To solve the problem, we start by determining the total cost of the case of fruit and how many good dozens remain after accounting for spoilage.
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Calculating the Total Cost: The grocer buys a case of fruit for $4.50.
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Total Dozens in the Case: The case contains 14 dozen.
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Spoilage: The grocer expects 4 dozen to spoil. Therefore, the number of good dozens left is: \[ 14 \text{ dozen} - 4 \text{ dozen} = 10 \text{ dozen} \]
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Calculating the Target Profit: The grocer wants to gain \( \frac{1}{3} \) of the total cost. Therefore, the desired profit can be calculated as follows: \[ \text{Profit} = \frac{1}{3} \times 4.50 = \frac{4.50}{3} = 1.50 \]
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Determining the Total Revenue Needed: To achieve this profit, the total revenue from selling the good fruit must be: \[ \text{Total Revenue} = \text{Total Cost} + \text{Profit} = 4.50 + 1.50 = 6.00 \]
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Calculating the Selling Price per Dozen: To find the price per dozen for the good fruit, we divide the total revenue needed by the number of good dozens: \[ \text{Price per dozen} = \frac{\text{Total Revenue}}{\text{Number of Good Dozens}} = \frac{6.00}{10} = 0.60 \]
Thus, the price per dozen that the grocer must sell the good fruit for in order to gain \( \frac{1}{3} \) of the whole cost is \(\boxed{0.60}\).
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