How to Solve a Property Distribution Problem Using Algebraic Methods

Introduction

In many real-life scenarios, particularly in the division of inheritance or property, algebra can be a powerful tool for solving complex distribution problems. This article provides a detailed step-by-step solution to a property distribution problem, breaking down the algebraic methods used and explaining the process in a clear, accessible manner.

Problem Statement

A man's property of Rs. 19,000 is to be divided among his wife, three sons, and two daughters. The wife receives twice what each son gets, and thrice what each daughter gets. The question is: what is each daughter's share?

Solution

To solve this problem, we will use algebraic methods. Let's denote the share of each son as S and the share of each daughter as D.

Step 1: Define the Variables

We know that:

The wife's share is twice what each son gets: Wife's share 2S The wife's share is also thrice what each daughter gets: Wife's share 3D This implies that: 2S 3D From this, we can express D in terms of S: D frac{2S}{3}

Step 2: Set Up the Equation

The total property is Rs. 19,000, which is divided among the wife, three sons, and two daughters. Therefore, the total can be expressed as:

Total Wife's share 3 × Sons share 2 × Daughters share

Substituting the expressions we have:

19,000 2S 3S 2D

Step 3: Substitute and Simplify

Now, substituting D (frac{2S}{3}) into the equation:

19,000 2S 3S 2((frac{2S}{3}))

This simplifies to:

19,000 5S (frac{4S}{3})

To combine the terms, convert 5S into thirds:

5S (frac{15S}{3})

Therefore:

19,000 (frac{15S}{3}) (frac{4S}{3}) (frac{19S}{3})

Solving for S:

19,000 (frac{19S}{3}) implies 19,000 × 3 19S implies 57,000 19S implies S (frac{57,000}{19}) 3,000

Step 4: Calculate the Daughter's Share

Now that we have S 3,000, we can find D:

Since D (frac{2S}{3}):

D (frac{2 times 3,000}{3}) (frac{6,000}{3}) 2,000

Therefore, each daughter's share is Rs. 2,000.

Verification

To verify, let's check the calculations:

Each son's share Rs. 3,000

The wife's share 2S 2 × 3,000 Rs. 6,000

Daughters' share 2D 2 × 2,000 Rs. 4,000

Total 6,000 3 × 3,000 2 × 2,000 6,000 9,000 4,000 Rs. 19,000

Conclusion

By using algebra, we can systematically solve complex property distribution problems. This method not only ensures accuracy but also provides a clear understanding of the relationships between different shares. Remember, algebra is a powerful tool in solving real-world distribution scenarios.