1. Find [tex]\frac{3}{7} + \left(\frac{-6}{11}\right) + \left(\frac{-81}{21}\right) + \left(\frac{-5}{22}\right)[/tex].

Solution:
[tex]\[
\begin{array}{l}
\frac{3}{7} + \left(\frac{-81}{21}\right) + \left(\frac{-6}{11}\right) + \left(\frac{-5}{22}\right) \\
= \left(\frac{3}{7} + \frac{-81}{21}\right) + \left(\frac{-6}{11} + \frac{-5}{22}\right) \\
= \left(\frac{9}{21} + \frac{-81}{21}\right) + \left(\frac{-12}{22} + \frac{-5}{22}\right) \\
= \left(\frac{9 - 81}{21}\right) + \left(\frac{-12 - 5}{22}\right) \\
= \frac{-72}{21} + \frac{-17}{22} \\
= \frac{-24 \times 3}{7 \times 3} + \frac{-17}{22} \\
= \frac{-24}{7} + \frac{-17}{22} \\
= \frac{-24 \times 22}{7 \times 22} + \frac{-17 \times 7}{22 \times 7} \\
= \frac{-528}{154} + \frac{-119}{154} \\
= \frac{-528 - 119}{154} \\
= \frac{-647}{154}
\end{array}
\][/tex]

Therefore, [tex]\frac{3}{7} + \left(\frac{-6}{11}\right) + \left(\frac{-81}{21}\right) + \left(\frac{-5}{22}\right) = \frac{-647}{154}[/tex].



Answer :

Let's break down the problem step-by-step to ensure clarity and accuracy.

We need to find the value of
[tex]\[ \frac{3}{7} + \left(\frac{-6}{11}\right) + \left(\frac{-81}{21}\right) + \left(\frac{-5}{22}\right). \][/tex]

First, let's look at each of these components individually:

1. [tex]\(\frac{3}{7}\)[/tex]
2. [tex]\(\frac{-6}{11}\)[/tex]
3. [tex]\(\frac{-81}{21}\)[/tex]: This can be simplified by dividing both numerator and denominator by their greatest common divisor (3):
[tex]\[ \frac{-81}{21} = \frac{-81 \div 3}{21 \div 3} = \frac{-27}{7}. \][/tex]

Next, let's work with pairs of terms to make the calculations manageable.

### Step 1: Combine [tex]\(\frac{3}{7}\)[/tex] and [tex]\(\frac{-27}{7}\)[/tex]

Since these have the same denominator already:
[tex]\[ \frac{3}{7} + \frac{-27}{7} = \frac{3 - 27}{7} = \frac{-24}{7} = -3.428571428571429. \][/tex]

### Step 2: Combine [tex]\(\frac{-6}{11}\)[/tex] and [tex]\(\frac{-5}{22}\)[/tex]

For these fractions, we need a common denominator. 22 is a common multiple of 11 and 22.

Rewrite [tex]\(\frac{-6}{11}\)[/tex]:
[tex]\[ \frac{-6}{11} = \frac{-6 \times 2}{11 \times 2} = \frac{-12}{22}. \][/tex]

Then,
[tex]\[ \frac{-12}{22} + \frac{-5}{22} = \frac{-12 - 5}{22} = \frac{-17}{22} = -0.7727272727272727. \][/tex]

### Step 3: Combine the results from Step 1 and Step 2

Now, add the two results:
[tex]\[ -3.428571428571429 + (-0.7727272727272727) = -4.201298701298701. \][/tex]

So, the final step-by-step solution yields the result:
[tex]\[ \frac{3}{7} + \left(\frac{-6}{11}\right) + \left(\frac{-81}{21}\right) + \left(\frac{-5}{22}\right) = -4.201298701298701. \][/tex]