Sure, let's break down the expression step-by-step.
We want to solve the following expression:
[tex]\[
\frac{\left(\frac{3}{4} + \frac{7}{8}\right)}{\left(\frac{2}{5} - \frac{8}{9}\right)}
\][/tex]
Step 1: Simplify the Numerator
First, let's simplify the numerator, which is the sum of [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{7}{8}\)[/tex].
To add these fractions, we need a common denominator. The least common multiple of 4 and 8 is 8.
[tex]\[
\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}
\][/tex]
Now add [tex]\(\frac{6}{8}\)[/tex] and [tex]\(\frac{7}{8}\)[/tex]:
[tex]\[
\frac{6}{8} + \frac{7}{8} = \frac{6 + 7}{8} = \frac{13}{8}
\][/tex]
So, the simplified numerator is [tex]\(\frac{13}{8}\)[/tex].
Step 2: Simplify the Denominator
Next, let's simplify the denominator, which is the difference of [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{8}{9}\)[/tex].
To subtract these fractions, we need a common denominator. The least common multiple of 5 and 9 is 45.
[tex]\[
\frac{2}{5} = \frac{2 \times 9}{5 \times 9} = \frac{18}{45}
\][/tex]
[tex]\[
\frac{8}{9} = \frac{8 \times 5}{9 \times 5} = \frac{40}{45}
\][/tex]
Now subtract [tex]\(\frac{40}{45}\)[/tex] from [tex]\(\frac{18}{45}\)[/tex]:
[tex]\[
\frac{18}{45} - \frac{40}{45} = \frac{18 - 40}{45} = \frac{-22}{45}
\][/tex]
So, the simplified denominator is [tex]\(\frac{-22}{45}\)[/tex].
Step 3: Divide the Simplified Numerator by the Simplified Denominator
Now we have:
[tex]\[
\frac{\frac{13}{8}}{\frac{-22}{45}}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. Thus:
[tex]\[
\frac{\frac{13}{8}}{\frac{-22}{45}} = \frac{13}{8} \times \frac{45}{-22}
\][/tex]
Now multiply the numerators together and the denominators together:
[tex]\[
\frac{13 \times 45}{8 \times -22} = \frac{585}{-176}
\][/tex]
Simplifying this fraction:
[tex]\[
\frac{585}{-176} = -\frac{585}{176}
\][/tex]
When you calculate this division, you get approximately:
[tex]\[
-3.3238636363636367
\][/tex]
Thus, the final answer is:
[tex]\[
\frac{\left(\frac{3}{4} + \frac{7}{8}\right)}{\left(\frac{2}{5} - \frac{8}{9}\right)} = -3.3238636363636367
\][/tex]
