Solving a Share Distribution Word Problem

This problem asks us to find the share of one person (B) when a total amount is distributed among three people (A, B, and C) with specific relationships between their shares. We are given that the total amount is Rs. 120.

Setting up the Equations for Shares

Let's represent the share of each person using variables. A good approach is to express all shares in terms of the person whose share we need to find or one which simplifies the relationships.

• Let B's share be represented by \(x\) Rupees.

Now, let's use the given relationships to express A's and C's shares in terms of \(x\):

• A's share is Rs. 20 more than B's share. So, A's share = B's share + 20 = \(x + 20\).
• A's share is Rs. 20 less than C's share. So, A's share = C's share - 20. This means C's share = A's share + 20.

Substitute the expression for A's share into the equation for C's share:

• C's share = \((x + 20) + 20 = x + 40\).

So, we have the shares expressed as:

• B's share = \(x\)
• A's share = \(x + 20\)
• C's share = \(x + 40\)

Formulating the Total Amount Equation

The total amount distributed among A, B, and C is Rs. 120. This means the sum of their shares must equal 120.

Total Amount = A's share + B's share + C's share

Using the expressions in terms of \(x\), we get the equation:

\((x + 20) + x + (x + 40) = 120\)

Solving the Equation for B's Share

Now, we need to solve the linear equation for \(x\):

\(x + 20 + x + x + 40 = 120\)

Combine the like terms (the terms with \(x\) and the constant terms):

\((x + x + x) + (20 + 40) = 120\)

\(3x + 60 = 120\)

To isolate the term with \(x\), subtract 60 from both sides of the equation:

\(3x + 60 - 60 = 120 - 60\)

\(3x = 60\)

To find the value of \(x\), divide both sides by 3:

\(\frac{3x}{3} = \frac{60}{3}\)

\(x = 20\)

Since we defined B's share as \(x\), B's share is Rs. 20.

Verifying the Shares

Let's check if these shares add up to the total and satisfy the given conditions:

• B's share = Rs. 20
• A's share = B's share + 20 = 20 + 20 = Rs. 40
• C's share = A's share + 20 = 40 + 20 = Rs. 60

Now, let's check the total:

A + B + C = 40 + 20 + 60 = 120

The total matches the given amount Rs. 120. The relationships (A is 20 more than B, A is 20 less than C) are also satisfied.

Therefore, B's share is Rs. 20.

Revision Table: Key Concepts

Additional Information: Solving Share Problems

Problems involving distributing a sum of money or items based on given conditions are common in mathematics. These often require setting up equations based on the relationships provided.

Key steps usually involve:

1. Assigning a variable to an unknown quantity, often the one you need to find or one that simplifies the relationships.
2. Writing expressions for other quantities based on the given relationships involving the variable.
3. Forming an equation using a given total or equality condition.
4. Solving the equation to find the value of the variable.
5. Calculating the values of all quantities if required and verifying the solution against the original problem statement.

Understanding how to translate words into mathematical expressions is crucial for solving such share distribution problems.