To find the total number of carafes of coffee sold by The Organic Cafe, we need to add the amounts of caffeinated and decaffeinated coffee sold.
The caffeinated coffee sold is \( 8 \frac{1}{12} \) carafes, and the decaffeinated coffee sold is \( 9 \frac{2}{3} \) carafes.
First, we convert these mixed numbers into improper fractions.
For the caffeinated coffee: \[ 8 \frac{1}{12} = \frac{8 \times 12 + 1}{12} = \frac{96 + 1}{12} = \frac{97}{12} \]
For the decaffeinated coffee: \[ 9 \frac{2}{3} = \frac{9 \times 3 + 2}{3} = \frac{27 + 2}{3} = \frac{29}{3} \]
Now, we need to add \( \frac{97}{12} \) and \( \frac{29}{3} \). To do this, we first find a common denominator. The least common multiple of 12 and 3 is 12.
We convert \( \frac{29}{3} \) to a fraction with a denominator of 12: \[ \frac{29}{3} = \frac{29 \times 4}{3 \times 4} = \frac{116}{12} \]
Now we can add the two fractions: \[ \frac{97}{12} + \frac{116}{12} = \frac{97 + 116}{12} = \frac{213}{12} \]
Next, we convert \( \frac{213}{12} \) to a mixed number. To do this, we divide 213 by 12: \[ 213 \div 12 = 17 \quad \text{with a remainder of } 9 \] So, \( \frac{213}{12} = 17 \frac{9}{12} \).
We can simplify \( \frac{9}{12} \) to \( \frac{3}{4} \): \[ \frac{9}{12} = \frac{3}{4} \]
Thus, the total number of carafes sold is: \[ 17 \frac{3}{4} \]
Therefore, The Organic Cafe sold a total of \( \boxed{17 \frac{3}{4}} \) carafes of coffee.
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