To solve the expression \(\frac{\left(\frac{3}{4}+\frac{7}{8}\right)}{\left(\frac{2}{5}-\frac{8}{9}\right)}\), let's break it down step-by-step.
### Step 1: Calculate the Numerator
The numerator of the expression is \(\frac{3}{4} + \frac{7}{8}\).
First, find a common denominator for \(\frac{3}{4}\) and \(\frac{7}{8}\). The least common denominator (LCD) of 4 and 8 is 8.
Convert \(\frac{3}{4}\) to an equivalent fraction with the denominator 8:
[tex]\[
\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}
\][/tex]
Now, add \(\frac{6}{8}\) and \(\frac{7}{8}\):
[tex]\[
\frac{6}{8} + \frac{7}{8} = \frac{6 + 7}{8} = \frac{13}{8}
\][/tex]
### Step 2: Calculate the Denominator
The denominator of the expression is \(\frac{2}{5} - \frac{8}{9}\).
First, find a common denominator for \(\frac{2}{5}\) and \(\frac{8}{9}\). The least common denominator (LCD) of 5 and 9 is 45.
Convert \(\frac{2}{5}\) to an equivalent fraction with the denominator 45:
[tex]\[
\frac{2}{5} = \frac{2 \times 9}{5 \times 9} = \frac{18}{45}
\][/tex]
Convert \(\frac{8}{9}\) to an equivalent fraction with the denominator 45:
[tex]\[
\frac{8}{9} = \frac{8 \times 5}{9 \times 5} = \frac{40}{45}
\][/tex]
Now, subtract \(\frac{40}{45}\) from \(\frac{18}{45}\):
[tex]\[
\frac{18}{45} - \frac{40}{45} = \frac{18 - 40}{45} = \frac{-22}{45}
\][/tex]
### Step 3: Compute the Final Fraction
We have the numerator \(\frac{13}{8}\) and the denominator \(\frac{-22}{45}\). Now, we need to divide \(\frac{13}{8}\) by \(\frac{-22}{45}\).
Dividing by a fraction is the same as multiplying by its reciprocal. Therefore:
[tex]\[
\frac{\frac{13}{8}}{\frac{-22}{45}} = \frac{13}{8} \times \frac{45}{-22} = \frac{13 \times 45}{8 \times -22} = \frac{585}{-176}
\][/tex]
Simplify the fraction (if it can be simplified further):
[tex]\[
\frac{585}{-176} = -\frac{585}{176}
\][/tex]
Upon simplifying, we get the approximate result:
[tex]\[
-\frac{585}{176} \approx -3.3238636363636367
\][/tex]
Therefore, the final value of the expression [tex]\(\frac{\left(\frac{3}{4}+\frac{7}{8}\right)}{\left(\frac{2}{5}-\frac{8}{9}\right)}\)[/tex] is [tex]\(-3.3238636363636367\)[/tex].
