An actor read aloud and recorded a certain 420-page book on several audiotapes. Was the average recording time per page...

Understanding the Question

We have a 420-page book that was recorded on audiotapes. We need to determine whether the average recording time per page was less than 90 seconds.

This is a yes/no question - we need a definitive answer: either YES (average < 90 seconds) or NO (average ≥ 90 seconds).

What We Need to Determine

The key insight: Average recording time per page = \(\frac{\mathrm{Total\ recording\ time}}{420\ \mathrm{pages}}\)

So our question becomes: Is \(\frac{\mathrm{Total\ recording\ time}}{420} < 90\ \mathrm{seconds}\)?

Let's simplify by multiplying both sides by 420:
Total recording time < \(420 × 90 = 37,800\ \mathrm{seconds} = 630\ \mathrm{minutes}\)

Therefore, we need to determine: Is the total recording time less than 630 minutes?

To have sufficiency, we must be able to definitively place the total recording time either:

• Below 630 minutes → Answer: YES
• At or above 630 minutes → Answer: NO

Analyzing Statement 1

Statement 1: Each tape was at most 40 minutes

This tells us the maximum length of any individual tape. However, it doesn't tell us how many tapes were used. Without the number of tapes, we cannot determine the total recording time.

Let's test different scenarios:

• If 10 tapes were used: Total time ≤ \(10 × 40 = 400\ \mathrm{minutes}\) (< 630) → YES
• If 20 tapes were used: Total time ≤ \(20 × 40 = 800\ \mathrm{minutes}\) (could be > 630) → Possibly NO

Since we get different answers depending on the number of tapes, 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: The book was recorded on 15 tapes

This tells us exactly how many tapes were used, but nothing about the length of each tape. Without knowing tape lengths, we cannot determine the total recording time.

Let's test different scenarios:

• If each tape was 30 minutes: Total time = \(15 × 30 = 450\ \mathrm{minutes}\) (< 630) → YES
• If each tape was 50 minutes: Total time = \(15 × 50 = 750\ \mathrm{minutes}\) (> 630) → NO

Since we get different answers depending on tape lengths, Statement 2 is NOT sufficient.

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

Combining Statements

Now let's use both statements together:

From Statement 1: Each tape was at most 40 minutes
From Statement 2: The book was recorded on exactly 15 tapes

Together, these tell us:
Total recording time ≤ \(15 × 40 = 600\ \mathrm{minutes}\)

Since 600 minutes < 630 minutes, we can now definitively answer YES - the average recording time per page was less than 90 seconds.

The key insight: The combination gives us an upper bound (600 minutes) that's below our threshold (630 minutes). This means no matter what the actual recording time was, it must be less than 630 minutes.

[STOP - Sufficient!]

The Answer: C

Both statements together are sufficient to answer the question, but neither statement alone is sufficient.

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