Read to learn about Compound Interest, Formulae involved in compound interest and other important concepts around it.

Table of Contents

• What is Compound Interest?
• Compound Interest Formula
• Derivation of Compound Interest Formula
• Compound Interest Formula for Different Periods
• How to Calculate Compound Interest?
• Key Points on Compound Interest
• Applications of Compound Interest
• Challenges in Compound Interest Calculation
• Solved Examples of Compound Interest

In mathematics, there are two main types of interest: simple interest (SI) and compound interest (CI). When interest is calculated on the original amount (principal) for a set time and stays the same every year, it is called simple interest. However, in compound interest, the interest earned is added to the principal after each period. This new total becomes the principal for the next period. So, interest is calculated on both the original amount and the interest already earned. This process is called compounding, and it helps the money grow faster than simple interest.

In this topic, you will learn how to solve different types of compound interest problems, how to use formulas for different time periods like yearly, half-yearly, quarterly, or monthly, and understand real-life applications. Key formulas and steps will also be shared to help you calculate compound interest quickly and easily.

In maths and statistics, interest is the extra money paid for using someone else’s money. For example, when you deposit money in a bank or post office, they pay you interest as a reward for keeping your money with them. On the other hand, when you borrow money from a bank, you have to pay interest along with the money you borrowed.

Compound Interest (C.I.) is a type of interest where the interest is added to the principal (original amount) after each period. Then, in the next period, interest is calculated on this new total (principal + previous interest). This is why compound interest is often called “interest on interest.”

Compound interest is very common in finance and economics, such as in loans, savings, and investments. It helps your money grow faster than simple interest, because you earn interest not just on your money, but also on the interest earned earlier.

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.

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Compound Interest in Maths:

In Maths, Compound Interest is the extra money you earn or pay over time on the money you saved or borrowed. It is different from simple interest because, in compound interest, you also earn interest on the interest already added.

To calculate compound interest easily, we use a formula. But before that, you need to know two things:

• Principal – The starting amount of money
• Amount – The total money after adding interest over time

Now, Compound Interest = Amount – Principal

So basically, compound interest is just the difference between how much you have now (Amount) and how much you started with (Principal).

The amount grows faster in compound interest because interest is added every year (or month), and next time, you earn interest on that too.

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.=(P×R×T)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 When the Rate is Compounded 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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Quarterly Compound Interest Formula

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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Simple vs. Compound Interest 

Let’s say you invest Rs. 100 for 3 years at a rate of 10% per year. Now, we’ll compare how much money you earn using Simple Interest and Compound Interest.

Simple Interest

In simple interest, you earn interest only on the original amount (called the principal).

Year 1: Rs. 100 × 10% = Rs. 10
Year 2: Rs. 100 × 10% = Rs. 10
Year 3: Rs. 100 × 10% = Rs. 10

Total Simple Interest = Rs. 30
Total Amount = Rs. 100 + Rs. 30 = Rs. 130

Compound Interest

In compound interest, you earn interest on the principal and on the interest from previous years.

Year 1: Rs. 100 × 10% = Rs. 10 → New amount = Rs. 110
Year 2: Rs. 110 × 10% = Rs. 11 → New amount = Rs. 121
Year 3: Rs. 121 × 10% = Rs. 12.10 → Final amount = Rs. 133.10

Total Compound Interest = Rs. 133.10 - Rs. 100 = Rs. 33.10

Conclusion

• Simple Interest (SI) = Rs. 30
• Compound Interest (CI) = Rs. 33.10
• Difference = Rs. 33.10 – Rs. 30 = Rs. 3.10

So, compound interest gives you more money because it adds interest on top of interest

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.

• Demographic Dynamics: Increase or decrease in population. Compound interest finds application in demography when assessing changes in population over time. It aids in understanding population growth or decline, considering factors like birth rates, death rates, and migration patterns. The formula proves beneficial in projecting future population figures, contributing to strategic urban planning and resource allocation.
• Biological Growth and Microorganisms: The growth of a bacteria if the speed of growth is identified. In microbiology, compound interest is employed to model the growth of bacterial populations. Understanding the speed of bacterial growth is crucial in various fields, including medicine and environmental science. The compound interest formula facilitates predicting bacterial population sizes at different time points, guiding researchers in optimizing conditions for experiments or addressing public health concerns.
• Economic and Financial Dynamics: The value of an item, if its price increases/decreases in the intermediate years. Compound interest plays a pivotal role in economic contexts, particularly when evaluating the changing value of commodities or assets. If an item's price experiences fluctuations over intermediate years, the compound interest formula becomes a valuable tool. It helps calculate the compounded impact of price changes, assisting investors, economists, and policymakers in comprehending the real economic impact and making informed decisions.

Challenges in Compound Interest Calculation

• Even though it is very much in use, compound interest calculations bear some challenges and aspects that must be appreciated:
• Complicated Calculation in the Mind: Compound interest calculations can be complicated, particularly when more than one compounding period and different interest rates are involved. Compound interest mental calculations can be difficult, and people tend to use financial calculators or spreadsheet software for precision.
• Variable Interest Rates: In practice, interest rates can change over time. Compound interest under varying rates demands a dynamic solution, and methods such as the compound interest formula might have to be modified for variable rates.
• Influence of Economic Factors: Economic conditions may affect interest rates as well as compounding frequencies. It is essential for individuals and investors alike to understand how economic factors can affect compound interest in financial scenarios.
• Practical Application in Investments: Implementation of the concept of compound interest in investment choice needs a sophisticated grasp of the risk, conditions of the market, and horizon of time. Practical application includes aspects more than the simple formula, e.g., market fluctuations and possible returns.

Compound Interest Solved Examples

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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