Calculating Expenditure to Savings Ratio from Salary Data

The problem asks for the ratio of the expenditures to the savings of a person, C, given the tax and savings as fractions of the salary.

Let's break down the components of the salary. A person's salary is typically divided into tax, savings, and expenditure.

Given information:

• Tax on salary = \(\frac{1}{4}\) of the salary
• Savings from salary = \(\frac{1}{3}\) of the salary

Let's denote the salary of C as \(S\).

Based on the given information, we can write:

• Tax amount = \(\frac{1}{4} \times S\)
• Savings amount = \(\frac{1}{3} \times S\)

The total amount deducted from the salary for tax and savings is the sum of these two amounts:

Total deductions = Tax + Savings

Total deductions = \(\frac{1}{4} S + \frac{1}{3} S\)

To add these fractions, we find a common denominator, which is 12.

Total deductions = \(\frac{3}{12} S + \frac{4}{12} S = \frac{3+4}{12} S = \frac{7}{12} S\)

The remaining part of the salary after tax and savings is the expenditure. Salary is used for Tax, Savings, and Expenditure. So, Salary = Tax + Savings + Expenditure.

Expenditure = Salary - (Tax + Savings)

Expenditure = \(S - \frac{7}{12} S\)

To subtract the fraction from \(S\), we consider \(S\) as \(\frac{12}{12} S\).

Expenditure = \(\frac{12}{12} S - \frac{7}{12} S = \frac{12-7}{12} S = \frac{5}{12} S\)

Now, we need to find the ratio of the expenditures to the savings.

Ratio = \(\frac{\text{Expenditures}}{\text{Savings}}\)

Ratio = \(\frac{\frac{5}{12} S}{\frac{1}{3} S}\)

We can cancel out \(S\) from the numerator and the denominator (assuming salary \(S > 0\)).

Ratio = \(\frac{5/12}{1/3}\)

To divide by a fraction, we multiply by its reciprocal:

Ratio = \(\frac{5}{12} \times \frac{3}{1}\)

Ratio = \(\frac{5 \times 3}{12 \times 1}\)

Ratio = \(\frac{15}{12}\)

Now, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.

Ratio = \(\frac{15 \div 3}{12 \div 3} = \frac{5}{4}\)

So, the ratio of the expenditures to the savings is 5:4.

Step-by-Step Calculation

1. Assume salary \(S\).
2. Calculate Tax: Tax = \(\frac{1}{4}S\).
3. Calculate Savings: Savings = \(\frac{1}{3}S\).
4. Calculate Total Deductions (Tax + Savings): Total Deductions = \(\frac{1}{4}S + \frac{1}{3}S = \frac{7}{12}S\).
5. Calculate Expenditure: Expenditure = Salary - Total Deductions = \(S - \frac{7}{12}S = \frac{5}{12}S\).
6. Find the ratio of Expenditure to Savings: Ratio = \(\frac{\text{Expenditure}}{\text{Savings}} = \frac{\frac{5}{12}S}{\frac{1}{3}S}\).
7. Simplify the ratio: Ratio = \(\frac{5/12}{1/3} = \frac{5}{12} \times \frac{3}{1} = \frac{15}{12} = \frac{5}{4}\).

Understanding Salary Components

It is useful to understand how income (salary) is typically allocated. A simple model is:

Salary = Tax + Savings + Expenditure

In this problem, we are given the tax and savings as fractions of the salary and asked to find the ratio of expenditure to savings. By finding the fractions for each component relative to the salary, we can determine the ratio between any two components.

Verification with an Example Salary

Let's assume a specific salary that is easily divisible by 4 and 3, for instance, 1200 units.

• Salary = 1200
• Tax = \(\frac{1}{4} \times 1200 = 300\)
• Savings = \(\frac{1}{3} \times 1200 = 400\)
• Total Deductions = Tax + Savings = 300 + 400 = 700
• Expenditure = Salary - Total Deductions = 1200 - 700 = 500

The ratio of expenditures to savings is:

Ratio = \(\frac{\text{Expenditure}}{\text{Savings}} = \frac{500}{400} = \frac{5}{4}\)

This confirms the result obtained using fractions of \(S\).

Revision Table: Salary Components & Ratios

Additional Information: Fractional Calculations

Working with fractions is essential for solving problems like this. Remember these key points:

• Adding/Subtracting fractions: Need a common denominator. Find the Least Common Multiple (LCM) of the denominators.
• Multiplying fractions: Multiply numerators together and denominators together. \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\).
• Dividing fractions: Multiply the first fraction by the reciprocal of the second fraction. \(\frac{a/b}{c/d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}\).
• Simplifying fractions: Divide the numerator and denominator by their Greatest Common Divisor (GCD).

In this problem, we used these concepts to find the expenditure as a fraction of salary and then calculated the ratio of expenditure to savings.

The final answer is the ratio of expenditures to savings, which is 5:4.