Calculating Expenditure Ratio from Income and Savings

This problem involves working with ratios of income, savings, and expenditures for two individuals, A and B. We are given ratios for their incomes and savings, along with a condition relating the income of B and the savings of A. Our goal is to find the ratio of their expenditures.

Understanding Income, Savings, and Expenditure

The fundamental relationship between income, savings, and expenditure is:

Income = Expenditure + Savings

This can be rearranged to find expenditure:

Expenditure = Income - Savings

Setting up Variables Based on Ratios

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

• The ratio of incomes of A and B is 3 : 5. Let their monthly incomes be $3x$ and $5x$, where $x$ is a common positive constant.
• The ratio of their savings is 2 : 3. Let their monthly savings be $2y$ and $3y$, where $y$ is another common positive constant.

So, we have:

• Income of A = $3x$
• Income of B = $5x$
• Savings of A = $2y$
• Savings of B = $3y$

Using the Given Condition to Relate Variables

We are given that the income of B is equal to three times the savings of A. We can write this as an equation:

Income of B = 3 × Savings of A

Substitute the variable expressions:

$5x = 3 \times (2y)$

$5x = 6y$

Now, we can express one variable in terms of the other. Let's express $y$ in terms of $x$:

$y = \frac{5x}{6}$

Expressing Incomes and Savings in a Single Variable

Now that we have a relationship between $x$ and $y$, we can express all incomes and savings in terms of a single variable, $x$.

• Income of A = $3x$
• Income of B = $5x$
• Savings of A = $2y = 2 \left(\frac{5x}{6}\right) = \frac{10x}{6} = \frac{5x}{3}$
• Savings of B = $3y = 3 \left(\frac{5x}{6}\right) = \frac{15x}{6} = \frac{5x}{2}$

Calculating Expenditures

Using the formula Expenditure = Income - Savings, we can calculate the expenditure for both A and B:

Expenditure of A:

Expenditure of A = Income of A - Savings of A

Expenditure of A = $3x - \frac{5x}{3}$

To subtract, find a common denominator (which is 3):

$3x = \frac{9x}{3}$

Expenditure of A = $\frac{9x}{3} - \frac{5x}{3} = \frac{(9-5)x}{3} = \frac{4x}{3}$

Expenditure of B:

Expenditure of B = Income of B - Savings of B

Expenditure of B = $5x - \frac{5x}{2}$

To subtract, find a common denominator (which is 2):

$5x = \frac{10x}{2}$

Expenditure of B = $\frac{10x}{2} - \frac{5x}{2} = \frac{(10-5)x}{2} = \frac{5x}{2}$

Finding the Ratio of Expenditures

Now we need to find the ratio of the expenditures of A and B, which is $\frac{\text{Expenditure of A}}{\text{Expenditure of B}}$.

Ratio of Expenditures = $\frac{\frac{4x}{3}}{\frac{5x}{2}}$

To divide fractions, multiply the numerator fraction by the reciprocal of the denominator fraction:

Ratio of Expenditures = $\frac{4x}{3} \times \frac{2}{5x}$

Cancel out the $x$ term (since $x$ is a common positive constant, it's not zero):

Ratio of Expenditures = $\frac{4}{3} \times \frac{2}{5} = \frac{4 \times 2}{3 \times 5} = \frac{8}{15}$

So, the ratio of the expenditures of A and B is 8 : 15.

Let's summarize the calculated values in a table for clarity:

The ratio of Expenditures A : B is $\frac{4x}{3} : \frac{5x}{2}$.

To express this ratio with integers, we can multiply both parts by the least common multiple (LCM) of the denominators (3 and 2), which is 6.

Ratio = $\left(\frac{4x}{3} \times 6\right) : \left(\frac{5x}{2} \times 6\right)$

Ratio = $(4x \times 2) : (5x \times 3)$

Ratio = $8x : 15x$

Ratio = $8 : 15$

Final Answer

The ratio of the expenditures of A and B is 8 : 15.

Revision Table: Income, Savings, and Expenditure Ratios

Here's a quick recap of the key values and steps involved in solving this ratio problem:

Additional Information on Ratio and Proportion Problems

Ratio and proportion problems are common in quantitative aptitude tests. Understanding the relationship between different quantities, like income, savings, and expenditure, is crucial. Here are some related concepts:

• Ratio: A comparison of two quantities of the same kind, expressed as a : b or a/b.
• Proportion: An equality of two ratios.
• Percentage: Ratios can often be easily converted to percentages for comparison. Expenditure as a percentage of income, for example.
• Algebraic Methods: Using variables (like $x$ and $y$ in this solution) is a standard method to solve ratio problems with given conditions. It helps translate the word problem into equations that can be solved systematically.
• Direct and Inverse Proportion: Understanding how quantities change in relation to each other (e.g., if income increases, assuming savings are constant, expenditure must also increase) helps in setting up and solving problems.

This type of problem demonstrates how a condition linking different ratios can be used to find unknown values or ratios by setting up and solving equations.