Sure, let's solve each part of the question step-by-step.
### 1.) [tex]\(\frac{2}{8} + \frac{1}{3}\)[/tex]
Step 1: Simplify [tex]\(\frac{2}{8}\)[/tex].
[tex]\[ \frac{2}{8} = \frac{1}{4} \][/tex]
Step 2: Find a common denominator for [tex]\(\frac{1}{4}\)[/tex] and [tex]\(\frac{1}{3}\)[/tex]. The least common multiple (LCM) of 4 and 3 is 12.
Step 3: Convert each fraction to have the common denominator of 12.
[tex]\[ \frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \][/tex]
[tex]\[ \frac{1}{3} = \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \][/tex]
Step 4: Add the fractions.
[tex]\[ \frac{3}{12} + \frac{4}{12} = \frac{7}{12} \][/tex]
Thus, [tex]\(\frac{2}{8} + \frac{1}{3} = \frac{7}{12}\)[/tex].
### 2.) [tex]\(\frac{1}{2} + \frac{3}{8}\)[/tex]
Step 1: Find a common denominator for [tex]\(\frac{1}{2}\)[/tex] and [tex]\(\frac{3}{8}\)[/tex]. The least common multiple (LCM) of 2 and 8 is 8.
Step 2: Convert each fraction to have the common denominator of 8.
[tex]\[ \frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8} \][/tex]
Step 3: Add the fractions.
[tex]\[ \frac{4}{8} + \frac{3}{8} = \frac{7}{8} \][/tex]
Thus, [tex]\(\frac{1}{2} + \frac{3}{8} = \frac{7}{8}\)[/tex].
### 4.) [tex]\(\frac{5}{6} - \frac{4}{9}\)[/tex]
Step 1: Find a common denominator for [tex]\(\frac{5}{6}\)[/tex] and [tex]\(\frac{4}{9}\)[/tex]. The least common multiple (LCM) of 6 and 9 is 18.
Step 2: Convert each fraction to have the common denominator of 18.
[tex]\[ \frac{5}{6} = \frac{5 \times 3}{6 \times 3} = \frac{15}{18} \][/tex]
[tex]\[ \frac{4}{9} = \frac{4 \times 2}{9 \times 2} = \frac{8}{18} \][/tex]
Step 3: Subtract the fractions.
[tex]\[ \frac{15}{18} - \frac{8}{18} = \frac{7}{18} \][/tex]
Thus, [tex]\(\frac{5}{6} - \frac{4}{9} = \frac{7}{18}\)[/tex].
### 5.) [tex]\( 5 \cdot 4 \frac{1}{2} + 6 \frac{1}{3} \)[/tex]
Step 1: Convert the mixed numbers to improper fractions.
[tex]\[ 4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2} \][/tex]
[tex]\[ 6 \frac{1}{3} = \frac{6 \times 3 + 1}{3} = \frac{19}{3} \][/tex]
Step 2: Multiply [tex]\( \frac{9}{2} \)[/tex] by 5.
[tex]\[ 5 \cdot \frac{9}{2} = \frac{5 \times 9}{2} = \frac{45}{2} = 22.5 \][/tex]
Step 3: Add the two improper fractions.
Convert [tex]\( \frac{19}{3} \)[/tex] into a decimal to add it easily.
[tex]\[ \frac{19}{3} \approx 6.3333 \][/tex]
[tex]\[ 22.5 + 6.3333 \approx 28.8333 \][/tex]
Thus, [tex]\(5 \cdot 4 \frac{1}{2} + 6 \frac{1}{3} = 28.8333\)[/tex].
Summary:
1.) [tex]\( \frac{2}{8} + \frac{1}{3} = \frac{7}{12} \)[/tex]
2.) [tex]\( \frac{1}{2} + \frac{3}{8} = \frac{7}{8} \)[/tex]
3.) [tex]\( \frac{5}{6} - \frac{4}{9} = \frac{7}{18} \)[/tex]
4.) [tex]\( 5 \cdot 4 \frac{1}{2} + 6 \frac{1}{3} = 28.8333 \)[/tex]
