Calculating Percentage Savings Increase After Income and Expenditure Changes

This problem requires us to determine the percentage growth in savings when both income and expenditure are modified. We'll break down the calculation step-by-step.

Understanding the Initial Financial Situation

Let's denote the initial income as $I$.

The problem states that the initial expenditure is 60% of the income. We can represent this as:

Initial Expenditure ($E$) = $0.60 \times I$

Savings are what remain after expenditure. So, the initial savings ($S$) is:

Initial Savings ($S$) = Initial Income ($I$) - Initial Expenditure ($E$)

Plugging in the value for $E$:

$S = I - (0.60 \times I)$

$S = (1 - 0.60) \times I$

$S = 0.40 \times I$

This means the initial savings represent 40% of the initial income.

Calculating the New Financial Figures

Next, we calculate the new income and expenditure based on the given percentage increases.

New Income ($I'$):

The income increased by 30%.

$I' = I + (30\% \text{ of } I) = I + (0.30 \times I)$

$I' = (1 + 0.30) \times I = 1.30 \times I$

New Expenditure ($E'$):

The expenditure increased by 20%.

$E' = E + (20\% \text{ of } E) = E + (0.20 \times E)$

$E' = (1 + 0.20) \times E = 1.20 \times E$

Now, let's express the new expenditure ($E'$) in terms of the initial income ($I$), using $E = 0.60 \times I$:

$E' = 1.20 \times (0.60 \times I)$

$E' = 0.72 \times I$

Determining the New Savings

The new savings ($S'$) are calculated by subtracting the new expenditure from the new income.

New Savings ($S'$) = New Income ($I'$) - New Expenditure ($E'$)

$S' = (1.30 \times I) - (0.72 \times I)$

$S' = (1.30 - 0.72) \times I$

$S' = 0.58 \times I$

Calculating the Percentage Increase in Savings

To find the percentage increase in savings, we first find the absolute difference between the new and initial savings.

Increase in Savings = $S' - S$

Increase in Savings = $(0.58 \times I) - (0.40 \times I)$

Increase in Savings = $0.18 \times I$

Now, we calculate the percentage increase relative to the original savings ($S$):

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

Percentage Increase = $ \frac{0.18 \times I}{0.40 \times I} \times 100 $

The variable $I$ cancels out:

Percentage Increase = $ \frac{0.18}{0.40} \times 100 $

Percentage Increase = $ \frac{18}{40} \times 100 $

Simplify the fraction:

Percentage Increase = $ \frac{9}{20} \times 100 $

Percentage Increase = $0.45 \times 100$

Percentage Increase = $45\%$

Summary of Changes

The calculation shows that the person's savings increased by 45%.