The payment per hour for a daily-wage employee is increased by 33.33% and at the same time the working hours are reduced by 33.33%. What is the net percentage change in the employee's total income?

Introduction / Context:
This question tests the combined effect of two opposite percentage changes on total earnings. The wage rate per hour increases, but the number of hours worked decreases. Total income is the product of wage per hour and hours worked, so we must apply the changes multiplicatively, not add or subtract percentage changes directly. Such problems are classic examples of successive percentage changes in quantitative aptitude.

Given Data / Assumptions:

• Hourly wage rate is increased by 33.33% (approximately one-third).
• Working time is reduced by 33.33%.
• Total income = wage per hour * number of hours worked.
• We assume there are no other components of income.

Concept / Approach:
Let the initial hourly wage be W and the initial number of hours be H. Then initial income = W * H. After the changes, the new wage becomes W * (1 + 1 / 3) = (4 / 3) * W, and the new hours become H * (1 - 1 / 3) = (2 / 3) * H. New income is then (4 / 3) * W * (2 / 3) * H. Comparing new income with initial income allows us to find the net percentage change. This is a standard successive change problem where the combined factor is the product of individual change factors.

Step-by-Step Solution:
Step 1: Let hourly wage initially be W and hours worked be H. Step 2: Initial income = W * H. Step 3: Wage increases by 33.33%, which is approximately 1 / 3. So new wage = W * (1 + 1 / 3) = (4 / 3) * W. Step 4: Working time decreases by 33.33%, so new hours = H * (1 - 1 / 3) = (2 / 3) * H. Step 5: New income = (4 / 3) * W * (2 / 3) * H. Step 6: Simplify the product: (4 / 3) * (2 / 3) = 8 / 9. Step 7: So new income = (8 / 9) * W * H. Step 8: Therefore, new income is 8 / 9 of the old income, or approximately 0.8889 times the original. Step 9: Net percentage change in income = (new - original) / original * 100 = (8 / 9 - 1) * 100. Step 10: 8 / 9 - 1 = -1 / 9, so percentage change = (-1 / 9) * 100 ≈ -11.11%. Step 11: Hence the income decreases by about 11.11%.

Verification / Alternative check:
Take specific numbers for clarity. Suppose initially W = Rs. 30 per hour and H = 9 hours, so income = 30 * 9 = Rs. 270. After a 33.33% increase, new wage ≈ 40 (since one-third more of 30 is 10). After a 33.33% decrease, new hours ≈ 6 (one-third fewer than 9). New income = 40 * 6 = Rs. 240. The decrease in income is 270 - 240 = 30. Percentage change = 30 / 270 * 100 ≈ 11.11% decrease. This matches the earlier calculation.

Why Other Options Are Wrong:
11.11% increase would require the combined factor to be 10 / 9 instead of 8 / 9. 21.11% increase or decrease do not correspond to any natural product of the given change factors. The only correct combined effect of a one-third increase and a one-third decrease in two multiplicative components is an 11.11% decrease in their product.

Common Pitfalls:
A common error is to simply subtract 33.33% from 33.33% and conclude there is no net change, which is incorrect because the rate and time multiply, not add. Another mistake is to compute 33.33% of hours using the new wage as a base or vice versa. Each change must be applied to its own quantity, and only then should their combined effect on income be calculated.

Final Answer:
The employee's total income decreases by approximately 11.11%.