A man distributes ₹ 165000 among his daughter, wife and son in such a manner that $\frac{1}{2}$ of the daughter's share, $\frac{1}{4}$ of the wife's share and $\frac{1}{5}$ of the son's share are equal. Find the daughter's share.

### Concept & Equating Fractional Shares When fractions of different shares are declared equal, the standard technique is to set them all equal to a constant $k$. This allows you to express every individual's share in terms of that single variable $k$, easily yielding the underlying ratio. $$ \text{If } \frac{A}{x} = \frac{B}{y} = \frac{C}{z}, \text{ then } A : B : C = x : y : z $$ ### Step-by-Step Solution 1. **Set up the equality:** Let the shares of the daughter, wife, and son be $D$, $W$, and $S$ respectively. According to the problem: $\frac{1}{2}D = \frac{1}{4}W = \frac{1}{5}S$ 2. **Introduce a constant $k$:** Let $\frac{D}{2} = \frac{W}{4} = \frac{S}{5} = k$ From this, we extract each share: $D = 2k$ $W = 4k$ $S = 5k$ 3. **Determine the ratio:** The ratio of their shares $D : W : S$ is $2k : 4k : 5k$, which simplifies directly to $2 : 4 : 5$. 4. **Calculate the daughter's share:** Total ratio parts = $2 + 4 + 5 = 11$. Total amount distributed = ₹ 165000. Daughter's share = $\frac{2}{11} \times 165000$ $165000 \div 11 = 15000$. Daughter's share = $2 \times 15000 = 30000$. ### Exam Strategy & Shortcut Whenever you see "$\frac{1}{a}$ of A = $\frac{1}{b}$ of B = $\frac{1}{c}$ of C", the ratio of their shares is immediately exactly $a : b : c$. You can instantly write down the ratio $2 : 4 : 5$, sum it to $11$, divide $165000$ by $11$ to get $15000$, and multiply by $2$ to get the daughter's share. The whole problem takes less than 15 seconds. ### Common Pitfall A common mistake is inverting the fractions to form the ratio, assuming the ratio is $\frac{1}{2} : \frac{1}{4} : \frac{1}{5}$. This would require finding an LCM and yields an entirely incorrect distribution. ### Final Answer Therefore, the correct answer is **₹ 30000**.