Question

Calculate the amount and the compound interest on: Rs. 4600 in 2 years when the rates of interest of successive years are $10\% $ and $12\% $ respectively.
A) Rs.5,667.20 and Rs.1,067.20
B) Rs.5,600.20 and Rs.167.20
C) Rs.5,680.20and Rs.167.20
D) Rs.5,667.20 and Rs.1,000.20

Answer 1

Hint:
We can find the amount after the $1^{\text{st}}$ year using the given interest rate. Then we can find the amount after the $2^{\text{nd}}$ year with the $2^{\text{nd}}$ rate of interest with the amount after 1 year as the principal amount. Then we can subtract the given principal amount from the total amount to get the interest.

Complete step by step solution:
We have the principal amount as Rs. 4600. As it is compound interest, we can find the amount after $1^{\text{st}}$ year.
The principle for the $1^{\text{st}}$ year is Rs. 4600.
 $ \Rightarrow {P_1} = 4600$
The rate of interest of $1^{\text{st}}$ year is given as $10\% $
 $ \Rightarrow {R_1} = 10\% $
On converting the percentage to fraction, we get,
 $ \Rightarrow {R_1} = \dfrac{{10}}{{100}}$
Now the amount after 1 year is given by,
 ${A_1} = {P_1}{\left( {1 + {R_1}} \right)^1}$
On substituting the values, we get,
 $ \Rightarrow {A_1} = 4600\left( {1 + \dfrac{{10}}{{100}}} \right)$
On converting the fraction into decimal, we get,
 $ \Rightarrow {A_1} = 4600\left( {1 + 0.1} \right)$
On simplification we get,
 $ \Rightarrow {A_1} = 4600\left( {1.1} \right)$
On multiplication we get,
 $ \Rightarrow {A_1} = 5060$
Therefore, the amount after 1 year is Rs. 5060.
Now in the $2^{\text{nd}}$ year, the principal amount is Rs. 5060.
 $ \Rightarrow {P_2} = 5060$
The rate of interest of $2^{\text{nd}}$ year is given as $12\% $
 $ \Rightarrow {R_2} = 12\% $
On converting the percentage to fraction, we get,
 $ \Rightarrow {R_2} = \dfrac{{12}}{{100}}$
Now the amount after the $2^{\text{nd}}$ year is given by,
 ${A_2} = {P_2}{\left( {1 + {R_2}} \right)^1}$
On substituting the values, we get,
 $ \Rightarrow {A_2} = 5060{\left( {1 + \dfrac{{12}}{{100}}} \right)^1}$
On simplifying the fraction, we get,
 $ \Rightarrow {A_2} = 5060\left( {1 + 0.12} \right)$
On simplification, we get,
 $ \Rightarrow {A_2} = 5060\left( {1.12} \right)$
On multiplication we get,
 $ \Rightarrow {A_2} = 5667.2$
Therefore, the amount after two years will be RS.5667.2
The total interest is given by subtracting the given principal amount from the total amount after 2 years.
 $ \Rightarrow I = {A_2} - {P_1}$
On substituting the values, we get,
 $ \Rightarrow I = 5667.2 - 4600$
On simplification we get,
 $ \Rightarrow I = 1067.2$
So, the interest for 2 years is RS.1067.2

Therefore, the correct answer is option A, RS.5,667.20 and RS.1,067.20.

Note:
Alternatively, the amount after 2 years is given by,
 $A = P\left( {1 + {R_1}} \right)\left( {1 + {R_2}} \right)$
On substituting the values, we get,
 $ \Rightarrow A = 4600 \times \left( {1 + \dfrac{{10}}{{100}}} \right)\left( {1 + \dfrac{{12}}{{100}}} \right)$
On simplification, we get,
 $ \Rightarrow A = 4600 \times \left( {1 + 0.1} \right)\left( {1 + 0.12} \right)$
On simplification we get,
 \[ \Rightarrow A = 4600 \times \left( {1.1} \right)\left( {1.12} \right)\]
On multiplication we get,
 \[ \Rightarrow A = 4600 \times 1.232\]
On further simplification we get,
 \[ \Rightarrow A = 5667.2\]
Therefore, the amount after two years will be Rs.5667.2
The total interest is given by subtracting the given principal amount from the total amount after 2 years.
 $ \Rightarrow I = {A_2} - {P_1}$
On substituting the values, we get,
 $ \Rightarrow I = 5667.2 - 4600$
On simplification we get,
 $ \Rightarrow I = 1067.2$
So, the interest for 2 years is Rs.1067.2
Therefore, the correct answer is option A, Rs.5,667.20 and Rs.1,067.20.