Let's carefully break down and solve each part of the question using the numerical values that we've obtained.
1. Converting mixed fractions to improper fractions and performing the arithmetic operations:
- First, let's convert the mixed fraction [tex]\(2 \frac{4}{5}\)[/tex] to an improper fraction:
[tex]\[
2 \frac{4}{5} = 2 + \frac{4}{5} = 2 + 0.8 = 2.8
\][/tex]
- Then, convert the simple fraction [tex]\(\frac{5}{7}\)[/tex] to a decimal:
[tex]\[
\frac{5}{7} \approx 0.7142857142857143
\][/tex]
- Adding these two values together:
[tex]\[
2.8 + 0.7142857142857143 \approx 3.5142857142857142
\][/tex]
2. Calculating the sum of mixed fractions and simple fractions:
- Next, we consider the mixed fraction [tex]\(3 \cdot 1 \frac{2}{9}\)[/tex]:
[tex]\[
1 \frac{2}{9} = 1 + \frac{2}{9} = 1 + 0.2222222222 \approx 1.2222222222
\][/tex]
Therefore:
[tex]\[
3 \cdot 1.2222222222 \approx 3.666666666666667
\][/tex]
- Then, the mixed fraction [tex]\(6 \frac{1}{2}\)[/tex]:
[tex]\[
6 \frac{1}{2} = 6 + \frac{1}{2} = 6 + 0.5 = 6.5
\][/tex]
3. Combine all the results:
- Adding the three components:
- The result of [tex]\( 2 \frac{4}{5} + \frac{5}{7} \)[/tex] is [tex]\( 3.5142857142857142 \)[/tex]
- The result of [tex]\( 3 \cdot (1 \frac{2}{9}) \)[/tex] is [tex]\( 3.666666666666667 \)[/tex]
- Finally adding [tex]\(6 \frac{1}{2}\)[/tex] which is [tex]\( 6.5 \)[/tex]
- Summing these three values together:
[tex]\[
3.5142857142857142 + 3.666666666666667 + 6.5 \approx 13.68095238095238
\][/tex]
In summary, let's present this step-by-step solution systematically:
- Convert [tex]\( 2 \frac{4}{5} \)[/tex] to [tex]\(2.8\)[/tex].
- Evaluate [tex]\(\frac{5}{7} \approx 0.7142857142857143\)[/tex].
- Sum of first calculation:
[tex]\[
2.8 + 0.7142857142857143 \approx 3.5142857142857142
\][/tex]
- Convert [tex]\( 3 \cdot 1 \frac{2}{9} \)[/tex] to:
[tex]\[
1 \frac{2}{9} \approx 1.22222222 \quad \text{thus} \quad 3 \times 1.22222222 \approx 3.666666666666667
\][/tex]
- Convert [tex]\(6 \frac{1}{2}\)[/tex] to [tex]\(6.5\)[/tex].
- Total sum:
[tex]\[
3.5142857142857142 + 3.666666666666667 + 6.5 = 13.68095238095238
\][/tex]
Thus, the final result is approximately [tex]\( 13.68095238095238 \)[/tex].
