To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{1}{5} \) x \( \frac{4}{9} \) = \( \frac{1 x 4}{5 x 9} \) = \( \frac{4}{45} \) = \(\frac{4}{45}\)

To divide fractions, invert the second fraction and then multiply:

\( \frac{3}{9} \) ÷ \( \frac{3}{8} \) = \( \frac{3}{9} \) x \( \frac{8}{3} \)

To multiply fractions, multiply the numerators together and then multiply the denominators together:

\( \frac{3}{9} \) x \( \frac{8}{3} \) = \( \frac{3 x 8}{9 x 3} \) = \( \frac{24}{27} \) = \(\frac{8}{9}\)

To subtract these fractions, first find the lowest common multiple of their denominators. The first few multiples of 6 are [6, 12, 18, 24, 30, 36, 42, 48, 54, 60] and the first few multiples of 14 are [14, 28, 42, 56, 70, 84, 98]. The first few multiples they share are [42, 84] making 42 the smallest multiple 6 and 14 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{4 x 7}{6 x 7} \) - \( \frac{3 x 3}{14 x 3} \)

\( \frac{28}{42} \) - \( \frac{9}{42} \)

Now, because the fractions share a common denominator, you can subtract them:

\( \frac{28 - 9}{42} \) = \( \frac{19}{42} \) = \(\frac{19}{42}\)

To simplify this fraction, first find the greatest common factor between them. The factors of 36 are [1, 2, 3, 4, 6, 9, 12, 18, 36] and the factors of 68 are [1, 2, 4, 17, 34, 68]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{36}{68} \) = \( \frac{\frac{36}{4}}{\frac{68}{4}} \) = \( \frac{9}{17} \)

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 9 are [9, 18, 27, 36, 45, 54, 63, 72, 81, 90] and the first few multiples of 15 are [15, 30, 45, 60, 75, 90]. The first few multiples they share are [45, 90] making 45 the smallest multiple 9 and 15 share.

Next, convert the fractions so each denominator equals the lowest common multiple:

\( \frac{6 x 5}{9 x 5} \) + \( \frac{7 x 3}{15 x 3} \)

\( \frac{30}{45} \) + \( \frac{21}{45} \)

Now, because the fractions share a common denominator, you can add them:

\( \frac{30 + 21}{45} \) = \( \frac{51}{45} \) = 1\(\frac{2}{15}\)