A certain one-day seminar consisted of a morning session and an afternoon session. If each of the 128 people attending...

Understanding the Question

Let's break down what we're looking for: How many people attended the morning session only?

We have 128 people total, and each person attended at least one session. Think of this visually: we can split these 128 people into three non-overlapping groups:

• Morning only
• Afternoon only
• Both sessions

To find "morning only," we need to know either:

• How many attended both sessions (so we can determine what's left for the "only" groups), OR
• How many attended afternoon total (then whoever didn't attend afternoon must be morning only!)

This is a value question - we need to determine if we can find the exact number of morning-only attendees.

Analyzing Statement 1

Statement 1 tells us: 3/4 of the people attended both sessions

Let's think about this step by step. If 3/4 of 128 people attended both sessions, that's:

• \(\frac{3}{4} \times 128 = 96\) people who attended both sessions
• This leaves \(128 - 96 = 32\) people for the "only" categories combined

But here's the crucial problem - we can't determine how to split these 32 people between "morning only" and "afternoon only" without additional information.

Testing different scenarios:

• Scenario A: 20 morning-only and 12 afternoon-only (\(20 + 12 = 32\)) ✓
• Scenario B: 10 morning-only and 22 afternoon-only (\(10 + 22 = 32\)) ✓

Since we get different values for morning-only attendees (20 vs 10), Statement 1 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: 7/8 of the people attended the afternoon session

This is where the key insight emerges! Let's calculate:

• \(\frac{7}{8} \times 128 = 112\) people attended the afternoon session
• Therefore, \(128 - 112 = 16\) people did NOT attend the afternoon session

Here's the crucial logic: Since everyone attended at least one session, if someone didn't attend the afternoon session, they MUST have attended the morning session only. There's no other possibility!

So those 16 people who didn't attend afternoon are exactly our "morning only" group.

[STOP - Sufficient!] Statement 2 gives us the exact answer: 16 people attended morning only.

The Answer: B

Statement 2 alone gives us the exact number of morning-only attendees (16), while Statement 1 alone leaves us with multiple valid possibilities.

Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."