Understanding the Income, Expenditure, and Savings Ratio Problem

This question asks us to find the ratio of the savings of two individuals, X and Y, given information about the ratios of their monthly incomes and expenditures, and a specific condition relating Y's income to X's expenditure.

Setting up the Variables for Income and Expenditure

We are given the following ratios:

• Ratio of monthly income of X and Y is 5 ∶ 4.
• Ratio of monthly expenditures of X and Y is 9 ∶ 7.

Let's represent the incomes and expenditures using variables:

• Let the monthly income of X be $5a$.
• Let the monthly income of Y be $4a$. Here, $a$ is a positive constant representing the common multiple for incomes.
• Let the monthly expenditure of X be $9b$.
• Let the monthly expenditure of Y be $7b$. Here, $b$ is a positive constant representing the common multiple for expenditures.

Note that the constants $a$ and $b$ are generally different, which is why we use different letters.

Applying the Given Condition

The problem states that the income of Y is equal to the expenditure of X. We can write this as an equation using our variables:

\( \text{Income of Y} = \text{Expenditure of X} \)

\( 4a = 9b \)

From this equation, we can express one variable in terms of the other. Let's express $b$ in terms of $a$:

\( b = \frac{4a}{9} \)

This relationship between $a$ and $b$ is crucial for solving the problem.

Calculating the Savings for X and Y

Savings for any individual is calculated as Income minus Expenditure.

\( \text{Savings} = \text{Income} - \text{Expenditure} \)

Using our variables:

• Savings of X ($S_X$) = Income of X - Expenditure of X = \( 5a - 9b \)
• Savings of Y ($S_Y$) = Income of Y - Expenditure of Y = \( 4a - 7b \)

Now, we substitute the relationship \( b = \frac{4a}{9} \) into the savings equations:

Savings of X ($S_X$):

\( S_X = 5a - 9b = 5a - 9 \left( \frac{4a}{9} \right) \)

\( S_X = 5a - 4a \)

\( S_X = a \)

Savings of Y ($S_Y$):

\( S_Y = 4a - 7b = 4a - 7 \left( \frac{4a}{9} \right) \)

\( S_Y = 4a - \frac{28a}{9} \)

To subtract these terms, we find a common denominator, which is 9:

\( S_Y = \frac{4a \times 9}{9} - \frac{28a}{9} \)

\( S_Y = \frac{36a}{9} - \frac{28a}{9} \)

\( S_Y = \frac{36a - 28a}{9} \)

\( S_Y = \frac{8a}{9} \)

Finding the Ratio of Savings of X and Y

The ratio of the savings of X and Y is \( S_X : S_Y \).

We have calculated \( S_X = a \) and \( S_Y = \frac{8a}{9} \).

The ratio is \( a : \frac{8a}{9} \)

To simplify this ratio, we can multiply both parts by 9:

\( a \times 9 : \frac{8a}{9} \times 9 \)

\( 9a : 8a \)

Since $a$ is a positive constant (incomes must be positive), we can divide both parts by $a$:

\( 9 : 8 \)

Thus, the ratio of the savings of X and Y is 9 ∶ 8.

Summary of Solution Steps

1. Represent income and expenditure based on given ratios using different variables ($a$ and $b$).
2. Use the condition (Income of Y = Expenditure of X) to form an equation relating $a$ and $b$.
3. Solve the equation to express one variable in terms of the other.
4. Calculate the savings for X and Y using the formula Savings = Income - Expenditure.
5. Substitute the relationship between $a$ and $b$ into the savings expressions.
6. Simplify the savings expressions to find their values in terms of a single variable ($a$).
7. Form the ratio of the savings and simplify it to the simplest form.

Revision Table: Key Concepts in Ratio Problems

Additional Information on Solving Ratio Problems

Ratio and proportion problems are common in quantitative aptitude. They often involve setting up equations based on the given information and then solving for the required ratio or value. Here are some tips:

• Always use variables to represent the unknown quantities based on the ratios. Use different variables for different ratios if they are independent (like income ratio and expenditure ratio here).
• Carefully translate the given conditions into algebraic equations.
• Simplify the equations to find relationships between the variables.
• Substitute these relationships into the expressions you need to calculate (like savings).
• Pay attention to units if they are provided (though not in this specific problem).
• Ensure the final ratio is in its simplest form.

Understanding the concept of savings as the difference between income and expenditure is fundamental to this type of problem. The key step here was correctly using the condition "income of Y is equal to the expenditure of X" to link the 'a' and 'b' variables.