To determine how long it takes Meghan to read \( \frac{12}{13} \) of her book, we first need to find out how long it takes her to read \( \frac{1}{13} \) of the book based on the information given.
Meghan reads \( \frac{1}{13} \) of the book in the following time:
\[ \text{Time for } \frac{1}{13} = \frac{114 \text{ hours}}{13} \]
Calculating this:
\[ \frac{114}{13} = 8.76923076923077 \approx 8.76923077 \text{ hours} \]
Now, multiply this time by 12 to find out how long it takes her to read \( \frac{12}{13} \) of the book:
\[ \text{Time for } \frac{12}{13} = 12 \times \frac{114}{13} \]
Calculating that:
\[ 12 \times \frac{114}{13} = \frac{1368}{13} \]
Next, we perform the division:
\[ \frac{1368}{13} = 105.23076923076923 \approx 105.23076923 \text{ hours} \]
We can convert \( 105.23076923 \) into a mixed number. The whole number part is 105. Now, we will find the fractional part from \( 0.23076923 \).
To express \( 0.23076923 \) as a fraction, observe it closely. The decimal \( 0.230769230769 \) repeats, which means it can be converted to a fraction.
Let \( x = 0.230769230769 \).
Multiply both sides by 1000 (since the repeating part has 6 digits):
\[ 1000x = 230.76923076923077 \]
Subtract the original \( x \):
\[ 1000x - x = 230.76923076923077 - 0.230769230769 \]
This simplifies to:
\[ 999x = 230.5384615384615 \] \[ x = \frac{230.5384615384615}{999} \] This continues to approximate the fractional part until simplified. However, we only need the repeating sequence \( \frac{12}{52} = \frac{3}{13} \).
Thus, we express \( 0.23076923076923 \) as \( \frac{3}{13} \).
Now, combining our whole number and fractional part, we have:
\[ \text{Time for } \frac{12}{13} = 105 + \frac{3}{13} = \frac{105 \times 13 + 3}{13} = \frac{1365 + 3}{13} = \frac{1368}{13} \]
The simplified time is \( 105 \frac{3}{13} \).
Therefore, the final answer is:
\[ \boxed{105 \frac{3}{13}} \]