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.

The correct option is 5: 3.

Step-by-step Explanation:

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

We are given the following information:

Mean monthly compensation for men (${\overline{x}}_1$) = Rs. 20000
Mean monthly compensation for women (${\overline{x}}_2$) = Rs. 24000
Combined mean monthly compensation for all staff ($\overline{x}$) = Rs. 21500

Using the formula for the combined mean:

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

Substitute the given values into the formula:

$21500=\frac{n_1\left(20000\right)+n_2\left(24000\right)}{n_1+n_2}$

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

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

Expand the terms:

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

Rearrange the terms to group $n_1$ on one side and $n_2$ on the other side:

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

Simplify both sides:

$1500n_1=2500n_2$

Now, find the ratio of $n_1$ to $n_2$:

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

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