Machines P, Q, and R produce aluminum cans at their respective constant rates. Is machine P's rate greater than machine...

Understanding the Question

We need to determine whether machine P's production rate is greater than machine Q's production rate. Let's denote:

• P = machine P's rate (cans per unit time)
• Q = machine Q's rate (cans per unit time)
• R = machine R's rate (cans per unit time)

What we need to determine: Is \(\mathrm{P} > \mathrm{Q}\)?

This is a yes/no question. For a statement (or combination) to be sufficient, it must allow us to definitively answer either "Yes, \(\mathrm{P} > \mathrm{Q}\)" or "No, \(\mathrm{P} \leq \mathrm{Q}\)" in all possible cases.

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{P} > \mathrm{R}\)

This gives us the relationship between P and R, but tells us nothing about Q. Let's test different scenarios to see if we can answer whether \(\mathrm{P} > \mathrm{Q}\):

Scenario 1: Q could be the fastest

• If \(\mathrm{Q} = 100, \mathrm{P} = 50, \mathrm{R} = 30\)
• Here \(\mathrm{P} > \mathrm{R}\) ✓, but \(\mathrm{P} < \mathrm{Q}\) (answer: NO)

Scenario 2: Q could be in the middle

• If \(\mathrm{Q} = 40, \mathrm{P} = 50, \mathrm{R} = 30\)
• Here \(\mathrm{P} > \mathrm{R}\) ✓, and \(\mathrm{P} > \mathrm{Q}\) (answer: YES)

Since we get different answers to "Is \(\mathrm{P} > \mathrm{Q}\)?" in different scenarios, 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.

Statement 2 tells us: P and R working together produce 1,000 cans in half the time Q takes to produce 1,000 cans alone.

This means P and R's combined rate equals twice Q's rate: \(\mathrm{P} + \mathrm{R} = 2\mathrm{Q}\)

In other words, Q's rate is the average of P and R's rates. But without knowing the individual relationship between P and R, we can't determine if \(\mathrm{P} > \mathrm{Q}\).

Scenario 1: P could be the larger contributor

• If \(\mathrm{P} = 60\) and \(\mathrm{R} = 40\), then \(\mathrm{P} + \mathrm{R} = 100 = 2\mathrm{Q}\), so \(\mathrm{Q} = 50\)
• Here \(\mathrm{P} > \mathrm{Q}\) (answer: YES)

Scenario 2: R could be the larger contributor

• If \(\mathrm{P} = 40\) and \(\mathrm{R} = 60\), then \(\mathrm{P} + \mathrm{R} = 100 = 2\mathrm{Q}\), so \(\mathrm{Q} = 50\)
• Here \(\mathrm{P} < \mathrm{Q}\) (answer: NO)

Since we get different answers in different scenarios, Statement 2 is NOT sufficient.

[STOP - NOT Sufficient!] This eliminates choice B (and confirms D is already eliminated).

Combining Statements

Now let's use both statements together:

• From Statement 1: \(\mathrm{P} > \mathrm{R}\)
• From Statement 2: \(\mathrm{P} + \mathrm{R} = 2\mathrm{Q}\), which means \(\mathrm{Q} = \frac{\mathrm{P} + \mathrm{R}}{2}\)

Here's the key insight: Q equals the average of P and R. Since we know \(\mathrm{P} > \mathrm{R}\), we can apply a fundamental principle: when two numbers are unequal, the larger number is always greater than their average.

To understand why this is true, imagine P and R on a number line. Their average Q lies exactly in the middle between them. Since P is to the right of R (because \(\mathrm{P} > \mathrm{R}\)), P must also be to the right of their midpoint Q. Therefore, \(\mathrm{P} > \mathrm{Q}\).

Example to visualize: If \(\mathrm{P} = 70\) and \(\mathrm{R} = 30\):

• Their average \(\mathrm{Q} = \frac{70 + 30}{2} = 50\)
• Indeed, \(70 > 50\), confirming \(\mathrm{P} > \mathrm{Q}\)

The combined statements give us a definitive "Yes" answer to "Is \(\mathrm{P} > \mathrm{Q}\)?", making them sufficient.

[STOP - Sufficient!] This eliminates choice E.

The Answer: C

Both statements together are sufficient to determine that \(\mathrm{P} > \mathrm{Q}\), but neither statement alone is sufficient.

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