A company has a total of 1,650 employees, some of whom are office workers. How many of the employees are...

Understanding the Question

Let's break down what we're being asked:

• A company has 1,650 total employees
• Some employees are office workers (the rest are non-office workers)
• We need to find the exact number of employees enrolled in Medical Plan X

This is a value question - we need a specific number, not just whether something is possible.

What Makes an Answer Sufficient?

We need enough information to calculate one unique value for the number of employees enrolled in Medical Plan X. If we can determine multiple possible values or can't calculate any value at all, the information is insufficient.

Key Strategic Insight

Since this question asks for a specific numerical value and the statements provide mathematical relationships, we'll work with equations. The strategic approach is to quickly count equations versus unknowns:

• One equation with two unknowns = insufficient
• Two independent equations with two unknowns = potentially sufficient

Analyzing Statement 1

Statement 1 tells us: The number of employees not enrolled in the plan is 75 more than the number of office workers enrolled in the plan.

Let's define our variables:

• \(\mathrm{E}\) = total employees enrolled in Medical Plan X (what we want to find)
• \(\mathrm{E_o}\) = office workers enrolled in the plan

Statement 1 gives us:

• Employees not enrolled = \(1,650 - \mathrm{E}\)
• This equals: \(\mathrm{E_o} + 75\)

So we have: \(1,650 - \mathrm{E} = \mathrm{E_o} + 75\)

Rearranging: \(\mathrm{E} = 1,575 - \mathrm{E_o}\)

Critical observation: This is one equation with two unknowns (\(\mathrm{E}\) and \(\mathrm{E_o}\)). We cannot determine a unique value for \(\mathrm{E}\) because:

• If \(\mathrm{E_o} = 100\), then \(\mathrm{E} = 1,475\)
• If \(\mathrm{E_o} = 500\), then \(\mathrm{E} = 1,075\)
• Multiple values are possible

Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: The number of employees enrolled in the plan is twice the number of office workers enrolled in the plan.

Using our same variables:

• \(\mathrm{E}\) = total employees enrolled
• \(\mathrm{E_o}\) = office workers enrolled

Statement 2 gives us: \(\mathrm{E} = 2\mathrm{E_o}\)

Again, this is one equation with two unknowns. For example:

• If 100 office workers are enrolled → 200 total employees enrolled
• If 500 office workers are enrolled → 1,000 total employees enrolled

Many values are possible.

Statement 2 alone is NOT sufficient.

This eliminates choice B.

Combining Statements

Now let's use both statements together:

• From Statement 1: \(\mathrm{E} = 1,575 - \mathrm{E_o}\)
• From Statement 2: \(\mathrm{E} = 2\mathrm{E_o}\)

Since both expressions equal \(\mathrm{E}\), we can set them equal:
\(2\mathrm{E_o} = 1,575 - \mathrm{E_o}\)

Solving:

• \(3\mathrm{E_o} = 1,575\)
• \(\mathrm{E_o} = 525\)

Therefore: \(\mathrm{E} = 2(525) = 1,050\)

Let's verify this makes sense:

• Employees enrolled: 1,050
• Employees not enrolled: \(1,650 - 1,050 = 600\)
• Office workers enrolled: 525
• Check Statement 1: Is \(600 = 525 + 75\)? Yes ✓
• Check Statement 2: Is \(1,050 = 2 \times 525\)? Yes ✓

With both statements together, we have two equations and two unknowns, giving us exactly one value for the number of employees enrolled in Medical Plan X: 1,050.

[STOP - Sufficient!]

Both statements together ARE sufficient.

This eliminates choice E.

The Answer: C

We need both statements working together to create a system of two equations with two unknowns, which allows us to find the unique answer.

Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."