Yvonne purchased an item and the total cost, including tax, was less than $1. She gave the clerk a $1...

Understanding the Question

Let's break down what we're asked to find: the exact total value of change Yvonne received.

Given Information

• Yvonne bought an item costing less than $1 (including tax)
• She paid with a $1 bill
• She received exactly 4 coins as change
• Each coin must be a penny (1¢), nickel (5¢), dime (10¢), or quarter (25¢)

What We Need to Determine

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.

Key Insight

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.

Analyzing Statement 1

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:

• Scenario 1: \(1¢ + \text{three more pennies} = 4¢ \text{ total}\)
• Scenario 2: \(1¢ + 5¢ + 5¢ + 10¢ = 21¢ \text{ total}\)
• Scenario 3: \(1¢ + 10¢ + 10¢ + 25¢ = 46¢ \text{ total}\)

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.

Analyzing Statement 2

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:

• Two pennies: 2¢
• Penny + nickel: 6¢
• Two dimes: 20¢
• Dime + quarter: 35¢
• Two quarters: 50¢

(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:

• 20¢: Can be made with 2 coins \((10¢ + 10¢)\) AND with 4 coins \((5¢ + 5¢ + 5¢ + 5¢)\)
• 26¢: Can be made with 2 coins \((1¢ + 25¢)\) AND with 4 coins \((1¢ + 5¢ + 10¢ + 10¢)\)

Since multiple values satisfy both conditions, Statement 2 alone is NOT sufficient.

[STOP - Not Sufficient!]

This eliminates choice B.

Combining Statements

Now we need both constraints together:

1. The change includes at least one penny (Statement 1)
2. The change amount can be made with exactly 2 coins (Statement 2)

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:

• 20¢: The only way with 4 coins is \(5¢ + 5¢ + 5¢ + 5¢\) (no penny!) - This violates Statement 1
• 26¢: Can be made as \(1¢ + 5¢ + 10¢ + 10¢\) (includes a penny!) - This satisfies both statements

Let me verify this is the only possibility. For any 2-coin total to work with our combined constraints:

• It must be makeable with 4 coins
• Those 4 coins must include at least one penny
• The remaining 3 coins must sum to \((\text{Total} - 1¢)\)

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!]

The Answer: C

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."