The ratio of male to female workers in two companies A and B is 13 : 10 and 7 : 5, respectively. If both the companies have the same number of female workers, then what is the ratio of the total number of workers in A to those in B?

Correct Option: B (23 : 24)

The problem evaluates fundamental concepts of Ratio and Proportion, a foundational topic in the UPSC Civil Services Aptitude Test (CSAT).

Given the data:

• Company A (Male : Female) = 13 : 10
• Company B (Male : Female) = 7 : 5

The core condition states that the absolute number of female workers in both companies is equal. To equate the female ratio components (10 and 5) across both companies, we must determine their Least Common Multiple (LCM), which is mathematically 10.

• Company A's ratio already uses 10 for females, so it remains unchanged at 13 : 10.
• Company B's ratio must be proportionately scaled. Multiplying both the antecedent (7) and consequent (5) by 2 yields a new equivalent ratio of 14 : 10.

Now that the units share a unified, comparable scale:

• Total workers in Company A = 13 (Males) + 10 (Females) = 23 units.
• Total workers in Company B = 14 (Males) + 10 (Females) = 24 units. Therefore, the ratio of the total number of workers in Company A to Company B is exactly 23 : 24.

Incorrect Options:

• Option A (24 : 23) is incorrect. This represents the inverse ratio (Total B : Total A), which acts as a common trap for candidates who calculate correctly but fail to align their final answer with the sequence requested in the prompt.
• Option C (18 : 17) is mathematically invalid. It serves as a distractor for those who erroneously manipulate the raw figures without first establishing a common baseline.
• Option D (27 : 18) is incorrect. It is a random distractor likely derived by arbitrarily adding the unadjusted numbers across the two equations.

Concluding Takeaway: When comparing separate mathematical ratios that share a common baseline entity, always establish a unified comparative scale by finding the LCM of that common entity before executing any additions or comparisons.