## Simple interest

Interest on money is the cost to borrow money which is dependent on risks involved, time, economic conditions and many other factors.

Simple interest is the interest one pays/earns on principal amount when money is borrowed/lent over a period of time.

**Example**: If a man takes ₹ 1000 from a friend and after some time he return ₹ 1200 to his friend, he pays ₹ 200 as interest.

**Principal ( P ) : **Money is borrowed or deposited for a certain time, that money is called principal.

**Rate of Interest ( R ) :** The rate at which interest is charged on the principal amount.

**Time ( T ) :** The period for which the money is borrowed or deposited is called time.

**Simple Interest ( SI ):** Simple Interest is calculated on the original principal for certain period of time.

Simple Interest | = | Principal × Rate × Time |

100 |

SI | = | P × R × T |

100 |

**Example**: Find the simple interest on ₹ 2000 for 3 years at 7 % per year.
**Solution:** given P = ₹ 2000
R = 7 %
T = 3 Yrs

SI | = | P × R × T |

100 |

= | 2000 × 7 × 3 |

100 |

We know that, | S = | P × R × T |

100 |

Then, P | = | S × 100 |

R × T |

**Example**: Find the principal if simple interest is ₹ 600 in 3 years at 8 % per year.

**Solution :** Given , SI = ₹ 600, T = 3 yrs , R = 8 %

P | = | S × 100 |

R × T |

P = | 600 × 100 | = | 600 × 100 |

8 × 3 | 24 |

We know that, | SI = | P × R × T |

100 |

Then , R | = | SI × 100 |

P × T |

**Example:** A sum of ₹ 5000 borrowed after 4 years given an interest of ₹ 1000 at the rate of simple interest. Find the rate percent.

**Solution**: given, P = ₹ 5000
T = 4 Yrs , SI = ₹ 1000

R = | SI × 100 | = | 1000 × 100 |

P × T | 5000 × 4 |

R = | 100 | = 5 % |

20 |

We know that, | SI = | P × R × T |

100 |

Then , T | = | SI × 100 |

P × R |

**Example**: In what time, A sum of ₹ 4000 give an interest of ₹ 800 at the rate of 4 % per annum.

**Solution** : Given, P = ₹ 4000 , SI = ₹ 800 and R = 4%

T = | SI × 100 | = | 800 × 100 |

P × R | 4000 × 4 |

T = | 100 | = 5 years |

20 |

Total Amount = Sum of principal + interest A = P + SI

A = | P + | P × R × T | ||

100 |

Total Amount ( A ) = P | 1 + | R × T | ||

100 |

**Example: **A man takes ₹ 5000 from bank for 3 years at the rate of 6 % per annum. what amount will be paid by man after 3 years?

**Solution**:- Given, P = ₹ 5000, R = 6 % and T = 3 yrs

Total Amount ( A ) = P | 1 + | R × T | ||

100 |

A = 5000 | 1 + | 6 × 3 | ||

100 |

A = 5000 | 100 + 18 | ||

100 |

A = 5000 × | 118 | = ₹ 5900 |

100 |

**Some Important Point :-**

**1. For half-yearly :-**

Time = 2 times = 2T and Rate = half = | R | % |

2 |

**Example:** Find the simple interest on ₹ 500 for 2 yr at 8 % half yearly.

**Solution :-** Given , P = ₹ 500
T= 2 , T = 4 yr Rate = 8% (already half yearly rate is given)

SI = | P × R × T |

100 |

SI = | 500 × 4 × 8 | = ₹ 160 |

100 |

**2. For quaterly :-**

Time = 4 times = 4T and Rate = quarter = | R | % |

4 |

**Example:** Find the simple interest on ₹ 1500 for 1 yr,if the rate of interest at 12 % quarterly.

**Solution :** given, P = ₹ 1500
T = 4 times = 4 yr R = 12% (quarterly rate is given)

SI = | P × R × T |

100 |

SI = | 1500 × 12 × 4 | = ₹ 720 |

100 |

**3. For monthly :-**

Time = 12 times = 12 T , Rate = | R | % |

12 |

**Example:** Find the simple interest on ₹ 2000 for 3 month, it the rate of interest is 36 % yearly.

**Solution:** given, P = ₹ 2000
T = 3 months ,

Rate = | 36 | = 3% monthly |

12 |

SI = | P × R × T |

100 |

SI = | 2000 × 3 × 3 | = ₹180 |

100 |

### Some Useful Shortcut Method :-

1.If a sum of money becomes n times in T years at simple interest, thenA = P | 1 + | R × T | ||

100 |

⇒ nP = P | 1 + | R × T | ||

100 |

⇒ | nP | = | 1 + | R × T | ||

P | 100 |

⇒ n = | 100 + RT |

100 |

∴ R = | 100( n - 1 ) | % |

T |

**Example:** A sum of money becomes 5 times in 10 yr at simple interest. Find the rate of interest.

**Solution:** Given, T = 10 yrs and n = 5

∴ R = | 100( n - 1 ) | % |

T |

= | 100( 5 - 1 ) | = 40% |

10 |

2. If a sum of money at a certain rate of interest becomes x times in T_{1} years and y times in T_{2} years,then

T_{2} = |
( y - 1 ) | × T_{1} |

( x - 1 ) |

**Example:** A sum becomes 3 times in 6 years at a certain rate of interest, Find the time in which the same amount will be 9 times at the same rate of interest.

**Solution:** x = 3, y = 9 and T_{1} = 6

T_{2} = |
( y - 1 ) | × T_{1} |

( x - 1 ) |

T_{2} = |
( 9 - 1 ) | × 6 = 24 yrs |

( 3 - 1 ) |

3. If a sum of money in a certain time becomes x times at R_{1} rate of interest and y times in at R_{2} rate of interest, then

R_{2} = |
( y - 1 ) | × R_{1} |

( x - 1 ) |

**Example:** In a certain time, a sum becomes 5 times at the rate of 9 % per annum. At what rate of interest, the sum becomes 13 times in same time.

**Solution:** given, x = 5 , y = 13 and R_{1} = 9 %

R_{2} = |
( y - 1 ) | × R_{1} |

( x - 1 ) |

R_{2} = |
( 13 - 1 ) | × 9 = 27 % |

( 5 - 1 ) |