Calculating Percentage Increase in Income

This problem involves understanding the relationship between income, saving, and expenditure, and how changes in saving and expenditure affect income.

The fundamental relationship is:

\( \text{Income} = \text{Saving} + \text{Expenditure} \)

Initial Conditions Analysis

We are given that a person saves \(33\frac{1}{3}\)% of his income.

• \(33\frac{1}{3}\)% can be written as \(\frac{100}{3}\)%.
• As a fraction, this is \(\frac{100}{3} \div 100 = \frac{100}{300} = \frac{1}{3}\).

So, the person saves \(\frac{1}{3}\) of his income.

Let's assume an initial income to make calculations easier. A value divisible by 3 is convenient. Let the initial income be ₹ 300.

• Initial Saving = \(\frac{1}{3} \times \text{Initial Income}\)
• Initial Saving = \(\frac{1}{3} \times 300 = \) ₹ 100
• Initial Expenditure = Initial Income - Initial Saving
• Initial Expenditure = \(300 - 100 = \) ₹ 200

Calculating New Saving and Expenditure

Now, we are given information about the increase in saving and expenditure:

• Saving increases by 22%.
• Expenditure increases by 10%.

Let's calculate the new values:

• Increase in Saving = 22% of Initial Saving
• Increase in Saving = \(\frac{22}{100} \times 100 = \) ₹ 22
• New Saving = Initial Saving + Increase in Saving
• New Saving = \(100 + 22 = \) ₹ 122

• Increase in Expenditure = 10% of Initial Expenditure
• Increase in Expenditure = \(\frac{10}{100} \times 200 = \) ₹ 20
• New Expenditure = Initial Expenditure + Increase in Expenditure
• New Expenditure = \(200 + 20 = \) ₹ 220

Calculating the New Income

The new income is the sum of the new saving and the new expenditure:

• New Income = New Saving + New Expenditure
• New Income = \(122 + 220 = \) ₹ 342

Calculating the Percentage Increase in Income

The initial income was ₹ 300, and the new income is ₹ 342. The increase in income is:

• Increase in Income = New Income - Initial Income
• Increase in Income = \(342 - 300 = \) ₹ 42

The percentage increase in income is calculated using the formula:

\( \text{Percentage Increase} = \frac{\text{Increase}}{\text{Original Value}} \times 100 \% \)

• Percentage Increase in Income = \(\frac{\text{Increase in Income}}{\text{Initial Income}} \times 100 \% \)
• Percentage Increase in Income = \(\frac{42}{300} \times 100 \% \)
• Percentage Increase in Income = \(\frac{42}{3} \% \)
• Percentage Increase in Income = \(14 \% \)

Thus, the percentage increase in his income is 14%.

Revision Table: Income, Saving, and Expenditure Calculation

Additional Information on Percentage Problems

Percentage problems involving income, saving, and expenditure are common. It's helpful to remember:

• The relationship \( \text{Income} = \text{Saving} + \text{Expenditure} \) is always true.
• Assuming a base value (like 100 or 1000, or a value easily divisible by given fractions) for the initial income often simplifies calculations, especially when dealing with percentages.
• Percentage change is always calculated relative to the original value. The formula is \( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100 \% \) for an increase, or \( \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \times 100 \% \) for a decrease.
• \(33\frac{1}{3}\)% is equivalent to the fraction \(\frac{1}{3}\), \(66\frac{2}{3}\)% is \(\frac{2}{3}\), \(25\%\) is \(\frac{1}{4}\), \(50\%\) is \(\frac{1}{2}\), etc. Knowing common percentage-fraction conversions can speed up calculations.
• Always read the question carefully to identify which values are increasing or decreasing and by what percentage of their original amount.