Question

Divide Rs. 1301 between A and B so that the amount of A after 7 years is equal to the amount of B after 9 years, the interest being compounded at 4% per annum.

Answer 1

Hint: Assume the part that A has as x and that of B as y. Form the first equation by taking the sum of x and y and equating it with 1301. Use compound interest formula for the calculation of amount (A) given as \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\]. Here P is the principal amount, r is the rate per annum, t is the time in years and n is the number of times the interest is given in one year. Use the values P = x, r = $\dfrac{4}{100}$, t = 7 and n = 1 for A and P = y, r = $\dfrac{4}{100}$, t = 9 and n = 1 for B. Form the second equation and find the values of x and y by solving the two equations.

Complete step by step answer:
Here we have been provided with the amount of Rs. 1301 and we are asked to divide it among A and B with certain provided conditions regarding the compound interest.
 Now, let us assume that the amount is divided in such a manner that A has Rs. x and B has Rs. y. So the sum of these parts must be equal to Rs. 1301, therefore we get,
$\Rightarrow x+y=1301........\left( i \right)$
It is given that the amount of A after 7 years must be equal to the amount of B after 9 years with the interest being compounded at 4% per annum. We know that the Amount (A) as calculated due to the compound Interest is given as \[A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}\] where P is the principal amount, r is the rate per annum, t is the time in years and n is number of times the interest is given in one year.
(1) For A we have P = x, r = $\dfrac{4}{100}$, t = 7 and since the interest is being compounded annually so we have n = 1, substituting in the formula we get,
\[\begin{align}
  & \Rightarrow {{A}_{1}}=x{{\left( 1+\dfrac{4}{100\times 1} \right)}^{1\times 7}} \\
 & \Rightarrow {{A}_{1}}=x{{\left( 1+\dfrac{1}{25} \right)}^{7}} \\
 & \Rightarrow {{A}_{1}}=x{{\left( \dfrac{26}{25} \right)}^{7}} \\
\end{align}\]
(2) For B we have P = y, r = $\dfrac{4}{100}$, t = 9 and here also the interest is being compounded annually so we have n = 1, substituting in the formula we get,
\[\begin{align}
  & \Rightarrow {{A}_{2}}=y{{\left( 1+\dfrac{4}{100\times 1} \right)}^{1\times 9}} \\
 & \Rightarrow {{A}_{2}}=y{{\left( 1+\dfrac{1}{25} \right)}^{9}} \\
 & \Rightarrow {{A}_{2}}=y{{\left( \dfrac{26}{25} \right)}^{9}} \\
\end{align}\]
Now, according to the information given in the question we must have ${{A}_{1}}={{A}_{2}}$, so we get,
\[\Rightarrow x{{\left( \dfrac{26}{25} \right)}^{7}}=y{{\left( \dfrac{26}{25} \right)}^{9}}\]
Cancelling the common factors we get,
\[\Rightarrow x=y{{\left( \dfrac{26}{25} \right)}^{2}}..........\left( ii \right)\]
Substituting the value of x in terms of y from equation (i) in equation (ii) we get,
\[\begin{align}
  & \Rightarrow 1301-y=y{{\left( \dfrac{26}{25} \right)}^{2}} \\
 & \Rightarrow y\left( {{\left( \dfrac{26}{25} \right)}^{2}}+1 \right)=1301 \\
 & \Rightarrow y\left( \dfrac{1301}{625} \right)=1301 \\
\end{align}\]
Simplifying the above relation for the value of y we get,
\[\Rightarrow y=625\]
Substituting the above obtained value of y in equation (i) we get,
\[\begin{align}
  & \Rightarrow x=1301-625 \\
 & \Rightarrow x=676 \\
\end{align}\]
Hence, the given amount is divided such that A has Rs. 676 and B has Rs. 625.

Note: Remember the formula of compound interest and all the terms used in the formula. You must be careful while substituting the value of n in the formula. If the question says that the interest is compounded half yearly then take n = 2 and for quarterly take n = 4. Always divide the given rate (in percentage) by 100 and then substitute in the formula.