Let's go through each problem step-by-step and provide detailed explanations for the operations:
### 1) [tex]\(\frac{13}{15} - \frac{8}{15}\)[/tex]
Both fractions have the same denominator, so we can directly subtract their numerators:
[tex]\[
\frac{13}{15} - \frac{8}{15} = \frac{13 - 8}{15} = \frac{5}{15} = \frac{1}{3}
\][/tex]
The simplified result is [tex]\(\frac{1}{3}\)[/tex] or approximately [tex]\(0.33333333333333337\)[/tex].
### 2) [tex]\(2 \frac{6}{9}\)[/tex]
First, simplify the fractional part. The fraction [tex]\(\frac{6}{9}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[
\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}
\][/tex]
So, the mixed fraction becomes:
[tex]\[
2 \frac{6}{9} = 2 \frac{2}{3} = 2 + \frac{2}{3}
\][/tex]
The final value is [tex]\(2 \frac{2}{3}\)[/tex] or approximately [tex]\(2.6666666666666665\)[/tex].
### 3) [tex]\(6 \frac{1}{8}\)[/tex]
Convert the mixed fraction to a decimal. The fractional part is [tex]\(\frac{1}{8}\)[/tex], which equals:
[tex]\[
6 \frac{1}{8} = 6 + \frac{1}{8} = 6 + 0.125 = 6.125
\][/tex]
### 4) [tex]\(27 - \frac{1}{9}\)[/tex]
Convert the whole number to a fraction with the same denominator for subtraction:
[tex]\[
27 = \frac{27 \times 9}{9} = \frac{243}{9}
\][/tex]
Now subtract the fractions:
[tex]\[
27 - \frac{1}{9} = \frac{243}{9} - \frac{1}{9} = \frac{243 - 1}{9} = \frac{242}{9}
\][/tex]
Convert this back to a decimal:
[tex]\[
\frac{242}{9} \approx 26.88888888888889
\][/tex]
### 5) [tex]\(16 \frac{2}{5} + \frac{4}{10} + 2 \frac{2}{3}\)[/tex]
First, simplify and convert each part:
1. [tex]\(16 \frac{2}{5} = 16 + \frac{2}{5} = 16 + 0.4 = 16.4\)[/tex]
2. [tex]\(\frac{4}{10} = 0.4\)[/tex]
3. [tex]\(2 \frac{2}{3} = 2 + \frac{2}{3} = 2 + \frac{2}{3} \approx 2.6666666666666665\)[/tex]
Now, sum these values:
[tex]\[
16.4 + 0.4 + 2.6666666666666665 = 19.466666666666665 + 2 \approx 21.066666666666666
\][/tex]
The detailed step-by-step solutions ensure that the mathematical processes are clearly outlined, leading to the final results:
1. [tex]\(\frac{1}{3}\)[/tex] (approx. [tex]\(0.33333333333333337\)[/tex])
2. [tex]\(2 \frac{2}{3}\)[/tex] (approx. [tex]\(2.6666666666666665\)[/tex])
3. [tex]\(6.125\)[/tex]
4. [tex]\(26.88888888888889\)[/tex]
5. [tex]\(21.066666666666666\)[/tex]
