Analyze the details below and answer the question.

There are 450 employees in a company who use three (A, B and C) different communication systems. The ratio of the total number of employees who use only system A, only system B, and only system C is 5 : 7 : 3 respectively. Employees who use all three systems are 50% of the employees who use only system B. The sum of the employees who use only system B and system C and those who use only system A and system B is 50. The number of employees who use only system A and system C is 30.

If the ratio of the number of employees who use only system A and system B to the number of employees who use only system B and system C is 2 : 3, then determine the ratio of the number of employees who use system A to those who use system C.

Correct Answer: Option 22:19

Let us break down the given information step-by-step to find the total number of employees using system A and system C.

Total number of employees = 450.

Let the number of employees who use only system A, only system B, and only system C be represented in the ratio 5 : 7 : 3.
So, let:
Only system A = 5x
Only system B = 7x
Only system C = 3x

Employees who use all three systems (A, B, and C) = 50% of the employees who use only system B.
$\text{All three systems}=0.5\times 7x=3.5x$

We are given that the sum of employees who use (only A and B) and (only B and C) is 50.
The ratio of employees who use (only A and B) to those who use (only B and C) is given as 2 : 3.

Dividing 50 into the ratio 2 : 3:
$\text{Employees who use only A and B}=\frac{2}{5}\times 50=20$

$\text{Employees who use only B and C}=\frac{3}{5}\times 50=30$

The number of employees who use only system A and system C is given as 30.

Now, the total number of employees is the sum of all mutually exclusive regions in the Venn diagram:
Total = (Only A) + (Only B) + (Only C) + (Only A and B) + (Only B and C) + (Only A and C) + (All three)

Substituting the values into the equation:
$450=5x+7x+3x+20+30+30+3.5x$

$450=18.5x+80$

$18.5x=370$

$x=\frac{370}{18.5}=20$

Now, we calculate the individual counts using x = 20:
Only system A = 5 × 20 = 100
Only system B = 7 × 20 = 140
Only system C = 3 × 20 = 60
All three systems = 3.5 × 20 = 70

Next, find the total number of employees who use system A:
System A = (Only A) + (Only A and B) + (Only A and C) + (All three)
$\text{System A}=100+20+30+70=220$

Find the total number of employees who use system C:
System C = (Only C) + (Only B and C) + (Only A and C) + (All three)
$\text{System C}=60+30+30+70=190$

Finally, find the ratio of employees who use system A to those who use system C:
$\text{Ratio}=\frac{220}{190}=\frac{22}{19}=22:19$