The total emoluments of two persons are the same, but one gets allowances to the extent of 65% of his basic pay and the other gets allowances to the extent of 80% of his basic pay. The ratio of the basic pay of the former to the basic pay of the latter is:

The correct option is 12 : 11.

Let us understand how we arrive at this ratio step-by-step.

Let the basic pay of the first (former) person be $P_1$ and the basic pay of the second (latter) person be $P_2$.

We are given that the total emoluments for both persons are the same. Total emoluments consist of the basic pay plus the allowances.

For the first person:
Allowances = 65% of his basic pay = 0.65 × $P_1$
Total Emoluments of the first person = $P_1$ + 0.65 × $P_1$ = 1.65 × $P_1$

For the second person:
Allowances = 80% of his basic pay = 0.80 × $P_2$
Total Emoluments of the second person = $P_2$ + 0.80 × $P_2$ = 1.80 × $P_2$

Since the total emoluments of both persons are equal, we can write the equation:
$1.65\times P_1=1.80\times P_2$

To find the ratio of the basic pay of the former to the latter, we solve for $\frac{P_1}{P_2}$:
$\frac{P_1}{P_2}=\frac{1.80}{1.65}$

We simplify this fraction by multiplying the numerator and denominator by 100 to remove decimals:
$\frac{P_1}{P_2}=\frac{180}{165}$

Next, we divide both the numerator and the denominator by their greatest common divisor, which is 15:
$180÷15=12$
$165÷15=11$

Therefore, the ratio of their basic pay is:
$\frac{P_1}{P_2}=\frac{12}{11}$

This corresponds to the ratio 12 : 11.