Understanding the Ratio Problem

This problem involves the concept of ratios and how they relate to income, expenditure, and savings. We are given the ratio of the incomes of two employees and the ratio of their expenditures. We also know the exact amount each employee saves per month. Our goal is to find the total of their monthly incomes.

The fundamental relationship we will use is:

Savings = Income - Expenditure

Setting Up the Equations

Let the ratio of incomes be \(7x : 4x\). So, the monthly income of the first employee is \(7x\) and the monthly income of the second employee is \(4x\).

Let the ratio of expenditures be \(3y : 1y\). So, the monthly expenditure of the first employee is \(3y\) and the monthly expenditure of the second employee is \(y\).

We are told that each employee saves ₹4,800 per month.

Using the Savings = Income - Expenditure relationship, we can write two equations based on the information for each employee:

1. For the first employee: \(7x - 3y = 4800\)
2. For the second employee: \(4x - y = 4800\)

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 of equations using either substitution or elimination method. Let's use the substitution method.

From the second equation (\(4x - y = 4800\)), we can express \(y\) in terms of \(x\):

\(y = 4x - 4800\)

Now, substitute this expression for \(y\) into the first equation (\(7x - 3y = 4800\)):

\(7x - 3(4x - 4800) = 4800\)

Distribute the -3:

\(7x - 12x + 14400 = 4800\)

Combine the \(x\) terms:

\(-5x + 14400 = 4800\)

Subtract 14400 from both sides:

\(-5x = 4800 - 14400\)

\(-5x = -9600\)

Divide by -5 to find the value of \(x\):

\(x = \frac{-9600}{-5}\)

\(x = 1920\)

Calculating the Monthly Incomes

Now that we have the value of \(x\), we can find the monthly income of each employee using the income ratio \(7x : 4x\).

• First employee's monthly income = \(7x = 7 \times 1920\)
• Second employee's monthly income = \(4x = 4 \times 1920\)

Let's calculate these values:

• First employee's monthly income = \(13440\)
• Second employee's monthly income = \(7680\)

Finding the Sum of Their Monthly Incomes

The question asks for the sum of their monthly incomes. This is the sum of the two incomes we just calculated.

Sum of incomes = First employee's income + Second employee's income

Sum of incomes = \(13440 + 7680\)

Sum of incomes = \(21120\)

Alternatively, since the incomes are \(7x\) and \(4x\), their sum is \(7x + 4x = 11x\). We found \(x = 1920\), so the sum of incomes is:

Sum of incomes = \(11 \times 1920 = 21120\)

The sum of their monthly incomes is ₹21,120.

Revision Table: Income Expenditure Savings

Additional Information: Ratio and Proportion Basics

Ratio and proportion problems often require setting up algebraic equations. Here are some key points:

• When a ratio is given, like \(a:b\), it means the quantities are \(ak\) and \(bk\) for some constant \(k\). In our problem, incomes were \(7x\) and \(4x\), and expenditures were \(3y\) and \(y\), using different constants because the ratios are independent.
• Word problems should be translated carefully into mathematical equations. Identify the unknowns and the relationships between them.
• Problems involving income, expenditure, and savings typically follow the formula: Income - Expenditure = Savings.
• Solving systems of linear equations is a fundamental skill needed for such problems. Methods include substitution and elimination.

Understanding these basic concepts of ratios and algebra is crucial for solving problems like this income and expenditure ratio question.