Let the number of male and female workers in company $A$ be $13x$ and $10x$, respectively. Let the number of male and female workers in company $B$ be $7y$ and $5y$, respectively.

Given that both companies have the same number of female workers, we can equate the two: $10x = 5y$ $y = 2x$

Now, let's find the total number of workers in each company. Total workers in company $A = 13x + 10x = 23x$ Total workers in company $B = 7y + 5y = 12y$

Substitute $y = 2x$ into the total for company $B$: Total workers in company $B = 12(2x) = 24x$

The ratio of the total number of workers in company $A$ to those in company $B$ is: $\frac{23x}{24x} = 23 : 24$

Alternative Method: Ratio of male to female workers in $A = 13 : 10$ Ratio of male to female workers in $B = 7 : 5$

To make the number of female workers equal in both ratios, we can multiply the ratio for company $B$ by $2$: New ratio for company $B = (7 \times 2) : (5 \times 2) = 14 : 10$

Now, the number of female workers is $10$ units in both companies. Total workers in company $A = 13 + 10 = 23$ units Total workers in company $B = 14 + 10 = 24$ units

Ratio of total workers in $A$ to $B = 23 : 24$.