Simple Interest and Compound Interest

# Simple Interest and Compound Interest

## Simple Interest

Simple interest represents a fee you pay on a loan or income you earn on deposits:

• When borrowing money: You have to repay the amount you borrowed and pay extra money as an interest, which represents the cost of borrowing.

• When lending money: You typically set a rate and earn interest income for making your money available to other people.

• When depositing money: Interest-bearing accounts like savings accounts pay interest income because you are making money available to the bank to lend to others.

### How to Calculate Simple Interest

Interest is calculated based on the original sum of money, known as the principal.
In the simple interest, simple means you're working with the simplest way of calculating interest. Once you understand how to calculate simple interest, you can move on to other varieties, like annual percentage yield, annual percentage rate, and compound interest.
To calculate simple interest, use the following formula:

Simple Interest = (P × R × T)/100

For example, you invest . 100 (the principal) at a 5% annual rate for 1 year. The simple interest will be:

Simple Interest = ( 100) × (0.05) × (1) = ₹ 5 simple interest for one year.

Note that the interest rate (5%) is written as a decimal (0.05). To do your calculations, you'll need to convert percentages to decimals. Remember this easily by thinking of the word percent as "per 100." You can convert a percentage into its decimal form by dividing it by 100.

For example:
Convert 5% into decimal= 5/100 =0.05
You can use it as a fraction, 5% = 5/100

Multiple years: If you want to calculate simple interest over more than 1 year, calculate the interest earnings using the principal from the first year, multiplied by the interest rate and total number of years.

Simple Interest: (₹ 100) × (0.05) × 3 = ₹ 15 simple interest for three years

### Limitations of Simple Interest

The simple interest provides a basic way of looking at interest. In the real world, your interest, whether you are paying it or earning it, is usually calculated using slightly more complex methods. However, understanding simple interest gives you a good start, and it can give you a general idea of what a loan will cost or what an investment will return.

## Types of Interest

Interest is defined as the cost of borrowing money as in the case of interest charged on a loan balance. Conversely, interest can also be the rate paid for money on deposit as in the case of a certificate of deposit. Interest can be calculated in two ways, simple interest or compound interest.

1.   Simple interest is calculated on the principal or original amount of a loan.
2.   Compound interest is calculated on the principal amount and also on the accumulated interest of previous periods, and can thus be regarded as “interest on interest.”

There can be a big difference in the amount of interest payable on a loan if interest is calculated on a compound rather than a simple basis. On the positive side, the magic of compounding can work to your advantage when it comes to your investments and can be a potent factor in wealth creation.

While simple interest and compound interest are basic financial concepts, becoming thoroughly familiar with them may help you make more informed decisions when taking out a loan or investing.

## What is Compound Interest?

Compound interest is interest calculated on the initial principal, which also includes all of the accumulated interest of previous periods of a deposit or loan.

### KEY POINTS

• Compound interest is interest calculated on the initial principal, which also includes all of the accumulated interest of previous periods of a deposit or loan.
• Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one.
• Interest can be compounded on any given frequency schedule, from continuous to daily to annually.
• When calculating compound interest, the number of compounding periods makes a significant difference.

The rate at which compound interest accrues depends on the frequency of compounding, such that the higher the number of compounding periods, the greater the compound interest. Thus, the amount of compound interest accrued on 100 compounded at 10% annually will be lower than that on 100 compounded at 5% semi-annually over the same time period. Since the interest-on-interest effect can generate increasingly positive returns based on the initial principal amount, it has sometimes been referred to as the "miracle of compound interest."

## Calculating Compound Interest

Compound interest is calculated by multiplying the initial principal amount by one plus the annual interest rate raised to the number of compound periods minus one. The total initial amount of the loan is then subtracted from the resulting value.
The formula for calculating compound interest is:
Compound Interest
= [P(1 + R%)n] – P

= P[(1 + R%)n – 1]

(Where P = Principal, R = annual interest rate in percentage terms, and n = number of compounding periods.)

Take a three-year loan of 10,000 at an interest rate of 5% that compounds annually. What would be the amount of interest? In this case, it would be: 10,000 [(1 + 0.05)3 1] = 10,000 [1.157625 1] = 1,576.25.

