Question

Divide Rs. 28,730 between A and B so that when their shares are lent out at 10% compound interest compounded per year, the amount that A receives in 3 years is the same as what B receives in 5 years.

Answer 1

Hint: Assume that, A invests Rs. x and B invests Rs. y. Form the equations according to the conditions given in the question. Then solve the equations to find out the value of x, y.

Complete step-by-step answer:

First let’s understand what compound interest is.
Compound interest is an interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan.
Interest can be compounded on any given frequency schedule, from continuous to daily to annually.
Here it is given that the interest compounded per year.
Now, the formula for compound interest is:
$A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}........(1)$
Where, A = final amount.
P = initial principal balance.
r = interest rate.
n = number of time interest applied per time period.
t = number of time periods.
Now let us assume that A invests Rs. $x$ and B invests Rs. $y$ .
Together they invested Rs. 28,730.
That means:
$x+y=28730........(2)$
This is our first equation.
Now, the second condition is A receives the same amount of money in 3 years as B receives in 5 years.
The interest rate is 10%. It compounds annually. That means in one year it compounds only one time.
Therefore, for A:
P = x, r = 10%, n = 1, t = 3.
Let us assume that A receives the amount ${{A}_{1}}$ .
Accordingly to the formula (1):
${{A}_{1}}=x{{\left( 1+\dfrac{10}{100} \right)}^{1\times 3}}$
$\Rightarrow {{A}_{1}}=x{{\left( 1+\dfrac{10}{100} \right)}^{3}}$
Let us assume that B receives ${{A}_{2}}$ . Therefore,
$\begin{align}
  & {{A}_{2}}=y{{\left( 1+\dfrac{10}{100} \right)}^{1\times 5}} \\
 & \Rightarrow {{A}_{2}}=y{{\left( 1+\dfrac{10}{100} \right)}^{5}} \\
\end{align}$
It is given in the question that they receive the same amount of money. Therefore,
$\begin{align}
  & {{A}_{1}}={{A}_{2}} \\
 & \Rightarrow x{{\left( 1+\dfrac{10}{100} \right)}^{3}}=y{{\left( 1+\dfrac{10}{100} \right)}^{5}} \\
\end{align}$
Divide the both side by ${{\left( 1+\dfrac{10}{100} \right)}^{3}}$
$\Rightarrow x=y\dfrac{{{\left( 1+\dfrac{10}{100} \right)}^{5}}}{{{\left( 1+\dfrac{10}{100} \right)}^{3}}}$
$\Rightarrow x=y{{\left( 1+\dfrac{10}{100} \right)}^{5-3}}$
$\begin{align}
  & \Rightarrow x=y{{\left( 1+\dfrac{10}{100} \right)}^{2}} \\
 & \Rightarrow x=y{{\left( \dfrac{110}{100} \right)}^{2}} \\
 & \Rightarrow x=y\dfrac{12100}{10000} \\
 & \Rightarrow x=\dfrac{121}{100}y \\
\end{align}$
Let us put the value of x in equation (2) to get the value of y:
$\begin{align}
  & x+y=28730 \\
 & \Rightarrow y+\dfrac{121}{100}y=28730 \\
 & \Rightarrow \dfrac{100y+121y}{100}=28730 \\
 & \Rightarrow \dfrac{221}{100}y=28730 \\
 & \Rightarrow y=28730\times \dfrac{100}{221} \\
 & \Rightarrow y=130\times 100 \\
 & \Rightarrow y=13000 \\
\end{align}$
Now we have, x+y=28730
$\begin{align}
  & \Rightarrow x+13000=28730 \\
 & \Rightarrow x=28730-13000 \\
 & \Rightarrow x=15730 \\
\end{align}$
Therefore, A invests Rs. 15,730 and B invests Rs. 13,000.

Note: We generally make mistake while putting the value of r in the following formula
$A=P{{\left( 1+\dfrac{r}{n} \right)}^{nt}}$
Remember that r is the rate of interest. It is in the percentage form. So don’t forget to divide by 100.