To find the total amount of liquid needed for the experiment, we will add the amounts of distilled water, vinegar, and liquid detergent together.
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Distilled Water: \[ 3 \frac{2}{3} = 3 + \frac{2}{3} = \frac{9}{3} + \frac{2}{3} = \frac{11}{3} \]
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Vinegar: \[ 1 \frac{3}{4} = 1 + \frac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4} \]
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Liquid Detergent: \[ \frac{2}{3} \]
Next, we need to find a common denominator to add these fractions. The denominators here are 3, 4, and 3. The least common multiple of these denominators is 12.
Now, we'll convert each quantity to have a denominator of 12:
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For Distilled Water \(\frac{11}{3}\): \[ \frac{11}{3} = \frac{11 \times 4}{3 \times 4} = \frac{44}{12} \]
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For Vinegar \(\frac{7}{4}\): \[ \frac{7}{4} = \frac{7 \times 3}{4 \times 3} = \frac{21}{12} \]
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For Liquid Detergent \(\frac{2}{3}\): \[ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \]
Now, we'll add the converted fractions: \[ \frac{44}{12} + \frac{21}{12} + \frac{8}{12} = \frac{44 + 21 + 8}{12} = \frac{73}{12} \]
Now, we can convert \(\frac{73}{12}\) into a mixed number. \[ 73 \div 12 = 6 \quad \text{(with a remainder of 1)} \] So, \(\frac{73}{12} = 6 \frac{1}{12}\).
Thus, the total amount of liquid needed is \(6 \frac{1}{12}\) cups.
Therefore, the correct answer is: \[ \boxed{6 \frac{1}{12}} \]
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