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## How to find Compound Interest when Interest is Compounded Quarterly?

If the Rate of Interest is Annual and Interest is Compounded Quarterly then the number of years is multiplied by 4 i.e. 4n and the annual interest rate is cut down by one-fourth. In such cases, Formula for Quarterly Compound Interest is given as under

Let us assume the Principal = P, Rate of Interest = r/4 %, and time = 4n, Amount = A, Compound Interest = CI then

A = P(1+(r/4)/100)^{4n}

In the above formula rate of interest is divided by 4 whereas the time is multiplied by 4.

We know CI = A – P

= P(1+(r/4)/100)^{4n} – P

= P{1+(r/4)/100)^{4n} – 1}

If you are aware of any of the three then you can automatically find the other one.

### Solved Problems on finding Compound Interest when Compounded Quarterly

1. Find the compound interest when $1,00, 000 is invested for 6 months at 5 % per annum, compounded quarterly?

Solution:

Principal Amount = $1,00, 000

Rate of Interest = 5% per annum

n = 6 months = 1/2 year

Since Interest Rate is Compounded Quarterly divide the interest rate by 4 i.e. r/4 and multiply the time by 4 i.e. 4n

Amount A = P(1+(r/4)/100)^{4n}

Substitute the Inputs in the above formula to find the amount

A = 1,00,000(1+(5/4)/100)^{4*1/2}

= 1,00,000(1+5/400)^{2}

= $ 1,02,515

CI = A – P

= $ 1,02,515 – $ 1,00,000

=$2515

2. Find the amount and the compound interest on Rs. 12,000 compounded quarterly for 9 months at the rate of 10% per annum?

Solution:

Principal Amount = Rs.12, 000

Rate of Interest = 10% per annum

n = 9 months = 9/12 = 3/4 year

Since Interest Rate is Compounded Quarterly divide the interest rate by 4 i.e. r/4 and multiply the time by 4 i.e. 4n

Amount A = P(1+(r/4)/100)^{4n}

Substitute the Input Values in the above formula to find the amount

A= 12,000(1+(10/4)/100)^{4*3/4}

= 12,000(1+10/400)^{3}

= 12,000(1+0.025)^{3}

= 12,000(1.025)^{3}

= Rs. 12922

CI = A – P

= 12922 – 12000

= Rs. 922

3. Calculate the compound interest (CI) on Rs. 4000 for 1 year at 10% per annum compounded quarterly?

Solution:

Principal Amount = Rs. 4,000

Rate of Interest = 10% per annum = 10/4 %

n = 1 year

Since Interest Rate is Compounded Quarterly divide the interest rate by 4 i.e. r/4 and multiply the time by 4 i.e. 4n

Amount A = P(1+(r/4)/100)^{4n}

Substitute the Input Values in the above formula to find the amount

A = 4000(1+(10/4)/100)^{4*1}

= 4000(1+10/400)^{4}

= 4000(1.1038)

= Rs. 4415.25

CI = A – P

= 4415.25 – 4000

= Rs. 415.25