A company reduces the number of its employees in the ratio 14 : 12 and increases their individual wages in the ratio 16 : 18. Determine whether the total wage bill increases or decreases, and in what ratio the new wage bill compares to the original wage bill.

Introduction / Context:
In this aptitude question on ratio and proportion, we analyse how simultaneous changes in the number of employees and their individual wages affect the total wage bill of a company. The core idea is that the overall wage bill equals the number of employees multiplied by the wage per employee, so any percentage or ratio change in these components multiplies together.

Given Data / Assumptions:

• The number of employees is reduced in the ratio 14 : 12 (original : new).
• The individual wages are increased in the ratio 16 : 18 (original : new).
• We assume the original total wage bill is some convenient value, for example 1 unit, to compare old and new situations.
• We need to decide whether the wage bill increases or decreases and find the ratio new : old.

Concept / Approach:
The total wage bill is proportional to (number of employees) * (wage per employee). When data is given in ratios, we treat them as multiplicative factors. If the number of employees changes from 14 to 12, the factor is 12/14. If wages change from 16 to 18, the factor is 18/16. The combined effect is the product of these two factors. If the product is less than 1, the wage bill decreases; if greater than 1, it increases.

Step-by-Step Solution:
Let original number of employees be 14 units and original wage per employee be 16 units. Then original total wage bill = 14 * 16. New number of employees = 12 units (from the ratio 14 : 12). New wage per employee = 18 units (from the ratio 16 : 18). New total wage bill = 12 * 18. Ratio of new wage bill to old wage bill = (12 * 18) / (14 * 16). Simplify: (12 * 18) = 216 and (14 * 16) = 224. So ratio = 216 / 224 = 27 / 28. Since 27 / 28 is less than 1, the wage bill has decreased.

Verification / Alternative check:
We can also think in percentage terms. The number of employees moves from 14 to 12, which is a factor of 12 / 14 = 6 / 7, approximately 0.857. The wages move from 16 to 18, which is 18 / 16 = 9 / 8, equal to 1.125. The combined factor is (6 / 7) * (9 / 8) = 54 / 56 = 27 / 28, which confirms the earlier calculation and again shows that the wage bill reduces slightly, because the reduction due to fewer employees dominates the increase due to higher wages.

Why Other Options Are Wrong:
“Increases, 28 : 27” and “Increases, 27 : 28” are incorrect because the product of the two changes is less than 1, not greater. “Decreases, 28 : 27” is wrong because it reverses the correct ratio; 28 : 27 would mean the new bill is larger than the old one. “No change in total wage bill” is incorrect because the combined factor is not equal to 1.

Common Pitfalls:
Students often mistakenly add or subtract percentage or ratio changes instead of multiplying them. Here, one might wrongly think that a reduction and an increase of similar size cancel out. Another error is to treat 14 : 12 and 16 : 18 as percentage changes directly, rather than as multiplicative ratios. Careful use of ratio multiplication avoids these mistakes.

Final Answer:
The total wage bill decreases, and the ratio of the new wage bill to the old wage bill is 27 : 28.