To solve the expression:
[tex]\[
\frac{\frac{5}{8}+\frac{3}{4}}{\frac{-2}{3}-\frac{5}{6}}
\][/tex]
we need to break it down into smaller steps, starting with computing both the numerator and the denominator separately.
### Step 1: Calculate the Numerator
The numerator of our expression is [tex]\(\frac{5}{8} + \frac{3}{4}\)[/tex].
To add these fractions, we must have a common denominator. The least common denominator (LCD) for 8 and 4 is 8. Therefore, we convert [tex]\(\frac{3}{4}\)[/tex] to [tex]\(\frac{6}{8}\)[/tex]:
[tex]\[
\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}
\][/tex]
Now we can add the fractions:
[tex]\[
\frac{5}{8} + \frac{6}{8} = \frac{5 + 6}{8} = \frac{11}{8}
\][/tex]
Thus, the numerator is:
[tex]\[
1.375 \quad \text{(as } \frac{11}{8} = 1.375\text{)}
\][/tex]
### Step 2: Calculate the Denominator
The denominator of our expression is [tex]\(\frac{-2}{3} - \frac{5}{6}\)[/tex].
To subtract these fractions, we must also have a common denominator. The least common denominator (LCD) for 3 and 6 is 6. Therefore, we convert [tex]\(\frac{-2}{3}\)[/tex] to [tex]\(\frac{-4}{6}\)[/tex]:
[tex]\[
\frac{-2}{3} = \frac{-2 \times 2}{3 \times 2} = \frac{-4}{6}
\][/tex]
Now we can subtract the fractions:
[tex]\[
\frac{-4}{6} - \frac{5}{6} = \frac{-4 - 5}{6} = \frac{-9}{6}
\][/tex]
Simplify [tex]\(\frac{-9}{6}\)[/tex]:
[tex]\[
\frac{-9}{6} = -1.5
\][/tex]
Thus, the denominator is:
[tex]\[
-1.5
\][/tex]
### Step 3: Divide the Numerator by the Denominator
Now we divide the numerator by the denominator:
[tex]\[
\frac{1.375}{-1.5} = -0.9166666666666666 \quad \text{(which can be written as } -\frac{11}{12} \text{)}
\][/tex]
### Step 4: Compare with the Given Options
Let's focus on the relevant results:
1. [tex]\( 2 \frac{1}{16} = 2.0625 \)[/tex]
2. [tex]\( \frac{11}{12} = 0.9166666666666666 \)[/tex]
3. [tex]\( -\frac{11}{12} = -0.9166666666666666 \)[/tex]
4. [tex]\( -2 \frac{1}{16} = -2.0625 \)[/tex]
From these comparisons, we can see that:
[tex]\[
\frac{\frac{5}{8}+\frac{3}{4}}{\frac{-2}{3}-\frac{5}{6}} = -0.9166666666666666
\][/tex]
This corresponds to:
[tex]\[
-\frac{11}{12}
\][/tex]
Therefore, the correct answer is:
[tex]\[
-\frac{11}{12}
\][/tex]
