Compound Interest Formula

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What is Compound Interest?

• Let’s know what is compound interest. Compound interest is defined as the interest calculated on the principal and the interest accumulated over the previous period of time.

• Compound interest is different from the simple interest.

• In simple interest, the interest is not added to the principal while calculating the interest during the next period while in the compound interest the interest is added to the principal to calculate the interest.

• The formula for compound interest is,

 Compound Interest (CI) = Principal (1+Rate/100)n - Principal

where, P is equal to Principal ,  R is equal to Rate of Interest,  T is equal to Time (Period)

A Little More About Compound Interest

We will first understand the concept and what is compound interest and then move onto the compound interest formula. Now interest can be defined as the amount we calculate on the principal amount that is given to us. But in compound interest, we calculate the interest on the principal amount and the interest that has accumulated during the previous period.

Essentially, compound interest is the interest on the interest! So in this method, rather than paying out the interest, it is reinvested and becomes a part of the principal.

As you will have noticed in simple interest, the interest amount remains the same for every period. This is not the case in compound interest. Since the previous interest amount is reinvested, the interest amount increases marginally every year. This is why we have a whole separate compound interest formula to help us calculate the compound interest of any given year.

The compound interest formula in maths is:

 Amount = Principal (1+Rate/100)n

where, P is equal to Principal,  Rate is equal to Rate of Interest,  n is equal to the time (Period)

Compound Interest Formula Derivation

To Better our understanding of the concept, let us take a look at the compound interest formula derivation.Here we will take our principal to be Rupee.1/- and work our way towards the interest amounts of each year gradually.

Year 1

• The interest on Rupee 1/- for 1 year is equal to r/100 = i (assumed)

• Interest after Year 1 is equal to Pi

• FV (Final Value) after Year 1 is equal to P + Pi = P(1+i)

Year 2

• Interest for Year 2 = P(1+i) × i

• FV after year two is equal to P(1+i) + P(1+i) × i = P(1+i)²

Year ‘t’

• Final Value (Amount) after year “t” is equal to P(1+i)t

• Now substituting actual values we get Final Value is equal to ( 1 + R/100)t

• CI = FV – P is equal to P ( 1 + R/100)t – P

This is the Compound Interest Formula Derivation

Applications of Compound Interest

Some of the applications of compound interest are:

1. Increase in population or decrease in population.

2. Growth of bacteria.

3. Rise in the value of an item.

4. Depreciation in the value of an item.

Now that we have some clarity about the concept and meaning of compound interest and compound interest formula in maths, let us try some Compound interest problems with solutions to deepen our understanding of the subject.

Compound Interest Problems with Solutions

Question 1) The count of a certain breed of bacteria was found to increase at the rate of 5% per hour. What will be the growth of bacteria at the end of 3 hours if the count was initially 6000?

Solution) Since the population of bacteria increases at the rate of 5% per hour,

We know the formula for calculating the amount, compound interest formula in maths

Amount= Principal(1 + R/100)n

Thus, the population at the end of 3 hours = 6000(1 + 3/100)3

= 6000(1 + 0.03)3 = 6000(1.03)3 =  Rs 6556.36

Question 2) Mr A decided to open a bank account and opted for the Compound Interest Option at 10%. He invested 10,000 for 3 years. At the end of three years, how much money will he get, and what will be the interest amount. The interest is calculated annually.

Solution)  As we already have a formula for future value amount, let us substitute the values in compound interest formula in maths

FV = ( 1 + R/100)t

FV = 10000 ( 1 + 10/100)5

FV = 10000 ( 1.1)5

FV = 16,105

CI = FV – P = 6,105/-

Question 3) Mr B lent money to his son at 8% CI calculated semi-annually. If he lent 1000/- for 2 years, how much will he get back at the end of the 2 years?

Solution) Since the CI is calculated semi-annually

• t= 2t = 4

• r = r/2 = 4

Final Value = P ( 1 + R/100)t

FV = 1000 ( 1 + 4/100)4

FV = 1000 (1.17)

FV or Amount = 1170/-

Question 1) How do you calculate the Compound Interest?

Answer) We can calculate compound interest by multiplying the initial principal amount by adding one and the annual interest rate raised to the number of compound periods minus one.

Question 2) What is the difference between Compound Interest and Simple Interest?

Answer) The difference between Compound Interest and Simple Interest are as follows:

 Simple Interest Compound Interest The simple interest is the same for all the number of years. The compound interest is different for all the years. SI < CI. CI > SI. Simple Interest (SI) = (P×R×T)/100 CI = Principal (1+Rate/100)n - principal

Question 3) How is Compound Interest used in real life?

Answer) The idea of compound interest is appealing only when you are on the earning side of the financial balance. Banks typically pay compounded interest on deposits, a benefit for depositors.Credit card companies charge interest on the principal amount and the accumulated interest.

Question 4) Why is compound interest so powerful?

Answer) Compound interest basically makes a certain sum of money that grows at a faster rate than simple interest because, in addition to earning returns on the money you invest, you also earn returns on those returns at the end of each and every compounding period, which could be daily, monthly, quarterly or annually.

Question 5) Why do banks use Compound Interest?

Answer) Almost all banks compound interest. Compounding means a financial institution pays you interest not only on the amount you originally deposited,but also on the interest your deposit has earned over time.So the more frequently the money of a person or a customer is compounded, the more interest you'll earn.