**Simple & Compound Interest PDF**

**INTEREST**

- Interest is the fixed amount paid on borrowed money.
- The sum lent is called the
**Principal.** - The sum of the principal and interest is called the
**Amount.**

**Interest is of two kinds:**

(i) Simple interest (ii) Compound interest

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**(i) Simple interest: **When interest is calculated on the original principal for any length of time, it is called simple interest.

- Simple interest = (Principal×Time×Rate)/100

i.e. S.I. = (P×R×T)/100 - Amount = Principal + Interest

i.e. A=P+I=P+PRT/100 = P[1+RT/100] - Principal(P) = (100×S.I.)/(R×T)
- Rate(R) =(100×S.I.)/(T×P)
- Time(T)=(100×S.I.)/(P×R)

If rate of simple interest differs from year to year, then S.I.=P×((R1+R2+R3+⋯))/100

**Example: **Find the amount to be paid back on a loan of Find the amount to be paid back on a loan of `18,000 at 5.5% per annum for 3 years

**Solution: **P=`18000, R=5.5%, T=3 years

S.I. = (P×R×T)/100 = (18000×5.5×3)/100 = Rs.2970

Amount = P + I = 18000 + 2970 = **Rs.20970
**

**(ii) Compound interest: **Money is said to be lent at compound interest when at the end of a year or other fixed period, the interest that has become due is not paid to the lender, but is added to the sum lent, and the amount thus obtained becomes the principal in the next year or period. The process is repeated until the amount for the last period has been found. Hence, When the interest charged after a certain specified time period is added to form new principal for the next time period, the interest is said to be compounded and the total interest accurse is compounded and the total interest accrued is compound interest.

- C.I. = p[(1+r/100)^n-1];
- Amount(A) = P(1+r/100)^n, Where n is number of time period.
- If rate of compound interest differs from year to year, then

Amount = P(1+r_1/100)(1+r_2/100)(1+r_3/100)…..

**Example:**

If `60000 amounts to `68694 in 2 years then find the rate of interest.

**Solution:**

Given: A = `68694

P = `60000

n = 2 years

r=?

A = P(1+r/100)^n

68694 = 60000(1+r/100)^2

68694/60000 = (1+r/100)^2

11449/10000= (1+r/100)^2

1+r/100= √(11449/10000)= √1.1449

1+r/100= 1.07

r/100 = 1.07 – 1 = 0.07

r = 0.07×100 = 7%

**Compound Interest-when interest is calculated quarterly**

Since 1 year has 4 quarters, therefore rate of interest will become th of the rate of interest per annum, and the time period will be 4 times the time given in years. Hence, for quarterly interest,

**A=P(1+(r/4)/100)^(4×n)=P(1+r/400)^4n**

**Example: **Find the compound interest on `25625 for 12 months at 16% per annum, compound quarterly.

**Solution: **

Principal(P)= `25625

Rate(r) = 16% =16/4% = 4%

Time = 12 months = 4 quaters

A = 25625(1+4/100)^4 = 25625(26/25)^4= 25625×26/25×26/25×26/25×26/25=29977.62

C.I. =A-P = 29977.62-25625 = **4352.62
**

**EFFECTIVE RATE**

If `1 is deposited at 4% compounded quaterly, a calculator can be used to find that at the end of one year, the compound amount is ` 1.0406, an increase of 4.06% over the original `1. The actual in the money is somewhat higher than the stated increase of 4%. To differentiate between these two numbers, 4% is called the nominal or stated rate of interest, while 4.06% is called the effective rate. To avoid confusion between stated rates and effective rates, we shall continue to use r for the stated rate and we will use r_{e} for the effective rate.

**Example: **Find the effective rate corresponding to a stated rate of 6% compound semiannually.

**Solution:**

A calculator shows that `100 at 6% compounded semiannually will grow to

A=100(1+(.06)/2)^2= 100(1.03)2=$ 106.09

Thus, the actual amount of compound interest is

`106.09 – `100=`6.09. Now if you earn `6.09 interest on

`100 in 1 year with annual compounding, your rate is 6.09/100=.0609=6.09%

Thus, the effective rate is re = 6.09%

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