The monthly wage per labourer in a factory increases in the ratio 22 : 25, while the total number of labourers decreases in the ratio 15 : 11. If the present total wage bill is Rs. 5000, what was the original total wage bill before these changes?

Introduction / Context:
This problem involves ratio and proportion in the context of wage bills. Wage bill equals the wage per person multiplied by the number of labourers. When both the wage rate and the number of workers change in known ratios, we can compare the original and new wage bills using multiplicative factors. The question asks us to reconstruct the original wage bill given the current wage bill and the changes in these ratios.

Given Data / Assumptions:
• Wage per labourer increases in the ratio 22 : 25. • Number of labourers decreases in the ratio 15 : 11. • Present total wage bill = Rs. 5000. • No other changes in payment structure are assumed.

Concept / Approach:
If the original wage rate is W and original number of workers is N, then original bill = W * N. After the change, new wage rate = W * 25/22, and new number of workers = N * 11/15. The new wage bill is therefore original bill multiplied by (25/22) * (11/15). Once we know that this new bill equals Rs. 5000, we can invert the factor to find the original wage bill.

Step-by-Step Solution:
Step 1: Let the original wage bill be B. Step 2: New wage factor due to wage increase = 25/22. Step 3: New number of labourers factor = 11/15. Step 4: New wage bill = B * (25/22) * (11/15). Step 5: Compute the combined factor: (25 * 11) / (22 * 15) = 275 / 330. Step 6: Simplify 275 / 330 by dividing numerator and denominator by 55, or more simply by 11: 275 / 330 = 25 / 30 = 5 / 6. Step 7: So new bill = B * (5 / 6) = 5000. Step 8: Therefore B = 5000 * (6 / 5) = 5000 * 1.2 = 6000.

Verification / Alternative check:
We can verify by forward calculation. Start with original bill Rs. 6000. First apply wage increase: multiply by 25/22 gives 6000 * 25 / 22. Then apply reduction in workers: multiply that product by 11/15. Overall we multiply 6000 by 5/6, giving 6000 * 5 / 6 = 5000, which exactly matches the present wage bill given in the question.

Why Other Options Are Wrong:
• Rs. 5500 is too low; using it in place of B gives a present bill less than Rs. 5000. • Rs. 6200 and Rs. 6350 are too high; multiplying them by 5/6 produces a value greater than Rs. 5000, contradicting the data.

Common Pitfalls:
Students sometimes add or subtract the ratios instead of multiplying. Another frequent mistake is to reverse the direction of the ratios, such as treating 22 : 25 as a decrease instead of an increase. Forgetting that both the wage and the number of workers change simultaneously can also lead to an incorrect single-factor approach.

Final Answer:
Therefore, the original total wage bill of the factory was Rs. 6000.