Sure, let's simplify each fraction one by one:
### Part (a): Simplify [tex]\(\frac{31}{11} \times \frac{2}{5}\)[/tex]
To multiply two fractions, multiply the numerators together and the denominators together.
[tex]\[
\frac{31}{11} \times \frac{2}{5} = \frac{31 \times 2}{11 \times 5} = \frac{62}{55}
\][/tex]
Divide [tex]\(62\)[/tex] by [tex]\(55\)[/tex] to get the simplified result.
### Part (b): Simplify [tex]\(\frac{2}{6} \times \frac{4}{3}\)[/tex]
Again, multiply the numerators together and the denominators together.
[tex]\[
\frac{2}{6} \times \frac{4}{3} = \frac{2 \times 4}{6 \times 3} = \frac{8}{18}
\][/tex]
We can further simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
[tex]\[
\frac{8}{18} = \frac{8 \div 2}{18 \div 2} = \frac{4}{9}
\][/tex]
Divide [tex]\(4\)[/tex] by [tex]\(9\)[/tex] to get the simplified result.
### Part (c): Simplify [tex]\(\frac{5}{3} \times \frac{9}{13}\)[/tex]
As with the previous parts, multiply the numerators together and the denominators together.
[tex]\[
\frac{5}{3} \times \frac{9}{13} = \frac{5 \times 9}{3 \times 13} = \frac{45}{39}
\][/tex]
We can further simplify this by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
[tex]\[
\frac{45}{39} = \frac{45 \div 3}{39 \div 3} = \frac{15}{13}
\][/tex]
Divide [tex]\(15\)[/tex] by [tex]\(13\)[/tex] to get the simplified result.
### Final Results:
- (a) [tex]\(\frac{31}{11} \times \frac{2}{5} = 1.1272727272727274\)[/tex]
- (b) [tex]\(\frac{2}{6} \times \frac{4}{3} = 0.4444444444444444\)[/tex]
- (c) [tex]\(\frac{5}{3} \times \frac{9}{13} = 1.153846153846154\)[/tex]
