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]
