The lending and borrowing of money involves the concept of simple interest and compound interest. If you borrow money for certain period of time, you would have to return the this sum of money (Principal) with some extra money. this extra money is called **Interest**.

The money borrowed is called **principal**. The sum of interest and principal is called the **amount**. The time for which money is borrowed is called **period**.

**Amount = Principal + Interest**

The rate of interest is as per annum (unless indicated).

### Simple Interest

If the interest on a certain sum borrowed for a certain period is reckoned uniformly, then it is called Simple Interest and denoted as SI. Simple interest is simply calculated on principal amount using the following formula:

**SI = (P × R × T)/100**

where, P = principal, R = rate per annum, T = time in years

Amount can be calculated by adding interest to principal.

**Example 1: A certain sum of money invested at some rate of interest triple itself in 4 years. In how many years the principal will become 9 times of itself at the same rate?**

When the principal is in simple interest the interest for every year will be same. In 3 years the amount becomes 3 times the principal and we have

A = P + I

3P = P + I

⇒ I = 2P

The interest is 2 time the principal in 4 years or equal to principal in 2 years.

The interest will be equal to P in 2 years. So interest will be 8P in 16 years.

Amount after 16 years = P + 8P = 9P.

Hence the required answer will be 16 years.

### Compound Interest

When the borrower and the lender agree to fix up a certain unit of time (say yearly or half-yearly or quarterly) to settle the previous account. In such cases, the amount after the first unit of time becomes the principal for the second unit. The amount after second unit becomes the principal for the third unit and so on. After a specified period, the difference between the amount and the money borrowed is called Compound Interest for that period.

In case of compound interest, the total interest received in the present year will be added to the original principal and for the following year the principal will be Amount received (Principal + interest).

Suppose you lend Rs.10000 (principal) for 3 years at 10% per annum. So, you will get Rs.1000 as interest per annum (simple interest) For three years, interest will be Rs.3000 and thus you will get total amount of Rs.13000.

Compound interest involves interest on interest too, thus will give you better amount after 3 years. While calculating the compound interest, the principal amount keeps changing year after year (if the interest is compounded annually).

After 1 year: Interest = Rs.1000; New Principal = Rs.11000

After 2 years: Interest = Rs.1100; New Principal = Rs.12100

After 3 years: Interest = Rs.1210; You get Rs.13310. So, there is gain of Rs.310 if you lend at compound interest.

**A = P(1 + r/n) ^{nt}**

**Example 2: A certain sum of money doubles in 3 years, then in how many years it will become 8 times at compound interest.**

Think logically that in every 2 years, the principal becomes doubles of itself. So in 4 years it will be 4 times and in next two years it will be double of 4 times that is 8 times of original principal. So the required answer would be 8 years.

**Example 3: A man took Rs. 5000 at 10% simple interest and gave it to another person at 10% compound interest, which is being compounded annually. After 3 years, how much extra money he will get?**

Simple Interest on Rs.5000 = (5000 × 10 × 3)/100 = Rs. 1,500

He has to pay amount = Rs.5000 + Rs.1500 = Rs. 6500

Amount with compound interest = 5000(1 + 10/100)^{3} = Rs.6655

So, the answer is 6655 – 6500 = Rs 155.