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Yvonne purchased an item and the total cost, including tax, was less than \(\$1\). She gave the clerk a \(\$1\) bill and received the correct change, which consisted of exactly 4 coins, each with a value of \(\$0.01\), \(\$0.05\), \(\$0.10\) or \(\$0.25\). What is the total value of the change that Yvonne received?
Let's break down what we're asked to find: the exact total value of change Yvonne received.
For sufficiency in this value question, we need to find exactly ONE possible value for the change. If multiple change amounts are possible, the information is NOT sufficient.
The change must be less than $1 (since the item had some cost) and must be expressible as exactly 4 coins from our limited set. This already constrains our possibilities significantly.
Statement 1 tells us: One of the 4 coins is a penny (1¢).
This means we have 1¢ plus 3 other coins from our set {1¢, 5¢, 10¢, 25¢}.
Let's test a few scenarios to see if this pins down a unique value:
We can create many different totals while satisfying the constraint of having exactly one penny. Since we get different possible values (4¢, 21¢, 46¢, and many others), Statement 1 alone is NOT sufficient.
[STOP - Not Sufficient!]
This eliminates choices A and D.
Now let's forget Statement 1 completely and analyze Statement 2 independently.
Statement 2 tells us: The change amount could have been made with exactly 2 coins.
This is a powerful constraint! The change total must be achievable with just 2 coins from our set.
Let's think about what 2-coin totals are possible:
(and several others)
But here's the catch: Yvonne actually received 4 coins. So we need amounts that can be made BOTH ways.
Testing a couple of possibilities:
Since multiple values satisfy both conditions, Statement 2 alone is NOT sufficient.
[STOP - Not Sufficient!]
This eliminates choice B.
Now we need both constraints together:
Here's where it gets interesting. From our analysis of Statement 2, we found that amounts like 20¢ and 26¢ can be made with both 2 coins and 4 coins. But now we need to check which of these can be made with 4 coins INCLUDING at least one penny.
Let's check our examples:
Let me verify this is the only possibility. For any 2-coin total to work with our combined constraints:
Most 2-coin totals either can't be made with 4 coins at all, or when they can, they don't include a penny. The mathematical constraints are so tight that only 26¢ survives.
Therefore, Yvonne received exactly 26¢ in change. Together, the statements are sufficient.
[STOP - Sufficient!]
Both statements together uniquely determine that Yvonne received 26¢ in change, but neither statement alone was sufficient.
Answer Choice C: "Both statements together are sufficient, but neither statement alone is sufficient."