A man divides his property, so that his son's share to his wife's and wife's shares to his daughter's are both as in the ratio $3:1$. If the daughter gets ₹$10,000$ less than the son, the value (in₹) of the whole property is :

Understanding the Property Division Problem

This problem asks us to find the total value of a man's property based on how he divides it among his son, wife, and daughter. We are given the ratios between the shares of these family members and the difference in value between the son's and daughter's shares.

Analyzing the Given Ratios

Let S be the share of the son, W be the share of the wife, and D be the share of the daughter.

We are given two ratios:

- The ratio of the son's share to his wife's share is $3:1$. This can be written as: $\frac{S}{W}=\frac{3}{1}$ This implies $S=3W$.
- The ratio of the wife's share to his daughter's share is $3:1$. This can be written as: $\frac{W}{D}=\frac{3}{1}$ This implies $W=3D$.

Relating All Shares to One Variable

To solve this problem, it's helpful to express all shares in terms of a single person's share. Let's express the son's (S) and wife's (W) shares in terms of the daughter's share (D).

- We know $W=3D$.
- Now, substitute this into the equation for the son's share ($S=3W$): $S=3\times (3D)$ $S=9D$

So, we have the shares related as follows:

- Son's share ($S$) = $9D$
- Wife's share ($W$) = $3D$
- Daughter's share ($D$) = $D$

Using the Difference in Shares to Find the Value

We are told that the daughter gets ₹$10,000$ less than the son. This gives us another equation:

$S=D+10000$

Now, substitute the expression for S in terms of D ($S=9D$) into this equation:

$9D=D+10000$

Solving for the Daughter's Share (D)

Let's solve the equation for D:

1. Subtract D from both sides: $9D-D=10000$ $8D=10000$
2. Divide by 8 to find D: $D=\frac{10000}{8}$ $D=1250$

So, the daughter's share is ₹$1,250$.

Calculating the Other Shares

Now that we know the daughter's share, we can find the shares of the wife and son:

- Wife's share ($W$): $W=3D=3\times 1250=3750$ The wife's share is ₹$3,750$.
- Son's share ($S$): $S=9D=9\times 1250=11250$ The son's share is ₹$11,250$.

Check: The difference between the son's and daughter's share is $S-D=11250-1250=10000$. This matches the condition given in the problem.

Calculating the Total Value of the Property

The total value of the property is the sum of the shares of the son, wife, and daughter:

Total Property = $S+W+D$

Total Property = $11250+3750+1250$

Total Property = $15000+1250$

Total Property = $16250$

Therefore, the value of the whole property is ₹$16,250$.