Compound Interest: Formula, Derivation, How to Solve with Examples

There are two types of interest in mathematics; simple interest and compound interest. If the interest on a sum of money for a certain period is calculated uniformly, then it is called simple interest. In contrast to simple interest, in compound interest previously accumulated interest is added to the principal amount of the current period, leading to compounding which is not done in SI.

Through this article, you will learn the various pattern of compound interest questions, how to calculate them with the derivation and formula for different time periods like; half-yearly, quarterly, monthly and so on with applications, key points.

What is Compound Interest?

Interest in mathematics as well as in statistics is the extra or additional money spent by organisations like banks/post offices on money deposited with them at the same time interest is also returned by people if they borrow money.  C.I. is generally the addition of interest to the principal sum of a loan/deposit, or in other words, it is also identified as of interest on interest. C.I. is standard in finance and economics.

For example;

Ram’s father deposited some money in the post office for 4 years. Every year the money grows more than the earlier year.

Similarly, Ankur has some money in the bank and every year some interest is computed to it, which is displayed in the passbook. This interest is not the same, each year it increases.

Commonly, the interest given/charged is never simple. The interest is determined by the amount of the previous year. This is recognized as interest compounded/C.I.

Compound Interest Formula

When there is a situation where the amount of the first year becomes principal for the second year and the amount of the second year becomes the principal for the third year and so on. Then this is called compound interest. For better understanding let’s have a look at the formulas and their meaning.

Compound Interest = Amount – Principal

Or

CI = A – P

Here;

Amount(A) is given by the formula;

\(A=P\left(1+\frac{r}{n}\right)^{nt}\)

Where;

‘A’ stands for the amount.

‘P’ is the principal.

‘r’ denotes the rate of interest.

‘n’ is the number of times interest is compounded yearly.

‘t’ is the time in years.

Substituting these values in the CI formula we obtain:

CI = A – P

\(CI=P\left(1+\frac{r}{n}\right)^{nt}-P\)

The above formula is the general formula when the principal is compounded n times in a year. If in case the interest is compounded annually/yearly/per year, the amount and CI is given by the formula:

\(A=P\left(1+\frac{R}{100}\right)^T\)

Therefore CI is calculated by the formula;

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

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Derivation of Compound Interest Formula

The compound interest equation/formula can be derived with the help of simple interest formulas as shown below.

The formula for SI is:

\(S.I.=\frac{\left(P\times R\times T\right)}{100}\)

Where; P is the principal amount, R is the rate of interest and T denotes the time.

The simple interest= CI for one year

The SI for the first year is;

\(SI\text{ (first year)}=\frac{\left(P\times R\times T\right)}{100}\)

Amount= SI+P

Hence, the amount after the 1st year = P+SI(first year)

Amount= \(P+\frac{P\times R\times T}{100}=P\left(1+\frac{R\times T}{100}\right)=P\left(1+\frac{R}{100}\right)\)

Here T=1 as we are calculating for one year.

This total amount is now the principal for second year as per the CI concept:

Hence P( for second year)=\(P\left(1+\frac{R}{100}\right)\)

The SI for the second year is=\(\frac{\left(P\times R\times T\right)}{100}\)

Therefore, the amount after the 2nd year is again= SI+P=\(P\left(1+\frac{R}{100}\right)\)

But here P=\(P\left(1+\frac{R}{100}\right)\)

Hence, amount=\(P\left(1+\frac{R}{100}\right)\left(1+\frac{R}{100}\right)\)

Amount(after second year)=\(P\left(1+\frac{R}{100}\right)^2\)

Similarly for n years,

Amount(A)=\(P\left(1+\frac{R}{100}\right)^n\)

Now, CI = A – P

\(CI=P\left(1+\frac{R}{100}\right)^n-P\)

After simplification we get:

\(CI=P\left[\left(1+\frac{R}{100}\right)^n-1\right]\)

Hence proved.

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Compound Interest Formula for Different Time Periods

So far in the article, we read about the CI definition and formula along with the derivation on the yearly basis. The compound interest can also be calculated for half-yearly, quarterly, monthly and so on. Let us drive through these formulas as well:

Compound Interest Formula Half Yearly

When the interest is compounded half-yearly i.e. the interest is determined every six months or we can say the amount is compounded twice in a given year. The formula is as follows:

\(A=P\left(1+\frac{\left(\frac{R}{2}\right)}{100}\right)^{2T}\)

And CI = A – P therefore;

\(C.I=P\left(1+\frac{\left(\frac{R}{2}\right)}{100}\right)^{2T}-P\)

The point to note here is while calculating for half-yearly; in the actual form the rate of interest is divided by 2 and the time is multiplied by 2 or is doubled.

