Question

# Find the amount on Rs.10,000 after 2 years if the rates of interest are $3\%$ and $4\%$ for successive years.

Hint: Use the formula of compound interest for finding the amounts denoted by ${A_1}$ and ${A_2}$ for the rate of interest $3\%$ and $4\%$ respectively. After finding the amounts ${A_1}$ and ${A_2}$, calculate its average to get the total amount received after the 2 years.

Complete step-by-step solution:
We know that the general formula for calculating the compound interest is given by the equation
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^n}.............(1)$
Here A denotes the Amount received after n years when Principal P has compounded annually for n years with the rate of interest denoted as R.
Let us take the amount received for the rate of interest 3% to be denoted by $A_1$.
Here ${P_1} = 10000,{R_1} = 5\% ,{n_1} = 2years$
Now we will put it in equation (1), the formula becomes
${A_1} = {P_1}{\left( {1 + \dfrac{{{R_1}}}{{100}}} \right)^{{n_1}}}................(2)$
Hence by substituting the values, we get
${A_1} = 10000{\left( {1 + \dfrac{3}{{100}}} \right)^2}$
$A_1^{} = 10609.$
Now similarly we let the amount received for the rate of interest 4% be denoted as ${A_2}$
Here ${P_2} = 10000,{R_2} = 4\% ,{n_2} = 2years$
Now the formula to calculate $A_2$ is given by the equation
${A_2} = {P_2}{\left( {1 + \dfrac{{{R_2}}}{{100}}} \right)^{{n_2}}}..................(3)$
Hence by substituting the values we get
${A_2} = 10000{\left( {1 + \dfrac{4}{{100}}} \right)^2}$
${A_2} = 10816$ .
The total amount calculated for the 2 years is given by the average of ${A_1}$ and ${A_2}$.
Hence, the $Amount = \dfrac{{{A_1} + {A_2}}}{2}$ $= (\dfrac{{10609 + 10816}}{2})$ $= 10712$
Therefore, the amount on Rs.10,000 after 2 years if the rates of interest are 3% and 4% respectively for the successive year is equal to 10712.

Note: When the rate of interest are different for the successive years the problem can also be solved by the formula
$Amount = P\left( {1 + \dfrac{{{R_1}}}{{100}}} \right)\left( {1 + \dfrac{{{R_2}}}{{100}}} \right)\left( {1 + \dfrac{{{R_3}}}{{100}}} \right).........\left( {1 + \dfrac{{{R_n}}}{{100}}} \right)$
The student should be careful in using the formula as many students get confused and simply add the rate of interest ie. $3 + 4 = 7\%$ to get the combined rate of interest which is entirely a wrong approach.
So ,by substituting all the values in equation $\left( 3 \right)$, we get
$Amount = 10000 \times \left( {1 + \dfrac{3}{{100}}} \right)\left( {1 + \dfrac{4}{{100}}} \right) = 10712.$