Understanding the Relationship Between Income, Savings, and Expenditure

In personal finance, a fundamental relationship exists:

$$ \text{Income} = \text{Expenditure} + \text{Savings} $$

This means that whatever amount of money is earned (income) is either spent (expenditure) or kept aside for future use (savings).

Solving the Income, Expenditure, and Savings Problem

Let's break down the problem step-by-step to determine the percentage increase in Sudha's income.

Step 1: Assume an Initial Income

To make calculations easier, let's assume Sudha's initial income is a round number, say $100.

• Initial Income = $100

Step 2: Calculate Initial Savings and Expenditure

We are told that Sudha saves 15% of her income.

• Initial Savings = 15% of Initial Income
• Initial Savings = $\frac{15}{100} \times 100 = \text{\$}15$

Now we can find her initial expenditure using the basic relationship:

• Initial Expenditure = Initial Income − Initial Savings
• Initial Expenditure = $100 - 15 = \text{\$}85$

Step 3: Calculate New Expenditure and New Savings

The problem states that her expenditure increases by 20% and her savings increase by 60%.

Calculating New Expenditure:

• Increase in Expenditure = 20% of Initial Expenditure
• Increase in Expenditure = $\frac{20}{100} \times 85 = \frac{1}{5} \times 85 = \text{\$}17$
• New Expenditure = Initial Expenditure + Increase in Expenditure
• New Expenditure = $85 + 17 = \text{\$}102$

Calculating New Savings:

• Increase in Savings = 60% of Initial Savings
• Increase in Savings = $\frac{60}{100} \times 15 = \frac{3}{5} \times 15 = 3 \times 3 = \text{\$}9$
• New Savings = Initial Savings + Increase in Savings
• New Savings = $15 + 9 = \text{\$}24$

Step 4: Calculate New Income

Using the fundamental relationship again, the new income is the sum of the new expenditure and new savings.

• New Income = New Expenditure + New Savings
• New Income = $102 + 24 = \text{\$}126$

Step 5: Calculate the Percentage Increase in Income

The initial income was $100, and the new income is $126. The increase in income is:

• Increase in Income = New Income − Initial Income
• Increase in Income = $126 - 100 = \text{\$}26$

To find the percentage increase, we compare the increase to the initial income:

$$ \text{Percentage Increase} = \frac{\text{Increase in Income}}{\text{Initial Income}} \times 100 $$

$$ \text{Percentage Increase} = \frac{26}{100} \times 100 = 26\% $$

Therefore, Sudha's income has increased by 26%.

Conclusion on Income Increase

By starting with an assumed income and calculating the initial and new values for savings and expenditure based on the given percentages, we found that the new income is 126 when the initial income was 100. This represents a 26% increase in her income.

Revision Table: Key Calculations for Income Problem

Additional Information: Percentage Change Calculations

Percentage change is a common concept in quantitative aptitude problems. It is calculated using the formula:

$$ \text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 $$

Or, if you know the original value and the percentage change, the new value can be found as:

$$ \text{New Value} = \text{Original Value} \times \left(1 + \frac{\text{Percentage Increase}}{100}\right) $$

or

$$ \text{New Value} = \text{Original Value} \times \left(1 - \frac{\text{Percentage Decrease}}{100}\right) $$

In this problem, we used these principles to find the new expenditure and new savings amounts before determining the total new income and its overall percentage increase.