Question

In what time will Rs.8000 amount to Rs.8820 at $10\% $ p.a. interest compounded half yearly?
A. 2.5 years
B. 4 years
C. 3 years
D. 1 year

Answer 1

Hint: According to the question given in the question we have to determine the time when Rs.8000 amount to Rs.8820 at $10\% $ p.a. interest compounded half yearly. So, first of all we have to determine the interest rate for each conversion period.
Now, we have to use the formula to find the amount for the Rs.8820 at $10\% $p.a. interest compounded half yearly which is as explained below:

Formula used: $ \Rightarrow {A_n} = P{(I + i)^n}..................(1)$
Where, ${A_n}$is the compounded amount, P is the principal amount and I is the compounded interest and n is the time for which the principal amount is invested.
Now, on substituting all the values in the formula (1) above, we can easily determine the time period by comparing all values of n in the obtained equation.
After obtaining the time which is n as mentioned in the question that interest compounded half yearly so, we have to divide the obtained time which is n by 2.

Complete step-by-step solution:
Step 1: First of all we have to determine the interest rate for each conversion period which can be obtained by dividing all the given interest rates by 2. Hence,
$
   = \dfrac{{10\% }}{2} \\
   = 5\% \\
   = \dfrac{5}{{100}} \\
   = 0.05
 $
Step 2: Now, we have to use the formula (1) as mentioned in the solution hint to determine the value of the compounded amount which is already given in the question so, we can easily determine the unknown time period which is n. Hence,
$ \Rightarrow 8820 = 8000{(1 + 0.05)^n}$
Now, we have to solve the expression as obtained just above,
$
   \Rightarrow \dfrac{{8820}}{{8000}} = {(1 + 0.05)^n} \\
   \Rightarrow 1.1025 = {(1.05)^n}..............(2)
 $
Step 3: Now, to obtain the value of n we have to make both of the bases equal to each-other which can be obtained by finding the square to the left side of the expression (2) as obtained in the solution step 2. Hence,
$
   \Rightarrow {(1.05)^2} = {(1.05)^n} \\
   \Rightarrow n = 2
 $
Step 4: Now, as mentioned in the question that interest compounded half yearly hence, we have to divide the time period by 2.
$
   = \dfrac{n}{2} \\
   = \dfrac{2}{2} \\
   = 1
 $
Hence, with the help of the formula (1) as mentioned in the solution hint we have obtained the value of the time period which is 1 years.

Therefore option (D) is correct.

Note: The compound interest is the sum of interest to the principal amount of a deposit or loan or we can say that interest on interest.
Compound interest is the result of reinvesting interest, rather than paying it out, so that the interest in the next period is then earned on the principal amount previously accumulated interest.