Question

Calculate compound interest for Rs. 15,000 for 1 year at $16\% $ compounded semi-annually.
A) Rs.3172
B) Rs.2496
C) Rs.3000
D) Rs.2572

Answer 1

Hint:
We can calculate the simple interest for 6 months for the given amount and the given rate of interest. Then we can add the interest to the given amount. Then we can find the simple interest for this new amount with the same rate of interest for the next 6 months. Then we can add it with the 2nd principal amount to get the total amount. Then we can subtract the given principal amount from it to get the interest.

Complete step by step solution:
We have the principle amount as Rs. 15000.
It is given that the rate of interest is $16\% $ and is compounded semi-annually.
So, we can find the interest after 6 months.
We have, ${P_1} = 15000$ , $R = 16\% $ and $N = \dfrac{6}{{12}} = \dfrac{1}{2}$
So, the interest for the 6 months is given by,
 $ \Rightarrow {I_1} = {P_1} \times R \times N$
On substituting the values, we get,
 $ \Rightarrow {I_1} = 15000 \times \dfrac{{16}}{{100}} \times \dfrac{1}{2}$
On simplification, we get,
 $ \Rightarrow {I_1} = 150 \times 8$
On multiplication we get,
 $ \Rightarrow {I_1} = 1200$
So, the new principal amount will become the sum of this interest and the initial principal amount.
 ${P_2} = {P_1} + {I_1}$
On substituting the values, we get,
 $ \Rightarrow {P_2} = 15000 + 1200$
On adding we get,
 $ \Rightarrow {P_2} = 16200$
Now we can calculate the interest for the next 6 months.
We have, ${P_2} = 16200$ , $R = \dfrac{{16}}{{100}}$ and $N = \dfrac{6}{{12}} = \dfrac{1}{2}$ .
So, the interest for the 6 months is given by,
 $ \Rightarrow {I_2} = {P_2} \times R \times N$
On substituting the values, we get,
 $ \Rightarrow {I_2} = 16200 \times \dfrac{{16}}{{100}} \times \dfrac{1}{2}$
On simplification, we get,
 $ \Rightarrow {I_2} = 162 \times 8$
Hence, we have,
 $ \Rightarrow {I_2} = 1296$
So, the amount after 1 year is given by the sum of the 2nd principal amount and its interest.
 $ \Rightarrow A = {P_2} + {I_2}$
On substituting the values, we get,
 $ \Rightarrow A = 16200 + 1296$
On simplification we get,
 $ \Rightarrow A = 17496$
Now the total interest is given by subtracting the principal amount from the total amount.
 $ \Rightarrow CI = A - {P_1}$
On substituting the values, we get,
 $ \Rightarrow CI = 17496 - 15000$
On simplification we get,
 $ \Rightarrow CI = 2496$
Thus, the compound interest is Rs. 2496.

So, the correct answer is option B.

Note:
Alternate solution to this problem is given by,
We have, $P = 15000$
As the interest is compounded semi-annually, we can have the equivalent year or period as 6 months.
So, we have \[N = 2\] as in a year we have 2 periods of 6 months.
We are given the rate of interest which is for 1 year. Then the rate for 6 months will be half of that.
 $ \Rightarrow R = \dfrac{{16\% }}{2}$
On division we get,
 $ \Rightarrow R = 8\% $
As \[a\% = \dfrac{a}{{100}}\] , we get,
 $ \Rightarrow R = \dfrac{8}{{100}}$
Now the amount after the 2 complete periods is given by,
 $A = P{\left( {1 + R} \right)^N}$
On substituting the values, we get,
 $ \Rightarrow A = 15000{\left( {1 + \dfrac{8}{{100}}} \right)^2}$
On simplification, we get,
 $ \Rightarrow A = 15000{\left( {1 + 0.08} \right)^2}$
On simplification we get,
 $ \Rightarrow A = 15000{\left( {1.08} \right)^2}$
On squaring the term, we get,
 $ \Rightarrow A = 15000 \times 1.1664$
On multiplication we get,
 $ \Rightarrow A = 17496$
Now the total interest is given by subtracting the principal amount from the total amount.
 $ \Rightarrow CI = A - {P_1}$
On substituting the values, we get,
 $ \Rightarrow CI = 17496 - 15000$
On simplification we get,
 $ \Rightarrow CI = 2496$
Thus, the compound interest is Rs. 2496.