Last week the average (arithmetic mean) of the daily costs of John's lunch on Monday, Tuesday, and Wednesday was $6....

Understanding the Question

Let's clarify what we're looking for. The question asks: What was the median of the daily costs of John's lunch for the 3 days?

Given Information

• The average lunch cost for Monday, Tuesday, and Wednesday was \(\$6\)
• Since average = sum ÷ count, we have: \(\mathrm{(Mon + Tue + Wed) ÷ 3 = \$6}\)
• Therefore: \(\mathrm{Mon + Tue + Wed = \$18}\)

What We Need to Determine

We need to find the median—the middle value when the three costs are arranged in order. For sufficiency, we need to determine one unique value for the median.

Key Insight

The median of three numbers depends on their relative ordering. Since the three values sum to \(\$18\), finding the median requires either knowing all three values or having enough information to determine which value sits in the middle position.

Analyzing Statement 1

Statement 1 tells us: The cost of Monday's lunch was \(\$5\).

With \(\mathrm{Monday = \$5}\) and the total = \(\$18\), we can determine that \(\mathrm{Tuesday + Wednesday = \$13}\).

Testing Different Scenarios

Let's check if different combinations of Tuesday and Wednesday (that sum to \(\$13\)) give us the same median:

• Scenario 1: \(\mathrm{Tuesday = \$6, Wednesday = \$7}\)
  Ordered: \(\$5, \$6, \$7\) → Median = \(\$6\)
• Scenario 2: \(\mathrm{Tuesday = \$4, Wednesday = \$9}\)
  Ordered: \(\$4, \$5, \$9\) → Median = \(\$5\)

Since we get different medians (\(\$6\) vs \(\$5\)), we cannot determine a unique median value.

Conclusion

Statement 1 alone is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: The cost of Tuesday's lunch was \(\$6\).

With \(\mathrm{Tuesday = \$6}\) and the total = \(\$18\), we know that \(\mathrm{Monday + Wednesday = \$12}\).

The Critical Insight

Here's the key: Tuesday's cost (\(\$6\)) exactly equals the average of all three days. This creates a special mathematical relationship.

Since \(\mathrm{Monday + Wednesday = \$12}\), their average is also \(\$6\). This means:

• If Monday < \(\$6\), then Wednesday must be > \(\$6\) by the same amount (to maintain the sum of \(\$12\))
• If Monday > \(\$6\), then Wednesday must be < \(\$6\) by the same amount
• If Monday = \(\$6\), then Wednesday must also = \(\$6\)

Verifying with Examples

Let's confirm this pattern:

• If \(\mathrm{Monday = \$5}\), then \(\mathrm{Wednesday = \$7}\) → Ordered: \(\$5, \$6, \$7\) → Median = \(\$6\)
• If \(\mathrm{Monday = \$4}\), then \(\mathrm{Wednesday = \$8}\) → Ordered: \(\$4, \$6, \$8\) → Median = \(\$6\)
• If \(\mathrm{Monday = \$3}\), then \(\mathrm{Wednesday = \$9}\) → Ordered: \(\$3, \$6, \$9\) → Median = \(\$6\)

Notice that Tuesday (\(\$6\)) always lands in the middle position because the other two values must balance around it.

Conclusion

Statement 2 alone is sufficient — the median is \(\$6\).

[STOP - Sufficient!]

The Answer: B

Since Statement 2 alone is sufficient but Statement 1 alone is not sufficient, the answer is B.

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