Question

Arif took a loan of Rs. 80, 000 from a bank. If the rate of interest is 10% p.a., find the difference in amounts he would be paying after 1$\dfrac{1}{2}$ years if the interest is
i. compounded annually and
ii. compounded half-yearly

Answer 1

Hint: In this question you are given with the principal amount i.e. 80,000, the rate of interest 10% p.a. and the time duration 1$\dfrac{1}{2}$ years. You need to calculate the total amount if the interest is compounded annually and if it is compounded half-yearly, to do this use the simple formula for amount i.e. A = $P{(1 + \dfrac{R}{{100}})^n}$ where P is the principal amount, A is the amount after the rate of interest is added, R is the rate of interest and n is the time duration.

Complete step-by-step answer:
Case (i) When the interest is compounded annually.
We are given that P = Rs. 80,000
R=10% p.a.
and
T=1$\dfrac{1}{2}$years
So, n= 1+$\dfrac{1}{2}$
So, amount for the first year = A = $P{(1 + \dfrac{R}{{100}})^n}$
A=80,000(1+$\dfrac{{10}}{{100}}$)
A= Rs. 88,000
Now calculate the S.I. for the rest 6 months.
So, S.I. = $\dfrac{{P \times R \times T}}{{100}}$
S.I. = $\dfrac{{88000 \times 10 \times 1}}{{100 \times 2}}$
S.I. = Rs. 4400
So the total amount = Rs. 88000 + Rs.4400 = Rs. 92400
Case (ii) When the interest is compounded half yearly.
P=80,000
R=10% p.a. = 5% per half yearly
T=1$\dfrac{1}{2}$ years
So n = 3(since the interest is compounded half yearly and there are 3 half years in the time duration of 1$\dfrac{1}{2}$years)
As, we know A = $P{(1 + \dfrac{R}{{100}})^n}$
So, A= 80,000(1+$\dfrac{5}{{100}}{)^3}$
So, A = 80,000${(\dfrac{{105}}{{100}})^3}$
So, A = Rs. 92610.
Thus, the difference between the two amounts which is asked in the question will be = Rs. 92610- Rs. 92400 = Rs. 210

Note: Compound interest is the addition of interest to the principal amount of the loan or deposit or, in other words, interest on debt. It is the result of reinvesting the interest, rather than paying it out, so that the interest in the next period is earned on the principal amount plus the previously accumulated interest for example, in the first part of the solution we separated the 1 year and $\dfrac{1}{2}$year tenure, the S.I. was calculated on a principal amount of 88,000 instead of 80,000.