To find the total amount of lemonade Beatrice made, we need to add the two amounts together: \(7 \frac{2}{3}\) cups and \(10 \frac{3}{5}\) cups.
First, we can convert the mixed numbers to improper fractions.
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Convert \(7 \frac{2}{3}\): \[ 7 \frac{2}{3} = 7 \times 3 + 2 = 21 + 2 = \frac{23}{3} \]
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Convert \(10 \frac{3}{5}\): \[ 10 \frac{3}{5} = 10 \times 5 + 3 = 50 + 3 = \frac{53}{5} \]
Now, we need to add \(\frac{23}{3}\) and \(\frac{53}{5}\). To do this, we need a common denominator. The least common multiple of 3 and 5 is 15.
Now we convert each fraction to have a denominator of 15:
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For \(\frac{23}{3}\): \[ \frac{23}{3} = \frac{23 \times 5}{3 \times 5} = \frac{115}{15} \]
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For \(\frac{53}{5}\): \[ \frac{53}{5} = \frac{53 \times 3}{5 \times 3} = \frac{159}{15} \]
Now we can add the two fractions: \[ \frac{115}{15} + \frac{159}{15} = \frac{115 + 159}{15} = \frac{274}{15} \]
Next, we convert \(\frac{274}{15}\) back to a mixed number. We divide: \[ 274 \div 15 = 18 \quad \text{(whole part)} \] with a remainder of: \[ 274 - (15 \times 18) = 274 - 270 = 4 \]
So, \(\frac{274}{15} = 18 \frac{4}{15}\).
Therefore, the total amount of lemonade Beatrice made is: \[ \boxed{18 \frac{4}{15}} \text{ cups} \]
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