To find the total amount of lemonade Beatrice made, we need to add \( 7 \frac{2}{3} \) cups and \( 10 \frac{3}{5} \) cups.
First, let's convert the mixed numbers to improper fractions.
- For \( 7 \frac{2}{3} \):
\[ 7 \frac{2}{3} = 7 \times 3 + 2 = 21 + 2 = 23 \quad \text{(the denominator remains 3)} \]
So,
\[ 7 \frac{2}{3} = \frac{23}{3} \]
- For \( 10 \frac{3}{5} \):
\[ 10 \frac{3}{5} = 10 \times 5 + 3 = 50 + 3 = 53 \quad \text{(the denominator remains 5)} \]
So,
\[ 10 \frac{3}{5} = \frac{53}{5} \]
Next, we will 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 both fractions to have a denominator of 15.
- For \( \frac{23}{3} \):
\[ \frac{23}{3} \times \frac{5}{5} = \frac{115}{15} \]
- For \( \frac{53}{5} \):
\[ \frac{53}{5} \times \frac{3}{3} = \frac{159}{15} \]
Now we can add the two fractions:
\[ \frac{115}{15} + \frac{159}{15} = \frac{115 + 159}{15} = \frac{274}{15} \]
To convert \( \frac{274}{15} \) back to a mixed number, we divide 274 by 15:
\[ 274 \div 15 = 18 \quad \text{(whole number)} \quad \text{and a remainder of } 4. \]
Thus,
\[ \frac{274}{15} = 18 \frac{4}{15} \]
Therefore, the total amount of lemonade Beatrice made is
\[ \boxed{18 \frac{4}{15}} \text{ cups}. \]
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