Sure, let's go through the detailed step-by-step solution for the given problem.
We start with the mixed numbers [tex]\( 5 \frac{1}{9} \)[/tex] and [tex]\( 2 \frac{3}{6} \)[/tex].
1. Convert Mixed Numbers to Improper Fractions:
- For [tex]\( 5 \frac{1}{9} \)[/tex]:
- First, multiply the whole number by the denominator: [tex]\( 5 \times 9 = 45 \)[/tex].
- Then add the numerator: [tex]\( 45 + 1 = 46 \)[/tex].
- So, [tex]\( 5 \frac{1}{9} \)[/tex] becomes [tex]\( \frac{46}{9} \)[/tex].
- For [tex]\( 2 \frac{3}{6} \)[/tex]:
- First, simplify [tex]\( \frac{3}{6} \)[/tex] to [tex]\( \frac{1}{2} \)[/tex].
- Multiply the whole number by the simplified denominator: [tex]\( 2 \times 2 = 4 \)[/tex].
- Then add the numerator: [tex]\( 4 + 1 = 5 \)[/tex].
- So, [tex]\( 2 \frac{3}{6} \)[/tex] becomes [tex]\( \frac{5}{2} \)[/tex].
2. Multiply the Improper Fractions:
- Now multiply the two fractions: [tex]\( \frac{46}{9} \)[/tex] and [tex]\( \frac{5}{2} \)[/tex].
[tex]\[
\frac{46}{9} \times \frac{5}{2} = \frac{46 \times 5}{9 \times 2} = \frac{230}{18}
\][/tex]
3. Simplify the Result:
- To simplify [tex]\( \frac{230}{18} \)[/tex], first find the greatest common divisor (GCD) of the numerator and denominator.
- The GCD of 230 and 18 is 2.
- Divide both the numerator and the denominator by their GCD:
[tex]\[
\frac{230}{18} = \frac{230 \div 2}{18 \div 2} = \frac{115}{9}
\][/tex]
Therefore, the simplified product is [tex]\( \frac{115}{9} \)[/tex].
