Solving the Percentage Problem on Salary and Expenditure

The question asks us to find the new cost of living expenditure as a percentage of the new increased salary, given initial expenditure percentage, salary increments, and a percentage increase in savings.

Let's break down the problem step-by-step:

Step 1: Define Initial Conditions

Let the initial salary of the man be \(S\).

• Initial expenditure on cost of living = \(60\%\) of initial salary.
• Initial expenditure = \(0.60 \times S = 0.60S\).
• Initial savings = Initial salary - Initial expenditure.
• Initial savings = \(S - 0.60S = 0.40S\).

Step 2: Calculate the New Increased Salary

The man gets two increments in his salary: first \(30\%\), then \(60\%\) on the new salary after the first increment.

• Salary after the first increment of \(30\%\):
  New salary \(S_1 = S + 30\%\) of \(S = S + 0.30S = 1.30S\).
• Salary after the second increment of \(60\%\) on \(S_1\):
  New salary \(S_2 = S_1 + 60\%\) of \(S_1 = S_1 + 0.60S_1 = 1.60S_1\).
• Substitute the value of \(S_1\):
  \(S_2 = 1.60 \times (1.30S)\).
• Calculate the final increased salary:
  \(S_2 = (1.60 \times 1.30)S = 2.08S\).

The new increased salary is \(2.08S\).

Step 3: Calculate the New Savings

His saving increases by \(250\%\) of his initial saving.

• Increase in saving = \(250\%\) of initial saving.
• Increase in saving = \(\frac{250}{100} \times 0.40S = 2.5 \times 0.40S = 1.00S\).
• New saving = Initial saving + Increase in saving.
• New saving = \(0.40S + 1.00S = 1.40S\).

Step 4: Calculate the New Cost of Living Expenditure

The total salary is now divided between expenditure and savings. So, the new expenditure is the new salary minus the new savings.

• New expenditure = New salary - New saving.
• New expenditure = \(2.08S - 1.40S = 0.68S\).

Step 5: Calculate the New Expenditure as a Percentage of Increased Salary

We need to find what percentage the new cost of living expenditure (\(0.68S\)) is of the increased salary (\(2.08S\)).

Percentage = \(\left(\frac{\text{New expenditure}}{\text{Increased salary}}\right) \times 100\%\)

Percentage = \(\left(\frac{0.68S}{2.08S}\right) \times 100\%\)

The variable \(S\) cancels out:

Percentage = \(\left(\frac{0.68}{2.08}\right) \times 100\%\)

Percentage = \(\left(\frac{68}{208}\right) \times 100\%\)

Simplify the fraction \(\frac{68}{208}\):

• Divide both by 4: \(\frac{68 \div 4}{208 \div 4} = \frac{17}{52}\).

Percentage = \(\left(\frac{17}{52}\right) \times 100\%\)

Now, perform the calculation:

\(\frac{17}{52} \times 100 \approx 0.326923 \times 100 \approx 32.6923\%\)

The question asks for the percentage rounded off to the nearest integer, but the options are given with one decimal place. Rounding \(32.6923\%\) to one decimal place gives \(32.7\%\).

Thus, the new cost of living expenditure is approximately \(32.7\%\) of the increased salary.

Revision Table: Summary of Calculations

Additional Information on Percentage Changes

Understanding consecutive percentage changes is important in such problems.

• If a value increases by \(x\%\), the new value is the original value multiplied by \(\left(1 + \frac{x}{100}\right)\).
• In this problem, the salary first increased by \(30\%\), so the multiplier was \(1 + \frac{30}{100} = 1.30\).
• Then, it increased by \(60\%\), so the multiplier on the new value was \(1 + \frac{60}{100} = 1.60\).
• The total multiplier for consecutive increases is the product of individual multipliers. Here, the total multiplier was \(1.30 \times 1.60 = 2.08\).
• This means the final salary is \(2.08\) times the original salary, which is a \(108\%\) increase overall (\(2.08 - 1\) = \(1.08\), which is \(108\%\)).
• An increase of \(250\%\) means the value becomes \(100\% + 250\% = 350\%\) of the original value, or \(3.5\) times the original value. In this case, the initial saving (\(0.40S\)) increased by \(250\%\), so the increase amount was \(2.50 \times 0.40S = 1.00S\). The new saving is the original plus the increase: \(0.40S + 1.00S = 1.40S\), which is indeed \(3.5 \times 0.40S\).

These concepts are fundamental when dealing with percentage-based financial problems.