Sure, let's work through each problem step-by-step.
### Problem 1:
[tex]\[ 7 \frac{1}{2} \times 8 \frac{5}{9} \][/tex]
First, we convert the mixed fractions to improper fractions.
1. [tex]\( 7 \frac{1}{2} \)[/tex]:
[tex]\[
7 \frac{1}{2} = \frac{7 \times 2 + 1}{2} = \frac{14 + 1}{2} = \frac{15}{2}
\][/tex]
2. [tex]\( 8 \frac{5}{9} \)[/tex]:
[tex]\[
8 \frac{5}{9} = \frac{8 \times 9 + 5}{9} = \frac{72 + 5}{9} = \frac{77}{9}
\][/tex]
Next, we multiply the improper fractions:
[tex]\[ \frac{15}{2} \times \frac{77}{9} = \frac{15 \times 77}{2 \times 9} = \frac{1155}{18} \][/tex]
Now, we simplify the fraction:
- The greatest common divisor (GCD) of 1155 and 18 is 3.
[tex]\[ \frac{1155 \div 3}{18 \div 3} = \frac{385}{6} \][/tex]
Thus, the product in simplest form is:
[tex]\[ \boxed{\frac{385}{6}} \][/tex]
### Problem 2:
[tex]\[ 7 \frac{2}{5} \times 2 \frac{1}{5} \][/tex]
First, we convert the mixed fractions to improper fractions.
1. [tex]\( 7 \frac{2}{5} \)[/tex]:
[tex]\[
7 \frac{2}{5} = \frac{7 \times 5 + 2}{5} = \frac{35 + 2}{5} = \frac{37}{5}
\][/tex]
2. [tex]\( 2 \frac{1}{5} \)[/tex]:
[tex]\[
2 \frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{10 + 1}{5} = \frac{11}{5}
\][/tex]
Next, we multiply the improper fractions:
[tex]\[ \frac{37}{5} \times \frac{11}{5} = \frac{37 \times 11}{5 \times 5} = \frac{407}{25} \][/tex]
This fraction cannot be simplified further since the GCD of 407 and 25 is 1.
Thus, the product in simplest form is:
[tex]\[ \boxed{\frac{407}{25}} \][/tex]
### Problem 3:
[tex]\[ 6 \frac{1}{6} \times 5 \frac{2}{4} \][/tex]
First, we convert the mixed fractions to improper fractions.
1. [tex]\( 6 \frac{1}{6} \)[/tex]:
[tex]\[
6 \frac{1}{6} = \frac{6 \times 6 + 1}{6} = \frac{36 + 1}{6} = \frac{37}{6}
\][/tex]
2. [tex]\( 5 \frac{2}{4} \)[/tex]:
[tex]\[
5 \frac{2}{4} = \frac{5 \times 4 + 2}{4} = \frac{20 + 2}{4} = \frac{22}{4}
\][/tex]
We can simplify [tex]\( \frac{22}{4} \)[/tex] to [tex]\( \frac{11}{2} \)[/tex]
Next, we multiply the improper fractions:
[tex]\[ \frac{37}{6} \times \frac{11}{2} = \frac{37 \times 11}{6 \times 2} = \frac{407}{12} \][/tex]
This fraction cannot be simplified further since the GCD of 407 and 12 is 1.
Thus, the product in simplest form is:
[tex]\[ \boxed{\frac{407}{12}} \][/tex]
