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

\( \frac{4}{6} \) ÷ \( \frac{4}{8} \) = \( \frac{4}{6} \) x \( \frac{8}{4} \)

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

\( \frac{4}{6} \) x \( \frac{8}{4} \) = \( \frac{4 x 8}{6 x 4} \) = \( \frac{32}{24} \) = 1\(\frac{1}{3}\)

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 12 are [12, 24, 36, 48, 60, 72, 84, 96]. The first few multiples they share are [12, 24, 36, 48, 60] making 12 the smallest multiple 6 and 12 share.

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

\( \frac{6 x 2}{6 x 2} \) - \( \frac{7 x 1}{12 x 1} \)

\( \frac{12}{12} \) - \( \frac{7}{12} \)

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

\( \frac{12 - 7}{12} \) = \( \frac{5}{12} \) = \(\frac{5}{12}\)

To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 4 are [4, 8, 12, 16, 20, 24, 28, 32, 36, 40] and the first few multiples of 8 are [8, 16, 24, 32, 40, 48, 56, 64, 72, 80]. The first few multiples they share are [8, 16, 24, 32, 40] making 8 the smallest multiple 4 and 8 share.

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

\( \frac{8 x 2}{4 x 2} \) + \( \frac{7 x 1}{8 x 1} \)

\( \frac{16}{8} \) + \( \frac{7}{8} \)

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

\( \frac{16 + 7}{8} \) = \( \frac{23}{8} \) = 2\(\frac{7}{8}\)

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

\( \frac{3}{8} \) x \( \frac{4}{5} \) = \( \frac{3 x 4}{8 x 5} \) = \( \frac{12}{40} \) = \(\frac{3}{10}\)

To simplify this fraction, first find the greatest common factor between them. The factors of 40 are [1, 2, 4, 5, 8, 10, 20, 40] and the factors of 80 are [1, 2, 4, 5, 8, 10, 16, 20, 40, 80]. They share 8 factors [1, 2, 4, 5, 8, 10, 20, 40] making 40 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{40}{80} \) = \( \frac{\frac{40}{40}}{\frac{80}{40}} \) = \( \frac{1}{2} \)