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.

The correct option is 22:19.

Step 1: Understand the given data and set up variables.

Total number of employees = 450.

Let the communication systems used by employees be represented in a 3-set Venn Diagram with sets A, B, and C.

Let:

• Only A = $5x$

• Only B = $7x$

• Only C = $3x$

Employees who use all three systems (A ∩ B ∩ C) = 50% of employees who use only system B

= 50% of $7x$ = $\mathrm{3.5}x$

We are given:

• Employees who use only system A and system C (Only A & C) = 30

• Sum of employees who use (Only B & C) and (Only A & B) = 50

Step 2: Calculate the number of employees for each intersection region.

The question states that the ratio of (Only A & B) to (Only B & C) is 2 : 3.

Since the sum of these two categories is 50:

• Only A & B = $\frac{2}{2+3}\times 50=20$

• Only B & C = $\frac{3}{2+3}\times 50=30$

Step 3: Solve for $x$.

The total number of employees is the sum of all mutually exclusive regions of the Venn Diagram:

Total = (Only A) + (Only B) + (Only C) + (Only A & B) + (Only B & C) + (Only A & C) + (All three)

$450=5x+7x+3x+20+30+30+3.5x$

Combine the terms involving $x$ and constant terms:

$450=18.5x+80$

$18.5x=370$

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

Step 4: Find the total number of employees using System A and System C.

Substitute $x=20$ into the respective sets:

• Only A = $5\times 20=100$

• Only B = $7\times 20=140$

• Only C = $3\times 20=60$

• All three (A ∩ B ∩ C) = $3.5\times 20=70$

Total employees using System A:

System A = (Only A) + (Only A & B) + (Only A & C) + (All three)

$\text{System A}=100+20+30+70=220$

Total employees using System C:

System C = (Only C) + (Only B & C) + (Only A & C) + (All three)

$\text{System C}=60+30+30+70=190$

Step 5: Calculate the required ratio.

Ratio of employees using System A to System C:

$\text{Ratio}=\frac{220}{190}=\frac{22}{19}=22:19$

Thus, the required ratio is 22:19.