Sure! Let's solve each of these multiplication problems involving mixed numbers step by step.
### 1. [tex]\(7 \frac{1}{2} \times 8 \frac{5}{9}\)[/tex]
1. Convert each mixed number to an improper fraction:
- [tex]\(7 \frac{1}{2}\)[/tex] becomes [tex]\(\frac{7 \cdot 2 + 1}{2} = \frac{15}{2}\)[/tex].
- [tex]\(8 \frac{5}{9}\)[/tex] becomes [tex]\(\frac{8 \cdot 9 + 5}{9} = \frac{77}{9}\)[/tex].
2. Multiply the improper fractions:
- [tex]\(\frac{15}{2} \times \frac{77}{9} = \frac{15 \cdot 77}{2 \cdot 9} = \frac{1155}{18}\)[/tex].
So, [tex]\(7 \frac{1}{2} \times 8 \frac{5}{9} = \frac{1155}{18}\)[/tex].
### 2. [tex]\(7 \frac{2}{5} \times 2 \frac{1}{5}\)[/tex]
1. Convert each mixed number to an improper fraction:
- [tex]\(7 \frac{2}{5}\)[/tex] becomes [tex]\(\frac{7 \cdot 5 + 2}{5} = \frac{37}{5}\)[/tex].
- [tex]\(2 \frac{1}{5}\)[/tex] becomes [tex]\(\frac{2 \cdot 5 + 1}{5} = \frac{11}{5}\)[/tex].
2. Multiply the improper fractions:
- [tex]\(\frac{37}{5} \times \frac{11}{5} = \frac{37 \cdot 11}{5 \cdot 5} = \frac{407}{25}\)[/tex].
So, [tex]\(7 \frac{2}{5} \times 2 \frac{1}{5} = \frac{407}{25}\)[/tex].
### 3. [tex]\(6 \frac{1}{6} \times 5 \frac{2}{4}\)[/tex]
1. Convert each mixed number to an improper fraction:
- [tex]\(6 \frac{1}{6}\)[/tex] becomes [tex]\(\frac{6 \cdot 6 + 1}{6} = \frac{37}{6}\)[/tex].
- [tex]\(5 \frac{2}{4}\)[/tex] becomes [tex]\(\frac{5 \cdot 2 + 2}{2} = \frac{12}{2} = 6\)[/tex] (note that [tex]\(5 \frac{2}{4}\)[/tex] simplifies to 6).
2. Multiply the improper fractions:
- [tex]\(\frac{37}{6} \times 6 = \frac{37 \cdot 6}{6 \cdot 1} = 37\)[/tex].
So, [tex]\(6 \frac{1}{6} \times 5 \frac{2}{4} = 37\)[/tex].
### 4. [tex]\(1 \frac{3}{4} \times 2 \frac{1}{2}\)[/tex]
1. Convert each mixed number to an improper fraction:
- [tex]\(1 \frac{3}{4}\)[/tex] becomes [tex]\(\frac{1 \cdot 4 + 3}{4} = \frac{7}{4}\)[/tex].
- [tex]\(2 \frac{1}{2}\)[/tex] becomes [tex]\(\frac{2 \cdot 2 + 1}{2} = \frac{5}{2}\)[/tex].
2. Multiply the improper fractions:
- [tex]\(\frac{7}{4} \times \frac{5}{2} = \frac{7 \cdot 5}{4 \cdot 2} = \frac{35}{8}\)[/tex].
So, [tex]\(1 \frac{3}{4} \times 2 \frac{1}{2} = \frac{35}{8}\)[/tex].
### 5. [tex]\(3 \frac{5}{6} \times 2 \frac{4}{7}\)[/tex]
1. Convert each mixed number to an improper fraction:
- [tex]\(3 \frac{5}{6}\)[/tex] becomes [tex]\(\frac{3 \cdot 6 + 5}{6} = \frac{23}{6}\)[/tex].
- [tex]\(2 \frac{4}{7}\)[/tex] becomes [tex]\(\frac{2 \cdot 7 + 4}{7} = \frac{18}{7}\)[/tex].
2. Multiply the improper fractions:
- [tex]\(\frac{23}{6} \times \frac{18}{7} = \frac{23 \cdot 18}{6 \cdot 7} = \frac{414}{42}\)[/tex].
So, [tex]\(3 \frac{5}{6} \times 2 \frac{4}{7} = \frac{414}{42}\)[/tex].
So, the solutions to the problems are:
1. [tex]\(\frac{1155}{18}\)[/tex]
2. [tex]\(\frac{407}{25}\)[/tex]
3. [tex]\(37\)[/tex]
4. [tex]\(\frac{35}{8}\)[/tex]
5. [tex]\(\frac{414}{42}\)[/tex]
