To determine the correct equation representing the situation described, we need to calculate the total cost of the oranges and potatoes, and then relate that to the $10.00 bill with the amount of change (x) Katherine should receive.
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Cost of oranges: Katherine bought \(3 \frac{1}{2}\) pounds of oranges at $1.45 per pound. \[ \text{Cost of oranges} = 3.5 \times 1.45 \]
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Cost of potatoes: She bought \(4 \frac{3}{4}\) pounds of potatoes that cost $1.69 for every 2 pounds. First, we need to find the cost per pound: \[ \text{Cost per pound for potatoes} = \frac{1.69}{2} = 0.845 \] Then, calculate the cost: \[ \text{Cost of potatoes} = 4.75 \times 0.845 \]
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Total cost of both items: \[ \text{Total cost} = (3.5 \times 1.45) + (4.75 \times 0.845) \]
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Change received: Given that Katherine pays with a $10.00 bill, we can express the situation with the equation: \[ 10 - \text{Total cost} = x \] Rearranging gives us: \[ \text{Total cost} + x = 10 \]
Now let's look at the available options:
- Option 1: \( 3 \frac{1}{2} (1.45) + 4 \frac{3}{4} (1.692) - x = 10 \) (Incorrect format, also uses incorrect cost per pound for potatoes)
- Option 2: \( 3 \frac{1}{2} (1.45) + 4 \frac{3}{4} (1.69) + x = 10 \) (This captures the relationship but adds instead of subtracting)
- Option 3: \( (3 \frac{1}{2} + 4 \frac{3}{4}) \cdot (1.45 + 1.69) + x = 10 \) (Incorrect, since it combines weights inappropriately)
- Option 4: \( 3 \frac{1}{2} (1.45) + 4 \frac{3}{4} (1.64 / 2) + x = 10 \) (This uses more incorrect values)
Since none of the statements match the correct arrangement derived, the second equation is the closest to being correct but should be rearranged.
Correct Equation (from analysis): \[ 3.5(1.45) + 4.75 \left(\frac{1.69}{2}\right) + x = 10 \]
However, it's actually better represented as: \[ 3.5(1.45) + 4.75(0.845) + x = 10 \]
If we need to pick the most appropriately structured statement based on the information given, the closest one is the second option. But do note that it needs to be rearranged or adjusted based on expressions provided above to be fully accurate:
Final Answer: \( \text{no option precisely matches, but Option 2 is closest and can be modified for accuracy.} \)