To find out how many mugs Ron can fill with the hot chocolate he made, you need to divide the total amount of hot chocolate by the amount each mug holds.
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Total hot chocolate made: \[ 2 \frac{3}{4} \text{ quarts} = \frac{11}{4} \text{ quarts} \quad (\text{since } 2 = \frac{8}{4} \text{ and } \frac{8}{4} + \frac{3}{4} = \frac{11}{4}) \]
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Each mug holds: \[ 1 \frac{1}{3} \text{ quarts} = \frac{4}{3} \text{ quarts} \quad (\text{since } 1 = \frac{3}{3} \text{ and } \frac{3}{3} + \frac{1}{3} = \frac{4}{3}) \]
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Now, divide the total hot chocolate by the amount each mug holds: \[ \text{Number of mugs} = \frac{\text{Total hot chocolate}}{\text{Each mug holds}} = \frac{\frac{11}{4}}{\frac{4}{3}} \]
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To divide fractions, multiply by the reciprocal: \[ = \frac{11}{4} \times \frac{3}{4} = \frac{11 \times 3}{4 \times 4} = \frac{33}{16} \]
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Convert \(\frac{33}{16}\) to a mixed number: \[ 33 \div 16 = 2 \text{ with a remainder of } 1 \] So, \[ \frac{33}{16} = 2 \frac{1}{16} \]
Thus, Ron can fill 2 full mugs and will have some hot chocolate left over. Therefore, the total number of mugs he can fill completely is 2.