If Machine X, working alone at its constant rate, performs half of a certain task and then Machine Y, working...

Understanding the Question

Let's break down what we're being asked. We need to find how many hours it takes Machine X, working alone at its constant rate, to complete the entire task.

Given Information

• When Machine X does half the task, then Machine Y does the other half, the total time is 16 hours
• Both machines work at constant rates
• We need Machine X's time to do the whole task alone

What We Need to Determine

From the given setup, if Machine X takes X hours to do the whole task alone, it takes \(\mathrm{X/2}\) hours to do half. Similarly, if Machine Y takes Y hours to do the whole task alone, it takes \(\mathrm{Y/2}\) hours to do half. Since these times add up to 16 hours:

• \(\mathrm{X/2 + Y/2 = 16}\)
• Therefore: \(\mathrm{X + Y = 32}\)

This means the combined total time for both machines to each do the full task separately would be 32 hours.

To find a specific value for X, we need either another relationship between X and Y, or information that uniquely determines one of these values.

Analyzing Statement 1

Statement 1 tells us: When both machines work together, they complete the task in 6 hours.

This is revealing! Working together takes only 6 hours, while working sequentially (each doing half) takes 16 hours. Let's think about what this dramatic difference means.

Understanding the Time Difference

When machines work sequentially, each works for only half the task. But when they work together, both machines contribute throughout the entire task. The faster machine helps more, which is why the combined time is so much shorter.

Consider this: If both machines had equal speeds (both taking 32 hours alone), they would complete the task together in 16 hours. But they actually take only 6 hours together—much faster than expected! This tells us one machine must be significantly faster than the other.

Testing the Possibilities

Since we know \(\mathrm{X + Y = 32}\) and their combined work takes 6 hours, there are exactly two scenarios that fit:

• Scenario 1: \(\mathrm{X = 8}\) hours, \(\mathrm{Y = 24}\) hours (Machine X is three times faster)
• Scenario 2: \(\mathrm{X = 24}\) hours, \(\mathrm{Y = 8}\) hours (Machine Y is three times faster)

Both scenarios satisfy our conditions:

• Both give us \(\mathrm{X + Y = 32}\) ✓
• Both would result in very fast combined work ✓

Without additional information, we cannot determine which scenario is correct.

Conclusion

Statement 1 is NOT sufficient because we get two different possible values for X.

[STOP - Not Sufficient!] This eliminates answer choices A and D.

Analyzing Statement 2

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

Statement 2 provides: Machine X has a faster rate than Machine Y.

Having a faster rate means Machine X takes less time to complete the task (remember: faster rate = less time needed). So \(\mathrm{X < Y}\).

What This Tells Us

We know from the question that \(\mathrm{X + Y = 32}\). Combined with \(\mathrm{X < Y}\), this means:

• X must be less than 16 (the faster machine)
• Y must be greater than 16 (the slower machine)

But there are infinitely many pairs that satisfy these conditions:

• \(\mathrm{X = 10, Y = 22}\) ✓
• \(\mathrm{X = 12, Y = 20}\) ✓
• \(\mathrm{X = 15, Y = 17}\) ✓
• \(\mathrm{X = 8, Y = 24}\) ✓
• And countless others...

Conclusion

Statement 2 is NOT sufficient because many different values of X are possible.

[STOP - Not Sufficient!] This eliminates answer choice B.

Combining Statements

Now let's use both statements together.

Combined Information

From Statement 1, we determined there are exactly two possibilities:

• Either \(\mathrm{X = 8}\) and \(\mathrm{Y = 24}\)
• Or \(\mathrm{X = 24}\) and \(\mathrm{Y = 8}\)

From Statement 2, we know that \(\mathrm{X < Y}\) (Machine X is faster, so it takes less time).

Finding Our Answer

Let's check which possibility from Statement 1 satisfies Statement 2:

• If \(\mathrm{X = 8}\) and \(\mathrm{Y = 24}\), then \(\mathrm{X < Y}\) ✓ (This satisfies Statement 2)
• If \(\mathrm{X = 24}\) and \(\mathrm{Y = 8}\), then \(\mathrm{X > Y}\) ✗ (This contradicts Statement 2)

Therefore, we must have X = 8 hours.

The combination uniquely determines that Machine X takes 8 hours to complete the task alone. The statements together are sufficient.

[STOP - Sufficient!] This eliminates answer choice E.

The Answer: C

Both statements together give us exactly one value for X (8 hours), but neither statement alone is sufficient.

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

Key Takeaway

This problem demonstrates a classic GMAT pattern: when one statement gives you two symmetric possibilities (like 8,24 or 24,8), a simple comparison statement (like "X is faster") often provides just enough information to determine sufficiency.