Solving Income and Expenditure Ratio Problems

This problem involves the concepts of ratios, income, expenditure, and savings. We are given the ratios of the monthly incomes and expenditures of two individuals, A and B, along with their monthly savings. We need to find the difference between their monthly incomes.

Setting up Variables and Equations

Let's represent the incomes and expenditures using variables based on the given ratios:

• The ratio of the monthly incomes of A and B is 11 : 13. We can assume their monthly incomes are \(11x\) and \(13x\) rupees respectively, where \(x\) is a common multiplier.
• The ratio of their expenditures is 9 : 11. We can assume their monthly expenditures are \(9y\) and \(11y\) rupees respectively, where \(y\) is another common multiplier.

We know that Savings = Income - Expenditure. Both A and B manage to save Rs. 4,000 per month. This gives us two linear equations:

1. For A: Income of A - Expenditure of A = Savings of A
2. \($11x - 9y = 4000$\) (Equation 1)
3. For B: Income of B - Expenditure of B = Savings of B
4. \($13x - 11y = 4000$\) (Equation 2)

We now have a system of two linear equations with two variables, \(x\) and \(y\).

Solving the System of Equations

We can solve this system using the elimination method. Our goal is to find the value of \(x\) to determine the incomes. Let's eliminate \(y\).

• Multiply Equation 1 by 11:
• \($11 \times (11x - 9y) = 11 \times 4000$\)
• \($121x - 99y = 44000$\) (Equation 3)
• Multiply Equation 2 by 9:
• \($9 \times (13x - 11y) = 9 \times 4000$\)
• \($117x - 99y = 36000$\) (Equation 4)

Now, subtract Equation 4 from Equation 3:

\((121x - 99y) - (117x - 99y) = 44000 - 36000\)

\($121x - 99y - 117x + 99y = 8000$\)

\($121x - 117x = 8000$\)

\($4x = 8000$\)

\($x = \frac{8000}{4}$\)

\($x = 2000$\)

We have found the value of \(x\).

Calculating Incomes and Their Difference

Now we can find the monthly incomes of A and B using the value of \(x\):

• Monthly income of A = \(11x = 11 \times 2000 = 22000\) rupees.
• Monthly income of B = \(13x = 13 \times 2000 = 26000\) rupees.

The question asks for the difference in their incomes. The difference is:

Difference = Income of B - Income of A

Difference = \(26000 - 22000\)

Difference = \(4000\) rupees.

Thus, the difference in their monthly incomes is Rs. 4,000.

Revision Table: Key Information

Additional Information: Ratios and Financial Concepts

Understanding ratios and basic financial equations (Income - Expenditure = Savings) is crucial for solving such problems. Here's a little more detail:

• Ratio: A ratio is a comparison of two or more quantities of the same kind by division. If a ratio is \(a:b\), it means the first quantity is \(\frac{a}{b}\) times the second quantity. When solving problems, we often represent quantities in a given ratio \(a:b\) as \(ax\) and \(bx\), where \(x\) is a non-zero common factor.
• Income: The money one receives, usually periodically, in exchange for labor, goods, or services.
• Expenditure: The amount of money spent on goods and services.
• Savings: The portion of income that is not spent on current consumption. It's calculated as Income - Expenditure.
• System of Linear Equations: When you have two or more linear equations involving the same variables, it's called a system of linear equations. These can often be solved to find the values of the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and graphical methods. In this problem, we used the elimination method.

Problems involving income, expenditure, and savings ratios are common in competitive exams and test one's ability to translate word problems into mathematical equations and solve them.