Let's go through each of the problems step by step and calculate the results.
1. Problem 1: [tex]\(\frac{1}{6} \times \frac{1}{4}\)[/tex]
To multiply two fractions, multiply the numerators together and multiply the denominators together:
[tex]\[
\frac{1}{6} \times \frac{1}{4} = \frac{1 \times 1}{6 \times 4} = \frac{1}{24}
\][/tex]
2. Problem 2: [tex]\(\frac{4}{5} \times \frac{3}{8}\)[/tex]
Again, multiply the numerators together and the denominators together:
[tex]\[
\frac{4}{5} \times \frac{3}{8} = \frac{4 \times 3}{5 \times 8} = \frac{12}{40}
\][/tex]
Simplify [tex]\(\frac{12}{40}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is 4:
[tex]\[
\frac{12}{40} = \frac{12 \div 4}{40 \div 4} = \frac{3}{10}
\][/tex]
3. Problem 3: [tex]\(\frac{1}{8} + 1 \frac{1}{5}\)[/tex]
First, convert the mixed number [tex]\(1 \frac{1}{5}\)[/tex] to an improper fraction.
[tex]\[
1 \frac{1}{5} = 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5}
\][/tex]
Now, add [tex]\(\frac{1}{8}\)[/tex] and [tex]\(\frac{6}{5}\)[/tex]. Find a common denominator, which will be 40:
[tex]\[
\frac{1}{8} = \frac{5}{40}
\][/tex]
[tex]\[
\frac{6}{5} = \frac{48}{40}
\][/tex]
Now add the fractions:
[tex]\[
\frac{5}{40} + \frac{48}{40} = \frac{53}{40}
\][/tex]
4. Problem 4: [tex]\(\frac{2}{4} \times 4 \frac{2}{4}\)[/tex]
First, convert the mixed number [tex]\(4 \frac{2}{4}\)[/tex] to an improper fraction:
[tex]\[
4 \frac{2}{4} = 4 + \frac{2}{4} = \frac{16}{4} + \frac{2}{4} = \frac{18}{4}
\][/tex]
Now, multiply [tex]\(\frac{2}{4}\)[/tex] by [tex]\(\frac{18}{4}\)[/tex]:
[tex]\[
\frac{2}{4} \times \frac{18}{4} = \frac{2 \times 18}{4 \times 4} = \frac{36}{16}
\][/tex]
Simplify [tex]\(\frac{36}{16}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is 4:
[tex]\[
\frac{36}{16} = \frac{36 \div 4}{16 \div 4} = \frac{9}{4}
\][/tex]
5. Problem 5: [tex]\(1 \frac{5}{6} + 2 \frac{2}{3}\)[/tex]
First, convert each mixed number to an improper fraction. For [tex]\(1 \frac{5}{6}\)[/tex]:
[tex]\[
1 \frac{5}{6} = 1 + \frac{5}{6} = \frac{6}{6} + \frac{5}{6} = \frac{11}{6}
\][/tex]
For [tex]\(2 \frac{2}{3}\)[/tex]:
[tex]\[
2 \frac{2}{3} = 2 + \frac{2}{3} = \frac{6}{3} + \frac{2}{3} = \frac{8}{3}
\][/tex]
Now, add [tex]\(\frac{11}{6}\)[/tex] to [tex]\(\frac{8}{3}\)[/tex]. Find a common denominator, which will be 6:
[tex]\[
\frac{8}{3} = \frac{16}{6}
\][/tex]
Now add the fractions:
[tex]\[
\frac{11}{6} + \frac{16}{6} = \frac{27}{6}
\][/tex]
Simplify [tex]\(\frac{27}{6}\)[/tex] by dividing the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[
\frac{27}{6} = \frac{27 \div 3}{6 \div 3} = \frac{9}{2}
\][/tex]
So the final results are:
1. [tex]\(\frac{1}{24}\)[/tex]
2. [tex]\(\frac{3}{10}\)[/tex]
3. [tex]\(\frac{53}{40}\)[/tex]
4. [tex]\(\frac{9}{4}\)[/tex]
5. [tex]\(\frac{9}{2}\)[/tex]
