Alright, let's tackle each part of the question step-by-step.
### Step 1: Fraction Addition
First, we need to evaluate the fraction addition:
[tex]\[
\frac{1}{2} + \frac{2}{4}
\][/tex]
The fraction [tex]\(\frac{2}{4}\)[/tex] can be simplified to [tex]\(\frac{1}{2}\)[/tex]. Therefore, the expression becomes:
[tex]\[
\frac{1}{2} + \frac{1}{2} = 1.0
\][/tex]
So, the result of this calculation is [tex]\(1.0\)[/tex].
### Step 2: Complex Fraction Subtraction
Next, we need to evaluate the complex fraction subtraction presented in two parts:
[tex]\[
\frac{1-2}{4 \cdot 6 \cdot 8 \cdot 10}
\][/tex]
and
[tex]\[
4 - (4 \cdot 8 \cdot 12 \cdot 16 \cdot 20)
\][/tex]
For the first part:
[tex]\[
1 - 2 = -1
\][/tex]
[tex]\[
4 \cdot 6 \cdot 8 \cdot 10 = 1920
\][/tex]
Thus, we have:
[tex]\[
\frac{-1}{1920} = -0.0005208333 \text{ (approximately)}
\][/tex]
For the second part:
[tex]\[
4 \cdot 8 = 32
\][/tex]
[tex]\[
32 \cdot 12 = 384
\][/tex]
[tex]\[
384 \cdot 16 = 6144
\][/tex]
[tex]\[
6144 \cdot 20 = 122880
\][/tex]
Therefore:
[tex]\[
4 - 122880 = -122876
\][/tex]
So, the results of the complex fraction subtraction are:
[tex]\[
-0.0005208333 \text{ and } -122876
\][/tex]
### Step 3: Fraction Multiplication
Now, let's evaluate the multiplication of fractions:
[tex]\[
\frac{1}{2} \times \frac{3}{4}
\][/tex]
When we multiply two fractions, we multiply the numerators and the denominators:
[tex]\[
\frac{1 \times 3}{2 \times 4} = \frac{3}{8} = 0.375
\][/tex]
So, the result of this calculation is [tex]\(0.375\)[/tex].
### Step 4: Fraction Division
Finally, let's evaluate the division of fractions:
[tex]\[
\frac{1}{2} \div \frac{2}{4}
\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of [tex]\(\frac{2}{4}\)[/tex] is [tex]\(\frac{4}{2}\)[/tex], which simplifies to [tex]\(2\)[/tex]:
[tex]\[
\frac{1}{2} \times 2 = 1.0
\][/tex]
So, the result of this calculation is [tex]\(1.0\)[/tex].
### Summary of Results
Putting all the results together, we have:
1. Fraction Addition: [tex]\(1.0\)[/tex]
2. Complex Fraction Subtraction: [tex]\(-0.0005208333\)[/tex] and [tex]\(-122876\)[/tex]
3. Fraction Multiplication: [tex]\(0.375\)[/tex]
4. Fraction Division: [tex]\(1.0\)[/tex]
So, the final results are:
[tex]\[
(1.0, [-0.0005208333, -122876], 0.375, 1.0)
\][/tex]
