Question

Calculate the amount of Rs.15000 at the end of 2 years 4 months, compounded annually at $6\% $ p.a.

Answer 1

Hint:
We can find the amount after 2 years for the principal amount of Rs.15000 at $6\% $ p.a. Then we can find the simple interest for the amount after 2 years at the same rate of interest for 4 months. Then we can add this interest to the amount after 2 years to find the amount of interest at the end of 2 years 4 months.

Complete step by step solution:
We are given an amount of Rs.15000 to calculate the amount at the end of 2 years 4 months compounded annually at $6\% $ p.a.
As it is compound interest, we can find the amount after 2 years.
We have the principal amount as Rs.15000
 $ \Rightarrow P = 15000$
The number of years is 2.
 $ \Rightarrow N = 2$
The rate of interest is $6\% $
 $ \Rightarrow R = 6\% $
On converting the percentage to fraction, we get,
  $ \Rightarrow R = \dfrac{6}{{100}}$
We know that the amount after interest for principal amount after N years for interest compounded annually at the rate R is given by,
 $A = P{\left( {1 + R} \right)^N}$
On substituting the values of P, R and N, we get,
 $ \Rightarrow A = 15000{\left( {1 + \dfrac{6}{{100}}} \right)^2}$
On converting the fraction to decimal, we get,
 $ \Rightarrow A = 15000{\left( {1 + 0.06} \right)^2}$
On simplification we get,
 $ \Rightarrow A = 15000{\left( {1.06} \right)^2}$
On calculating the square, we get,
 $ \Rightarrow A = 15000 \times 1.1236$
On multiplication we get,
 $ \Rightarrow A = 16854$
Now at the end of 2 years, the amount will become Rs. 16854.
Now we can find the simple interest for the next 4 months for this amount.
So, the new principle is Rs. 16854
 $ \Rightarrow P' = 16854$
We need to find interest at end of 4 months. So, the number of years is given by,
 $ \Rightarrow N' = \dfrac{4}{{12}}$
On simplification, we get,
 $ \Rightarrow N' = \dfrac{1}{3}$
The rate of interest will be the same.
 $ \Rightarrow R = \dfrac{6}{{100}}$
We know the simple interest is given by the equation,
 $I = P \times N \times R$
On substituting the values, we get,
 $ \Rightarrow I = 16854 \times \dfrac{1}{3} \times \dfrac{6}{{100}}$
On cancelling the common factors, we get,
 $ \Rightarrow I = \dfrac{{16854}}{{50}}$
On simplification we get,
 $ \Rightarrow I = 337.08$
Now the total amount at the end of 2 years 4 months is given by,
 $A' = P' + I$
On substituting the values, we get,
 $ \Rightarrow A' = 16854 + 337.08$
On adding we get,
 $ \Rightarrow A' = 17191.08$
On rounding of the decimal, we get,
 $ \Rightarrow A' = 17191$

So, the required amount is Rs. 17191.

Note:
We must take the compound interest, not the simple interest. While calculating the compound interest we can calculate only for complete years, not for any number of months. While calculating the interest of 4 months, we must take the principle amount as the amount after 2 years. The compound interest equation will give the amount after the required years. But the equation for simple interest will give only the amount of interest. We need to add the interest with the principal amount to get the required amount.