Question

A man wants to divide a sum of \[Rs{\text{ }}37500\] among his son and daughter aged $12$ years and $14$ years respectively in such a way that on attaining the age of $18$ years each gets the same amount at $5\% $ p.a. simple interest. The son's share is :
A. $Rs.$ $19500$
B. $Rs.$ $18000$
C. $Rs.$ $15000$
D. $Rs.$$20000$

Answer 1

Hint- To deal with this problem first we will assume the share of son as a further variable according to the statement provided we will measure the amount earned by son and daughter on turing $18$ years. We will use the simple interest formula which is mentioned in solution to calculate the amount.

Complete step-by-step answer:
Given the argument, a man wants to divide a total of \[Rs{\text{ }}37500\] between his son and daughter aged $12$ years and $14$ years respectively in such a way that at the age of $18$ years each receives the same amount at $5\% $ p.a. Simple value.
Here total amount is \[Rs{\text{ }}37500\]
Let son's share is $Rs$ $X$
so daughter's share will be $Rs$ \[37500 - X\]
As we know that simple interest formula is given as
\[S.I.{\text{ }} = {\text{ }}\dfrac{{P \times R \times T}}{{100}}\]
Where \[P\] is principal amount , \[R\] is rate in percentage and \[T\] is time in years
Now we will calculate amount received by son and daughter on turning $18$
Amount received by the son on turning \[18 = \]
Son's share \[ + \] simple interest at $5\% $
\[X + \dfrac{{(X \times 6 \times 5)}}{{100}} = \dfrac{{130X}}{{100}} = \dfrac{{13X}}{{10}}\]
Amount received by the daughter on turning \[18 = \]
Daughter's share \[ + \]simple interest at $5\% $
\[
  (37500 - X) + \dfrac{{((37500 - X) \times 5 \times 4)}}{{100}} \\
   = (37500 - X) + \dfrac{{(37500 - X) \times 20}}{{100}} \\
   = (37500 - X)\left[ {1 + \dfrac{{20}}{{100}}} \right] \\
   = (37500 - X)\left[ {\dfrac{{120}}{{100}}} \right] \\
   = \dfrac{{(37500 - X) \times 6}}{5} \\
\]
It is given that amount divided in such a way that on attaining the age of \[18\]years each gets the same amount at \[5\% \] p.a. simple interest.
So
 \[\dfrac{{13X}}{{10}} = \dfrac{{(37500 - X) \times 6}}{5}\]
Now we will simply above equation by cross multiplying , so we have
\[
  65X = (37500 - X) \times 60 \\
    \\
\]
Further solving it
\[
  65X = (37500 - X) \times 60 \\
   \Rightarrow 65X = 2250000 - 60X \\
   \Rightarrow 125X = 2250000 \\
   \Rightarrow X = \dfrac{{2250000}}{{125}} \\
   \Rightarrow X = 18,000 \\
\]
So we get the value of \[X = 18000\]
Hence son's share is $Rs.$ $18000$ So the correct answer is option B.

Note- Simple Interest(S.I.) depends on the principal amount of income while Compound Interest(C.I.) is based on the principal amount and the interest that accumulates in each time span. Compound interest is interest applied to the principal value of a loan or deposit or, in other words, return on debt.