Question

A sum of Rs. \[18750\] is left by will by a father to be divided between two sons, 12 and 14 years of age, so that when they attain maturity at 18, the amount received by each of them at 5 percent simple interest will be the same. Find the sum allotted at present to each son?

Answer 1

Hint:To find the sum allocated to each son we need to equate their simple interest added by their principal or in other terms the amount gained till they reach maturity and by taking the principal as Rs. \[x\] and Rs. \[18750-x\] and the time as \[\left( 18-12 \right)\] and \[\left( 18-14 \right)\] years.

The formula for the simple interest is given as:

\[=\dfrac{P\times R\times T}{100}\]
where \[P\] is the principal amount invested, \[R\] is the rate interest, \[T\] is the time taken.

Complete step by step solution:
Let us take the investment or the money received by the first son is Rs.\[x\].
The time taken for that sum to mature is till the first son’s 18 birthday, hence, the time taken is \[\left( 18- 12 \right)\] or \[6\] years.

The investment or the money received by the second son is Rs. \[18750-x\].

The time taken for that sum to mature is till the second son’s 18 birthday, hence, the time taken is \[\left( 18-14 \right)\] or \[4\] years.

The rate of interest is the same for both the sons as \[5%\].

The amount received by the first son is the sum of simple interest plus Rs. \[x\] (Principal he invested).
\[\Rightarrow x+\dfrac{x\times 6\times 5}{100}\]

The amount received by the second son is the sum of simple interest plus Rs. \[18750+x\] (Principal he invested).

\[\Rightarrow 18750-x+\dfrac{\left( 18750-x \right)\times 4\times 5}{100}\]

Equating both the amount that will be generated when they both become \[18\] is given as:
\[\Rightarrow x+\dfrac{x\times 6\times 5}{100}=18750-x+\dfrac{\left( 18750-x \right)\times 4\times
5}{100}\]
\[\Rightarrow 2x+\dfrac{30x}{100}=18750+\dfrac{\left( 18750 \right)\times 20}{100}-\dfrac{20x}{100}\]
\[\Rightarrow \dfrac{250x}{100}=18750+3750\]
\[\Rightarrow x=\dfrac{22500\times 100}{250}\]
\[\Rightarrow x=Rs.9000\]

Therefore, if the first son has been allocated Rs. \[9000\] from his father then the second son has Rs. \[18750-9000=9750\].

Hence, the investment of the first and the second son is Rs.\[9000\] and Rs.\[9750\].

Note: Students may go wrong if they equate the simple interest instead of amount as the interest will not give the amount they invested as each year the interest is added with principal simultaneously thereby increasing the principal value. Hence, the equation of amount received after they become 18 has to be given in this question.