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Compound Interest Quarterly

In the last section, we learn how to calculate the CI for half-yearly or semi-annually, in the continuation let us learn the formula on a quarterly basis.

Similar to half-yearly; the rate of interest r in the quarterly format is divided by 4 and the time is multiplied by 4. The formulas are listed below:

\(A=P\left(1+\frac{\left(\frac{R}{4}\right)}{100}\right)^{4T}\)

CI = A – P

\(C.I=P\left(1+\frac{\left(\frac{R}{4}\right)}{100}\right)^{4T}-P\)

Monthly Compound Interest Formula

Similar to half-yearly and quarterly calculations, we can compound the data monthly as well. The formula for the same is as follows:

\(C.I=P\left(1+\frac{\left(\frac{R}{12}\right)}{100}\right)^{12T}-P\)

For the monthly compound interest calculation, we divide the rate by 12 and multiply the time by 12 as per the month as n=12.

How to Calculate Compound Interest?

We can understand C.I. as the outcome of reinvesting interest, rather than spending it out so that interest in the succeeding period is then received on the principal sum plus previously accumulated interest. Until now we are clear with the various formulas relating to the CI calculation varying from yearly, half-yearly, quarterly and monthly as well. Let us now understand the compound interest formula with a solved example.

An amount of 25000 is deposited in ICICI Bank for 2 years, obtaining the interest compounded annually at the rate of 10%.

Given:

P = 25000

R = 10%

T = 2 years

According to formula;

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

Substituting the value of P, R and T in the formula:

\(C.I.=25000\left(1+\frac{10}{100}\right)^2-25000\)

C.I.=30250-25000=5250

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Key Points on Compound Interest

Compounded interest is determined on Principal + Accumulated Interest periodically. Some of the key points relating to the topic are as follows:

• The principal amount on compounded interest continues to change during the tenure.
• Returns on C.I. are relatively high.
• The calculation for compound interest is more complex as compared to simple interest.
• The CI relies on the amount collected at the end of the earlier tenure and not on the initial principal when compared to SI. 
• The interest rate for the first year in C.I. as well as in simple interest is the same.
• Leaving the first year calculations, the interest compounded annually > simple interest for the given data.

Applications of Compound Interest

There are some conditions where we could use the compound interest formula. Three of them are listed below.

• Increase or decrease in population.
• The growth of a bacteria if the speed of growth is identified.
• The value of an item, if its price increases/decreases in the intermediate years.

Solved Examples on Compound Interest

In any type of interest whether it be simple or compound apart from the definition and related concepts, formulas associated with examples play a major role. We have been through various types of formulas, now let’s practice some solved examples for the same.

Solved Examples 1: A invested Rs. 3000 on compound interest at a rate of interest 10% for 2 years and B invested Rs. 3200 on compound interest at a rate of interest 15% for 3 years. Find total C.I . (compounded annually).

Solution:

Given:

Sum of Rs. 3000 invested at rate = 10% for 2 years

Sum of Rs. 3200 invested at rate = 15% for 3 years

Formula:

Let P = Principal, R = rate of interest and T = time period

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

Calculation:

C.I after 2 years = \(3000\left(1+\frac{10}{100}\right)^2-3000\)= Rs. 630

C.I after 3 years = \(3200\left(1+\frac{15}{100}\right)^3-3200\)= Rs. 1666.8

Total C.I = 630 + 1666.8 = Rs. 2296.8

∴ Total C.I. is Rs. 2296.8

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Solved Examples 2: Find the C. I at the rate of 20% for 3 years on that principal which in 2 years at the rate of 10% per annum gives Rs.10500 as compound interest. (when compounded annually)

Solution:

Let P = Principal, R = rate of interest and T = time period

Annual compound interest formula is:

\(C.I.=P\left(1+\frac{R}{100}\right)^T-P\)

Given,

R = 10% and T = 2

⇒ \(10500=P\left(1+\frac{10}{100}\right)^2-P\)

⇒ 10500 = 0.21P

⇒ P = 50000

⇒ Principal = Rs. 50000

Then,

R = 20% and T = 3

C.I.

= \(50000\left(1+\frac{20}{100}\right)^3-50000\)

= \(50000\left(1.2\right)^3-50000\)

= 36400

Solved Examples 3: Calculate the compound interest/CI on 10000 rupees, for 2 years duration when the rate of 4% is given, and the interest is being compounded half-yearly.

Solution:

P = 10000

R = 4%

T = 2 years

Being compounded half-yearly the rate will get divided by 2, and time will get multiplied by 2, by a formula

\(A=P\left(1+\frac{\left(\frac{R}{2}\right)}{100}\right)^{2T}\)

\(A=10000\left(1+\frac{2}{100}\right)^4=10824.32\\\)

\(C.I.=10824.32-10000=824.32\)

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Compound Interest FAQs

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