Understanding the Money Division Problem

The problem describes how a total amount of Rs. 820 is divided among a group of men, women, and boys. We are given specific relationships between the amounts received by individuals from each group. Our goal is to find the total amount received by all the women.

Defining Variables and Setting Up Equations

Let's represent the amount received by one person from each group:

• Let \(B\) be the amount received by one boy.
• Let \(M\) be the amount received by one man.
• Let \(W\) be the amount received by one woman.

Based on the problem statement, we can write down the relationships:

1. Total amount: The total money distributed is the sum of the amounts received by all individuals. There are 6 men, 8 women, and 12 boys.
   Total Amount $= (\text{Number of men} \times M) + (\text{Number of women} \times W) + (\text{Number of boys} \times B)$
   \(6M + 8W + 12B = 820\) (Equation 1)
2. Relationship between woman's share, man's share, and boy's share: Every woman gets an amount equal to that received by one man and one boy combined.
   \(W = M + B\) (Equation 2)
3. Relationship between man's share and boy's share: Every man gets one and a half times the amount received by a boy.
   \(M = 1.5 \times B\)
   \(M = \frac{3}{2} B\) (Equation 3)

Solving the System of Equations

Now we have a system of three equations with three variables (\(M\), \(W\), \(B\)). We can use substitution to solve for the individual amounts.

Substitute Equation 3 (\(M = \frac{3}{2} B\)) into Equation 2 (\(W = M + B\)):

\(W = \frac{3}{2} B + B\)

To add the terms on the right side, find a common denominator:

\(W = \frac{3}{2} B + \frac{2}{2} B\)

\(W = \frac{3B + 2B}{2}\)

\(W = \frac{5}{2} B\) (Equation 4)

Now we have expressions for \(M\) (Equation 3) and \(W\) (Equation 4) in terms of \(B\). Substitute these expressions into Equation 1 (\(6M + 8W + 12B = 820\)):

\(6 \left(\frac{3}{2} B\right) + 8 \left(\frac{5}{2} B\right) + 12B = 820\)

Simplify the terms:

\(\left(\frac{6 \times 3}{2}\right) B + \left(\frac{8 \times 5}{2}\right) B + 12B = 820\)

\(\left(\frac{18}{2}\right) B + \left(\frac{40}{2}\right) B + 12B = 820\)

\(9B + 20B + 12B = 820\)

Combine the terms involving \(B\):

\((9 + 20 + 12) B = 820\)

\(41B = 820\)

Now, solve for \(B\):

\(B = \frac{820}{41}\)

\(B = 20\)

So, the amount received by one boy is Rs. 20.

Calculating Individual Shares

Now that we know the value of \(B\), we can find the values of \(M\) and \(W\) using the relationships we found:

• Amount received by one man (\(M\)):
  \(M = \frac{3}{2} B = \frac{3}{2} \times 20\)
  \(M = 3 \times 10 = 30\)
  So, one man receives Rs. 30.
• Amount received by one woman (\(W\)):
  \(W = \frac{5}{2} B = \frac{5}{2} \times 20\)
  \(W = 5 \times 10 = 50\)
  So, one woman receives Rs. 50.
  Alternatively, using \(W = M + B\): \(W = 30 + 20 = 50\). This confirms our result.

Verification of Total Amount

Let's check if these individual amounts add up to the total amount of Rs. 820:

• Total amount for 6 men = \(6 \times M = 6 \times 30 = 180\)
• Total amount for 8 women = \(8 \times W = 8 \times 50 = 400\)
• Total amount for 12 boys = \(12 \times B = 12 \times 20 = 240\)

Total distributed amount = \(180 + 400 + 240 = 820\).

This matches the given total amount, confirming our calculations for individual shares are correct.

Finding the Total Amount Received by 8 Women

The question asks for the total amount received by 8 women. Since each woman receives Rs. 50, the total amount for 8 women is:

Total amount for 8 women = \(8 \times W = 8 \times 50 = 400\)

The total amount received by 8 women is Rs. 400.

Conclusion

By setting up equations based on the given information and solving them, we found the amount received by each boy, man, and woman. The total amount received by 8 women is Rs. 400.

Revision Table: Money Division Concepts

Additional Information: Proportional Distribution Problems

This problem is an example of a proportional distribution question. These problems involve dividing a total quantity among different categories based on given ratios or relationships between the categories. The key steps usually involve:

1. Identifying the quantities to be distributed and the categories among which it is divided.
2. Defining variables for the amounts received by individuals or units in each category.
3. Translating the given relationships and the total quantity into a system of equations.
4. Solving the system of equations to find the value of each variable, often by expressing all variables in terms of a single variable.
5. Calculating the specific amount requested in the question using the values of the variables.

Understanding how to set up and solve systems of linear equations is crucial for these types of problems. Sometimes, the relationships are given directly as ratios (e.g., man:boy ratio is 3:2), which can also be used to set up the equations.