A man wants to distribute money. He gives $\frac{2}{5}$ to his wife and divides the rest equally among his three sons. However, he first gives $\frac{1}{6}$ of the initial amount to his brother and then distributes the rest as originally intended. We need to find: (i) What fraction of the initial amount did the wife receive? (ii) What fraction of the initial amount did he have remaining after giving to his brother and wife? (iii) Given a son receives 40000 rupees less than he was to receive originally, find the initial amount.

1. Problem Description

A man wants to distribute money. He gives $\frac{2}{5}$ to his wife and divides the rest equally among his three sons. However, he first gives $\frac{1}{6}$ of the initial amount to his brother and then distributes the rest as originally intended. We need to find:

(i) What fraction of the initial amount did the wife receive?

(ii) What fraction of the initial amount did he have remaining after giving to his brother and wife?

(iii) Given a son receives 40000 rupees less than he was to receive originally, find the initial amount.

2. Solution Steps

(i) The wife receives $\frac{2}{5}$ of the initial amount.

So the answer is $\frac{2}{5}$.

(ii) The man gives $\frac{1}{6}$ of the initial amount to his brother and $\frac{2}{5}$ of the initial amount to his wife.

The total fraction given away is:

$\frac{1}{6} + \frac{2}{5} = \frac{5}{30} + \frac{12}{30} = \frac{17}{30}$

Therefore, the remaining fraction is:

$1 - \frac{17}{30} = \frac{30}{30} - \frac{17}{30} = \frac{13}{30}$.

(iii) Let the initial amount be $x$.

Originally, the wife received $\frac{2}{5}x$. The remaining amount was $x - \frac{2}{5}x = \frac{3}{5}x$. Each son would receive $\frac{1}{3} \times \frac{3}{5}x = \frac{1}{5}x$.

After giving $\frac{1}{6}x$ to his brother, the remaining amount is $x - \frac{1}{6}x = \frac{5}{6}x$.

The wife receives $\frac{2}{5}x$. The amount left for the sons is $\frac{5}{6}x - \frac{2}{5}x = \frac{25}{30}x - \frac{12}{30}x = \frac{13}{30}x$.

Each son receives $\frac{1}{3} \times \frac{13}{30}x = \frac{13}{90}x$.

We are given that a son received 40000 rupees less than he was to receive originally.

Therefore, $\frac{1}{5}x - \frac{13}{90}x = 40000$.

$\frac{18}{90}x - \frac{13}{90}x = 40000$

$\frac{5}{90}x = 40000$

$\frac{1}{18}x = 40000$

$x = 40000 \times 18$

$x = 720000$

3. Final Answer

(i) $\frac{2}{5}$

(ii) $\frac{13}{30}$

(iii) 720000 rupees