Machines X and Y work at their respective constant rates. How many more hours does it take machine Y, working...

Understanding the Question

We have two machines, X and Y, working at constant rates. The question asks: How many more hours does machine Y take than machine X to complete the same production order?

In other words, if machine X takes some number of hours and machine Y takes a different number of hours, what's the exact difference?

For this to be sufficient, we need to determine a specific numerical value for the time difference. A relationship like "Y takes 3 hours more than X" won't be enough - we need the actual number of hours.

Key Insight: What We Need

To find a specific time difference, we need either:

• The actual size of the production order, OR
• The actual working rates of the machines

Relationships between the machines (like "twice as fast" or "\(\frac{2}{3}\) the time") alone won't give us specific numerical answers.

Analyzing Statement 1

What Statement 1 tells us: When machines X and Y work together, they complete the order in \(\frac{2}{3}\) the time that machine X would take working alone.

This gives us a relationship between their speeds. Since the combined work takes only \(\frac{2}{3}\) of X's time, Y must be contributing something meaningful, but X must still be the faster machine.

The critical question: Does this tell us the actual time difference?

Let's test with concrete examples:

• Scenario 1: If X takes 3 hours alone, then together they take 2 hours. The time difference depends on these specific values.
• Scenario 2: If X takes 30 hours alone, then together they take 20 hours. The time difference would be completely different.

Without knowing the actual production order size or machine rates, we cannot determine the specific number of hours difference.

Statement 1 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates choices A and D.

Analyzing Statement 2

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

What Statement 2 tells us: Machine Y takes twice as long as machine X to complete the same order.

This reveals an important relationship:

• Time for X = \(\mathrm{T}\) hours
• Time for Y = \(2\mathrm{T}\) hours
• Time difference = \(2\mathrm{T} - \mathrm{T} = \mathrm{T}\) hours

So the difference equals exactly the time X takes. But here's the crucial point: we don't know what T is.

Let's test scenarios:

• If X takes 5 hours: Y takes 10 hours → difference is 5 hours
• If X takes 50 hours: Y takes 100 hours → difference is 50 hours

Different values of T give us different answers to our question.

Statement 2 is NOT sufficient. [STOP - Not Sufficient!]

This eliminates choice B (which we already eliminated with choice D).

Combining Statements

Now let's see what happens when we use both statements together.

From Statement 1: Working together takes \(\frac{2}{3}\) of X's time alone
From Statement 2: Y takes twice as long as X

These two pieces of information are consistent with each other and tell us the complete relationship between the machines' speeds and times. However, they still don't tell us the actual values.

Here's why this matters:

• If the production order is 100 units: We get one set of specific times
• If the production order is 1000 units: We get a completely different set of times

The relationships stay the same (Y always takes twice as long as X, and together they take \(\frac{2}{3}\) of X's time), but the actual time difference in hours changes based on the order size.

To illustrate: The difference will always equal X's time (from Statement 2), but without knowing what X's time actually is, we can't answer "how many hours?"

Even with both statements, we cannot determine the specific number of hours difference.

The combined statements are NOT sufficient. [STOP - Not Sufficient!]

This eliminates choice C.

The Answer: E

The statements together are not sufficient because they only provide relationships between the machines, not the actual values needed to calculate a specific time difference.

Strategic Insight for DS

When GMAT asks for a specific numerical value ("how many hours/units/dollars"), relationships alone are rarely sufficient. You need either:

1. Actual values (not just ratios or relationships), OR
2. Enough constraints to force a unique numerical answer

In this problem, we have plenty of relationships but no actual values - hence, not sufficient.

Answer Choice E: Statements (1) and (2) TOGETHER are NOT sufficient.