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?

The correct option is B - $23:24$.

[as per provisional answerkey]

Solution

We are given the ratios of Male (M) to Female (F) workers in two companies, A and B.
Company A (M : F) = $13:10$
Company B (M : F) = $7:5$

Step 1: Equate the number of female workers.
The problem states that both companies have the same number of female workers. To compare the total workers, we must make the "Female" part of both ratios equal.
In Company A, the female part is 10.
In Company B, the female part is 5.
The Least Common Multiple (LCM) of 10 and 5 is 10.

Step 2: Adjust the ratios.
Company A ratio remains $13:10$ (Total units = $13+10=23$).
To make the female part in Company B equal to 10, we multiply the entire ratio ($7:5$) by 2:
Company B ratio = $(7\times 2):(5\times 2)=14:10$ (Total units = $14+10=24$).

Step 3: Calculate the ratio of total workers.
Total workers in A = 23 units
Total workers in B = 24 units
Ratio of Total A : Total B = $23:24$.

Why the other options are incorrect

• Option (a) - 24 : 23: This is the inverse ratio (B : A). The question specifically asks for the ratio of A to B.
• Option (c) - 18 : 17: This result would occur if one incorrectly added the individual components of the ratios (13+5 and 10+7) without normalizing the common variable (females).
• Option (d) - 27 : 18: This ratio does not correspond to any logical derivation from the given numbers 13, 10, 7, and 5 under the constraint of equal female workers.

Key Concept

Ratio Normalization: When comparing two groups based on a common attribute, you must scale the ratios so that the common attribute is represented by the same numerical value in both sets.