In the context of compound interset, sometimes there arises a situation when the borrower and the lender fix up a certain unit of time (like yearly or half-yearly or multiples of n where n is the no of years infractions)

In such case, the amount becomes the principle for the second unit of time after the first unit of time(n=1)## Different cases are as follows:-

### Case I

when the compound interest is calculated half-yearlyif the rate is r% per annum and time is n years then the corresponding rate and time are n/2 and 2*n respectively.

A = P[1 + (r²)/100]²n

where A= amount

P=principle

r=rate

t=time in years

#### Illustration

P= Rs 15000r=10%

t=1 Year

compounded half yearly

then according to above formula

r=10/2 = 5%

t=2 x 1=2 half year

A = 15000 x [(1 + 5/100)²] = Rs 16537.50

### Case 2

when comound interst is calculated quarterlyin this case rate = r/4% and time 4 x n quarter years

A= P[1+(r/4)/100]⁴ x n

#### llustratin

P = Rs 15625

t = 9 months = 3 quarters

r= 16/4 = 4%

A= 15625 x (1+4/100)³ = Rs 1951

t = 9 months = 3 quarters

r= 16/4 = 4%

A= 15625 x (1+4/100)³ = Rs 1951

#### Important note :

The difference between the compound interest and simple interst over a period of two year is given by

[C.I - S.I = P(r/100)²]

where symbols have their usual meanings.

Amount = P(1 + r/100)³ x (1+ (2/5r)/100)

C.I(yearly) = 5000 x (1+4/100) x (1+0.5 x 4/100)

= Rs 5304

C.I(half yearly) = 5000 x (1+2/100)³

= Rs 5306.04

Difference = 5306.04-5304 = RS 2.04

A = x/(1 + r/100)ⁿ

[C.I - S.I = P(r/100)²]

where symbols have their usual meanings.

### Case 3

When interset is compounded annualy but time is in fraction says yearsAmount = P(1 + r/100)³ x (1+ (2/5r)/100)

#### Illustration

calculate difference between compound interest on Rs 5000 for 4 % per annum compounded yearly and half yearlyC.I(yearly) = 5000 x (1+4/100) x (1+0.5 x 4/100)

= Rs 5304

C.I(half yearly) = 5000 x (1+2/100)³

= Rs 5306.04

Difference = 5306.04-5304 = RS 2.04

### Case 4

Amount Due in n years for a sum of Rs.xA = x/(1 + r/100)ⁿ

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