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.

Calculate the combined number of employees who use only system C and only system A?

The correct answer is 160.

Let us break down the given information step-by-step using set theory and Venn diagram concepts for the 450 total employees using communication systems A, B, and C.

Let:

• Only system A = $a$
• Only system B = $b$
• Only system C = $c$
• Only systems A and B = $x$
• Only systems B and C = $y$
• Only systems A and C = $z$
• All three systems (A, B, and C) = $k$

Step 1: Express the given conditions in algebraic form.

1. Total number of employees:

$a+b+c+x+y+z+k=450$

2. Ratio of employees using only system A, only system B, and only system C is 5 : 7 : 3:

Let $a=5r$, $b=7r$, and $c=3r$, where $r$ is a common multiplier.

3. Employees using all three systems ($k$) are 50% of the employees using only system B ($b$):

$k=0.5\times b=0.5\times 7r=3.5r$

4. Sum of employees using only system B and C ($y$) and those using only system A and B ($x$) is 50:

$x+y=50$

5. Employees using only system A and system C ($z$) is 30:

$z=30$

Step 2: Substitute all values into the total equation to find $r$.

Substituting $a$, $b$, $c$, $(x+y)$, $z$, and $k$ into the total sum equation:

$\left(5r\right)+\left(7r\right)+\left(3r\right)+(x+y)+z+k=450$

$15r+50+30+3.5r=450$

$18.5r+80=450$

$18.5r=450-80$

$18.5r=370$

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

Step 3: Calculate the required values.

Now, calculate the number of employees using only system A ($a$) and only system C ($c$):

$a=5r=5\times 20=100$

$c=3r=3\times 20=60$

Step 4: Find the combined number of employees using only system C and only system A.

$\text{Combined number}=a+c=100+60=160$

Thus, the combined number of employees who use only system C and only system A is 160.