## Compounding Periods

When calculating compound interest, the number of compounding periods makes a significant difference. Generally, the higher the number of compounding periods, the greater the amount of compound interest. So for every 100 of a loan over a certain period, the amount of interest accrued at 10% annually will be lower than the interest accrued at 5% semi-annually, which will, in turn, be lower than the interest accrued at 2.5% quarterly.
In the formula for calculating compound interest, the variables “R” and “n” have to be adjusted if the number of compounding periods is more than once a year.
Therefore, for a 10-year loan at 10%, where interest is compounded semi-annually (number of compounding periods = 2), R = 5% (i.e. 10% / 2) and n = 20 (i.e.10 x 2).

## Solved Examples on Compound Interest

Example 1: Find the compound interest and amount on  800 for 3 years at 5% p.a.

Solution: Here P =  800, R = 5%, N = 3 years
Amount (A) = P(1 + R%)n
= 800(1 + 5%)3
= 800(1 + 5/100)3
= 800(21/20)3
=  926.10

C.I. = A – P =  926.10 –  800=  126.10
Hence, amount =  926.10 and compound interest =  126.10

Example 2: Find the compound interest and amount on  8000 for 1⅟2 years at 10% p.a., interest being compounded half-yearly.

Solution: Here P =  8000, R = 10% p.a. = 5% per half-year, N = 1⅟2 years = 3 half-years
Amount (A) = P(1 + R%)n
= 8000(1 + 5%)3
= 8000(1 + 5/100)3
= 8000(21/20)3
=  9261
C.I. = A – P =  9261 –  8000 =  1261
Hence, amount =  9261 and compound interest =  1261

Example 3: The compound interest on a sum of money for 2 years is  820 and the simple interest on the same sum for same period is  800. Find the rate of interest and the sum of money.

Solution:
Since S.I. for 2 years =  800

Ãž S.I. for 1st year =  400

Ãž C.I. for 1st year =  400

Ãž C.I. for 2nd year = ( 820 –  400) =  420

Difference of interest = ( 420 –  400) =  20
Now,  20 is the interest on  400 for one year.
Thus, rate of interest = (S.I × 100)/(P × T)
= (20 × 100)/(400 × 1)
= 5%

Again, principal = (S.I × 100)/(R × T)
= (400 × 100)/(5 × 1)
=  8000

## Growth

Certain things such as the height and weight of a child, the height of a tree, the population of a town etc., increase over a period of time. The increase in amount over a period of time is called growth. Thus, growth is the relative increase in a quantity over a period of time.

Rate of Growth: Growth per unit time is called the rate of growth.

### Formulas for Population Growth

Let the present population of a city be P, R is the constant rate of its growth per annum and N is the number of years, then

## Depreciation

The value of a machine or building or any other article subject to wear and tear decreases with time. The relative decrease in the value of a machine or building or any article over a period of time is called its depreciation.

Rate of Depreciation: Depreciation per unit time is called the rate of depreciation.

### Formulas for Depreciation

We can easily derive the following formulas by unitary method.
Let the present value of a machine be P.
Let it depreciates at the rate of R% per annum.
Then,

Solved Examples on Growth and Depreciation

Example 1: The population of a small village 2 years ago was 3125. Due to migration to towns, it decreases every year at the rate of 4% per annum. Find its present population.

Solution:
Present population = (Population 2 years) × P(1 – R%)2
= 3125(1 – 4/100)2
= 3125(1 – 0.04)2
= 3125(0.96)2
= 2880
The present population of the village = 2880.

Example 2: The population of a city increases by 10% every year. If its present population is 1210000, find the population after two years.

Solution: Here, P = 1210000; R = 10%, N = 2 years

Population after 2 years = P(1 + R%)N
= 1210000 (1 + 10/100)2
= 1210000 (1 + 1/10)2
= 1210000 (11/10)2
= 1464100
Thus, population after 2 years will be 1464100.

Example 3:
A machine was purchased 2 years ago. Its value depreciates by 10% every year. Its present value is  19083.60. For how much money was the machine purchased?

Solution: Let the machine be purchased for  P.
Its depreciated value P(1 – 10/100)2
81P/100

So, 81P/100 = 19083.60
P = (19083.60 × 100)/81
= 23560

Therefore, the machine was purchased for  23560.