The mean monthly compensation of staff members at a corporation is Rs. 21500. The mean monthly compensation for men is Rs. 20000 and for women is Rs. 24000. Determine the ratio of the number of men to women employed at the corporation.

Correct Answer: 5: 3

To find the ratio of the number of men to women employed at the corporation, we can use the formula for the weighted average or mean.

Let $n_1$ be the number of men and $n_2$ be the number of women employed at the corporation.

Given parameters are:
Mean monthly compensation for men, ${\overline{x}}_1=\text{Rs. 20000}$
Mean monthly compensation for women, ${\overline{x}}_2=\text{Rs. 24000}$
Overall mean monthly compensation for all staff members, $\overline{x}=\text{Rs. 21500}$

The weighted mean formula is given by:

$\overline{x}=\frac{n_1{\overline{x}}_1+n_2{\overline{x}}_2}{n_1+n_2}$

Substitute the given values into the formula:

$21500=\frac{20000n_1+24000n_2}{n_1+n_2}$

Multiply both sides by $(n_1+n_2)$:

$21500(n_1+n_2)=20000n_1+24000n_2$

Expand the left side:

$21500n_1+21500n_2=20000n_1+24000n_2$

Rearrange the terms to group $n_1$ and $n_2$ on opposite sides:

$21500n_1-20000n_1=24000n_2-21500n_2$

Simplify both sides:

$1500n_1=2500n_2$

Now, calculate the ratio of men to women ($\frac{n_1}{n_2}$):

$\frac{n_1}{n_2}=\frac{2500}{1500}=\frac{5}{3}$

Thus, the ratio of the number of men to women employed at the corporation is 5: 3.