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:

Let the basic pay of the first person be $B_1$ and that of the second person be $B_2$.
Total emoluments are calculated as:

$$\text{Total Emoluments}=\text{Basic Pay}+\text{Allowances}$$

* For the first person, total emoluments $=B_1+65\%\text{ of }{B}_1=1.65B_1$
* For the second person, total emoluments $=B_2+80\%\text{ of }{B}_2=1.80B_2$

Given that their total emoluments are equal:

$$1.65B_1=1.80B_2$$

We find the ratio $\frac{B_1}{B_2}$:

$$\frac{B_1}{B_2}=\frac{1.80}{1.65}=\frac{180}{165}$$

Dividing both the numerator and the denominator by 15:

$$\frac{180÷15}{165÷15}=\frac{12}{11}$$

Thus, the ratio of their basic pay is $12:11$.