Correct Option

Let the basic pay of the former person be x₁ and the basic pay of the latter person be x₂.

The former person receives allowances equivalent to 65% of their basic pay. Therefore, the total emoluments for the former person are calculated as:

$= x_{1} + \frac{65}{100} x_{1} = \frac{165}{100} x_{1}$

The latter person receives allowances equivalent to 80% of their basic pay. Consequently, the total emoluments for the latter person are:

$= x_{2} + \frac{80}{100} x_{2} = \frac{180}{100} x_{2}$

As per the problem statement, the total emoluments of both persons are identical. Thus, we can equate their total emoluments:

$\frac{165}{100} x_{1} = \frac{180}{100} x_{2}$

To determine the ratio of the basic pay of the former to the basic pay of the latter ( $x_{1} : x_{2}$ ), we rearrange the equation:

$\frac{x_{1}}{x_{2}} = \frac{180}{100} \times \frac{100}{165} = \frac{12}{11}$

Therefore, the ratio of the basic pay of the former to the basic pay of the latter is 12:11.

Incorrect Options

Options (1) 16:13, (2) 5:4, and (3) 7:5 are incorrect. These ratios do not satisfy the condition that the total emoluments of both individuals are equal, given that their allowances are 65% and 80% of their respective basic pays. The accurate calculation, derived from equating the total emoluments, demonstrates that a ratio of 12:11 is necessary for the stated condition to hold true.