Answer :
To solve the given expression [tex]\(\frac{\left(\frac{3}{4}+\frac{7}{8}\right)}{\left(\frac{2}{5}-\frac{8}{9}\right)}\)[/tex], we'll break it down step by step:
### Step 1: Calculate the Numerator
First, we need to compute the sum inside the numerator:
[tex]\[ \frac{3}{4} + \frac{7}{8} \][/tex]
To add these fractions, we need a common denominator. The least common multiple of 4 and 8 is 8. Rewriting [tex]\(\frac{3}{4}\)[/tex] with the denominator 8:
[tex]\[ \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \][/tex]
Now, add the two fractions:
[tex]\[ \frac{6}{8} + \frac{7}{8} = \frac{6 + 7}{8} = \frac{13}{8} \][/tex]
### Step 2: Calculate the Denominator
Next, we compute the difference in the denominator:
[tex]\[ \frac{2}{5} - \frac{8}{9} \][/tex]
To subtract these fractions, we again need a common denominator. The least common multiple of 5 and 9 is 45. Rewriting each fraction with the denominator 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 the two fractions:
[tex]\[ \frac{18}{45} - \frac{40}{45} = \frac{18 - 40}{45} = \frac{-22}{45} \][/tex]
### Step 3: Form the Fraction and Simplify
We now have the calculated numerator and denominator:
[tex]\[ \frac{\left(\frac{13}{8}\right)}{\left(\frac{-22}{45}\right)} \][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[ \frac{13}{8} \div \frac{-22}{45} = \frac{13}{8} \times \frac{45}{-22} = \frac{13 \times 45}{8 \times -22} = \frac{585}{-176} \][/tex]
[tex]\[ \frac{585}{-176} = -\frac{585}{176} \][/tex]
### Conclusion
Thus, the value of the given expression is:
[tex]\[ \frac{\left(\frac{3}{4}+\frac{7}{8}\right)}{\left(\frac{2}{5}-\frac{8}{9}\right)} = -\frac{585}{176} \][/tex]
Comparing this to the given choices:
A. [tex]\(-\frac{585}{176}\)[/tex]
B. [tex]\(-\frac{450}{192}\)[/tex]
C. [tex]\(\frac{585}{464}\)[/tex]
D. [tex]\(\frac{450}{320}\)[/tex]
The correct answer is:
A. [tex]\(-\frac{585}{176}\)[/tex]
### Step 1: Calculate the Numerator
First, we need to compute the sum inside the numerator:
[tex]\[ \frac{3}{4} + \frac{7}{8} \][/tex]
To add these fractions, we need a common denominator. The least common multiple of 4 and 8 is 8. Rewriting [tex]\(\frac{3}{4}\)[/tex] with the denominator 8:
[tex]\[ \frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \][/tex]
Now, add the two fractions:
[tex]\[ \frac{6}{8} + \frac{7}{8} = \frac{6 + 7}{8} = \frac{13}{8} \][/tex]
### Step 2: Calculate the Denominator
Next, we compute the difference in the denominator:
[tex]\[ \frac{2}{5} - \frac{8}{9} \][/tex]
To subtract these fractions, we again need a common denominator. The least common multiple of 5 and 9 is 45. Rewriting each fraction with the denominator 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 the two fractions:
[tex]\[ \frac{18}{45} - \frac{40}{45} = \frac{18 - 40}{45} = \frac{-22}{45} \][/tex]
### Step 3: Form the Fraction and Simplify
We now have the calculated numerator and denominator:
[tex]\[ \frac{\left(\frac{13}{8}\right)}{\left(\frac{-22}{45}\right)} \][/tex]
To divide by a fraction, multiply by its reciprocal:
[tex]\[ \frac{13}{8} \div \frac{-22}{45} = \frac{13}{8} \times \frac{45}{-22} = \frac{13 \times 45}{8 \times -22} = \frac{585}{-176} \][/tex]
[tex]\[ \frac{585}{-176} = -\frac{585}{176} \][/tex]
### Conclusion
Thus, the value of the given expression is:
[tex]\[ \frac{\left(\frac{3}{4}+\frac{7}{8}\right)}{\left(\frac{2}{5}-\frac{8}{9}\right)} = -\frac{585}{176} \][/tex]
Comparing this to the given choices:
A. [tex]\(-\frac{585}{176}\)[/tex]
B. [tex]\(-\frac{450}{192}\)[/tex]
C. [tex]\(\frac{585}{464}\)[/tex]
D. [tex]\(\frac{450}{320}\)[/tex]
The correct answer is:
A. [tex]\(-\frac{585}{176}\)[/tex]
