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Last week the average (arithmetic mean) of the daily costs of John's lunch on Monday, Tuesday, and Wednesday was \(\$6\). What was the median of the daily costs of John's lunch for the 3 days?
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?
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.
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.
Statement 1 tells us: The cost of Monday's lunch was \(\$5\).
With \(\mathrm{Monday = \$5}\) and the total = \(\$18\), we can determine that \(\mathrm{Tuesday + Wednesday = \$13}\).
Let's check if different combinations of Tuesday and Wednesday (that sum to \(\$13\)) give us the same median:
Since we get different medians (\(\$6\) vs \(\$5\)), we cannot determine a unique median value.
Statement 1 alone is NOT sufficient.
This eliminates choices A and D.
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 \(\mathrm{Monday + Wednesday = \$12}\).
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:
Let's confirm this pattern:
Notice that Tuesday (\(\$6\)) always lands in the middle position because the other two values must balance around it.
Statement 2 alone is sufficient — the median is \(\$6\).
[STOP - Sufficient!]
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."