# Simple Interest - Basic Concepts

The money borrowed or lent out for a certain period is called the principal. The money paid for using other's money is called interest.

If I borrow money from you for a certain time period, then at the end of the time period, I return not only the borrowed money but also some additional money. This additional money that a borrower pays is called interest. The actual borrowed money is called Principal. The interest is usually calculated as a percentage of the principal and this is called the interest rate. There is a well-accepted norm about the interest rate. It is always assumed to be per annum, i.e. for a period of one year, unless stated otherwise.

If the interest on a certain sum borrowed for a certain period is reckoned uniformly, then it is called simple interest.
Amount = Principal + S.I.

Simple Interest can be computed in two basic ways. The first way, with simple annual interest, the interest computed on the principal only and is equal to [(principal) Ã— (rate) Ã— (time)]/100.

Here rate is taken as percent per annum and time is taken in years. Sometimes the interest is given and time or one of other two items is missing. Then the formula for calculating the time becomes (interest Ã— 100)/(principal Ã— rate). And similarly the rate/principal can be calculated.

# Formula

Let principal = P, Rate = R% per annum and Time = T years. Then,

(i) Simple Interest (SI) = (P Ã— R Ã— T)/100Similarly,

(ii) P = 100 Ã— SI/(R Ã— T)

(iii) R = 100 Ã— SI/(P Ã— T)

(iv) T = 100 Ã— SI/(P Ã— R).

Note that simple interest does not change over time. If for example Rs 10,000 is invested at the rate of 10% per annum, then interest of Rs 1000 will paid annually for an indefinite period. Every year the investor will get Rs 1000 without any increment.

The second way is to add the interest earned to the principal, thereby increasing it, and subsequently adding interest on the higher amount. This is called compounding, explained in the second section.

## Illustrations :-

Example

The simple interest on Rs 600 for 3 years at 4% will be:
1. Rs. 60                      2. Rs. 70                      3. Rs. 72                      4. Rs. 65

Solution

Using the formula, we get (600 Ã— 3 Ã— 4)/100 = Rs 72.

Illustration:

Anand borrowed a loan at 14% per annum at simple interest. After four years, he repaid Rs. 11,700. Find the amount borrowed by him.
1. Rs. 8,500                 2. Rs. 7,500                 3. Rs. 11,000               4. Rs. 6,500

Solution

He repays 11,700, this means that the interest is 11,700 â€“ P.
Note that since the formula gives interest, we must subtract P from the amount in order to get the correct equation. A common mistake is to equate the amount as given.
Using the formula, we get (P Ã— 14 Ã— 4)/100 = 11700 â€“ P.
On solving the equation, we get: 56P = 1170000 â€“ 100P
156P = 1170000; or P = 1170000/156 = 7500.

Illustration:

A sum of Rs 3,800 is lent out in two parts in such a way that the interest on one part at 8% for 5 years is equal to that on another part at Â½ % for 15 years. Find the sum lent out at 8%.

Solution

Let P be the sum let out at 8% and Q be the sum let out at Â½ %.
So P Ã— 8 Ã— 5 =  Q Ã— Â½ Ã— 15.
Therefore P : Q = 15/ (2 Ã— 8 Ã— 5) = 3 : 16.
The total sum of Rs. 3,800 is to be divided in the ratio of 3 : 16.
Hence the first sum is 3800 Ã— 3/19 = Rs 600.

# Repayment in Equal Instalments

Illustration:

What annual installment will discharge a loan of Rs. 3540, due after 5 years. The rate of interest is 9% per annum?

Solution

Note here that Rs 3540 is NOT principal. It is the sum due after 5 years.
Instalment Years   0 (today)
x       x      x        x       x

1        2      3       4       5
The first instalment will be paid one year from now i.e. 4 years before it is actually due.
The second instalment will be paid two years from now i.e. 3 years before it is actually due.
The third instalment will be paid 2 years before it is actually due.
The fourth instalment will be paid 1 year before it is actually due.
The fifth instalment will be paid on the day the amount is actually due.
So on the first instalment the interest will be paid for 4 years, on the second for 3 years, on the third for 2 years, on the fourth for 1 year and on the fifth for 0 year.
Total interest for 10 years will be paid (4 + 3 + 2 + 1 + 0).
Let the instalment be x. Then 9% on x for 10 years equals the debt.
Hence interest = (x Ã— 10 Ã— 9)/100 = Rs. 0.9x.
Add to this the installments paid, and we get: 5x + 0.9x = 5.9x.
Now we can equate 5.9x = 3540; hence x = 3540/5.9 = 600.
Hence each installment should be Rs 600.

This method looks lengthy, but the student should arrive at the figure 5.9 mentally and divide, which makes it much quicker.

Otherwise the following straight method can also be applied, where the annual installment required is equal to

Illustration:

A person borrowed Rs. 2,500 from two money-lenders. For one loan he paid 8% per annum and for the other 6% per annum simple interest. If he paid Rs. 180 as total interest for one year, how much did he borrow at 8% per annum?
1. Rs. 1,000                 2. Rs. 1,200                 3.  Rs. 1,300                4. Rs. 1,500

Solution

Calculate the interest for one year. Let x be one part, then the other part is (2500 â€“ x).
Then: 8% of x + 6% of (2500 â€“ x) = 180.
Now we can solve this: 150 â€“ 2% of x = 180; 2% of x = 30; or x = 1500.

Illustration:

If Rs 8,000 is invested at 6 percent simple annual interest, how much interest is earned after 3 months?

Solution

Since the annual interest rate is, 6%, the interest for 3 months is
(8,000 Ã— 6 Ã— 3)/(100 Ã— 12) = Rs. 120.

Illustration:

A sum of Rs 3,800 is lent out in two parts in such a way that the interest on one part at 8% for 5 years is equal to that on another part at Â½ % for 15 years. Find the sum lent out at 8%.

Solution

Let P be the sum let out at 8% and Q be the sum let out at Â½ %.
So P Ã— 8 Ã— 5 =  Q Ã— Â½ Ã— 15. Therefore P : Q = 15 /  ( 2 x 8 x 5 ) = 3 : 16.
The total sum of Rs. 3,800 is to be divided in the ratio of 3 : 16.
This way the first sum is 3800 x 3/ 19 = Rs 600.

Sometimes instead of interest, the amount is given, then either you need to add the above simple interest formula in the principal again and then solving the equation. Otherwise the following straight formula can be applied.

.

Sometimes, the amount is given, but instead of principal, the simple interest is asked in the question. And the interest can be calculated with the help of the following straight formula

Simple interest .

The above-mentioned principal is also known as present worth of the amount. Similarly the simple interest is also known as true discount on the amount.