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Simplify the following expression:

[tex]\[ \frac{\left(\frac{3}{4}+\frac{7}{8}\right)}{\left(\frac{2}{5}-\frac{8}{9}\right)} \][/tex]


Sagot :

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]