8400 + 420 = Rs 8820

Total Interest given = I₁+ I₂ =Rs 400 + Rs 420 = Rs 820

Formula for CI

The amount at the end of n year,
A = P(1+^R/₁₀₀)ⁿ
CI = A − P

➢ Let's calculate the amount and compound interest on Rs 10,800 for 3 years at 12¹/₂% per annum compounded annually.
→ P = Rs 10800
     R = 12¹/₂% per annum
     T = 3 years
     Interest is compounded annually

A = P (1+^R/₁₀₀)ⁿ
    = 10800(1+²⁵/₂₀₀)³
    = 10800(²²⁵/₂₀₀)³
    = 10800(⁹/₈)³
    = 10800(⁷²⁸/₅₁₂)
A = Rs 15356.25
CI = Rs (15356.25 − 10800)
     = Rs 5556.25

When the interest rate is not compounded annually

Conversion Period

The time period after which the interest is added each time to form a new principal is called the conversion period.

When the interest is compounded half yearly 

➢ There are two conversion periods in a year each after 6 months.
➢ The half yearly rate will be half of the annual rate.
➢ The  half yearly time period will be double of the annual time.
R  → ^R/₂                    n → 2n
A = P(1+^R/₂₀₀)²ⁿ

When the interest is compounded quarterly

➢ There are four conversion periods in a year.
➢ The quarterly rate will be one-fourth of the annual rate.
➢ The quarterly time period will be four-times of the annual time.
R  → ^R/₄                    n → 4n
A = P(1+^R/₄₀₀)⁴ⁿ

Try These

Q1) Find the amount which Ram will get on Rs 4096, if he gave it for 18 months at 12¹/₂ % per annum, interest being compounded half yearly.

Solution

A1) P = Rs 4096
       R = 12¹/₂ % per annum = ²⁵/₂ % per annum
       T = 18 months =1 ³/₂ year = ³/₂ year
∵ Interest is compounded half-yearly
∴ R = ²⁵/_2×2 % half-yearly = ²⁵/₄ % half-yearly 
   T = 2׳/₂ half-year = 3 half-year
∴ Amount which Ram get at the end of 3 half-years = 4096(1+²⁵/₄₀₀)³
= 4096(⁴²⁵/₄₀₀)³
= 4096(¹⁷/₁₆)³
= 4096(⁴⁹¹³/4096)
= Rs 4913

Applications of Compound Interest Formula

Situations where we could use the formula for calculation of amount in CI.

➢ Increase (or decrease) in population.
➢ The growth of a bacteria if the rate of growth is known.
➢ The value of an item, if its price increases or decreases in the intermediate years.

Try These

Q1. A machinery worth Rs 10,500 depreciated by 5%. Find its value after one year.
Q2. Find the population of a city after 2 years, which is at present 12 lakh, if the rate of increase is 4%.

Solution

A1. Value of machinery after one year = 10,500(1+⁵/₁₀₀)¹
        = 10,500 × ¹⁰⁵/₁₀₀
        = Rs 11,025

A2. Population of the city after 2 years = 12,00,000(1+⁴/₁₀₀)²
       = 12,00,000 × ¹⁰⁴/₁₀₀
       = 12,48,000

Simple Interest
	Direct & Inverse Proportion ➤