To solve the given mathematical expression step by step, follow these instructions:
1. Understand the expression:
[tex]\[
\frac{\left(\frac{3}{4}+\frac{7}{8}\right)}{\left(\frac{2}{5}-\frac{8}{9}\right)}
\][/tex]
2. Calculate the numerator [tex]\(\left(\frac{3}{4} + \frac{7}{8}\right)\)[/tex]:
- Find the common denominator for fractions [tex]\(\frac{3}{4}\)[/tex] and [tex]\(\frac{7}{8}\)[/tex]. The common denominator is 8.
- Convert [tex]\(\frac{3}{4}\)[/tex] to an equivalent fraction with the denominator of 8: [tex]\(\frac{3 \times 2}{4 \times 2} = \frac{6}{8}\)[/tex].
- Add the two fractions:
[tex]\[
\frac{6}{8} + \frac{7}{8} = \frac{6 + 7}{8} = \frac{13}{8}
\][/tex]
- So, the numerator is [tex]\(\frac{13}{8} = 1.625\)[/tex].
3. Calculate the denominator [tex]\(\left(\frac{2}{5} - \frac{8}{9}\right)\)[/tex]:
- Find the common denominator for fractions [tex]\(\frac{2}{5}\)[/tex] and [tex]\(\frac{8}{9}\)[/tex]. The common denominator is 45.
- Convert [tex]\(\frac{2}{5}\)[/tex] to an equivalent fraction with the denominator of 45: [tex]\(\frac{2 \times 9}{5 \times 9} = \frac{18}{45}\)[/tex].
- Convert [tex]\(\frac{8}{9}\)[/tex] to an equivalent fraction with the denominator of 45: [tex]\(\frac{8 \times 5}{9 \times 5} = \frac{40}{45}\)[/tex].
- Subtract the two fractions:
[tex]\[
\frac{18}{45} - \frac{40}{45} = \frac{18 - 40}{45} = \frac{-22}{45}
\][/tex]
- So, the denominator is [tex]\(\frac{-22}{45} = -0.4888888888888888\)[/tex].
4. Calculate the division of the numerator by the denominator:
[tex]\[
\frac{\frac{13}{8}}{\frac{-22}{45}} = \frac{13}{8} \div \frac{-22}{45} = \frac{13}{8} \times \frac{45}{-22} = \frac{13 \times 45}{8 \times -22} = \frac{585}{-176} = -\frac{585}{176}
\][/tex]
Thus, the final result is:
[tex]\[
\boxed{-\frac{585}{176}}
\][/tex]
So, the correct answer is:
A. [tex]\(-\frac{585}{176}\)[/tex]
