To add these fractions, first find the lowest common multiple of their denominators. The first few multiples of 2 are [2, 4, 6, 8, 10, 12, 14, 16, 18, 20] and the first few multiples of 10 are [10, 20, 30, 40, 50, 60, 70, 80, 90]. The first few multiples they share are [10, 20, 30, 40, 50] making 10 the smallest multiple 2 and 10 share.

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

\( \frac{6 x 5}{2 x 5} \) + \( \frac{5 x 1}{10 x 1} \)

\( \frac{30}{10} \) + \( \frac{5}{10} \)

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

\( \frac{30 + 5}{10} \) = \( \frac{35}{10} \) = 3\(\frac{1}{2}\)

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

\( \frac{2}{5} \) ÷ \( \frac{1}{7} \) = \( \frac{2}{5} \) x \( \frac{7}{1} \)

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

\( \frac{2}{5} \) x \( \frac{7}{1} \) = \( \frac{2 x 7}{5 x 1} \) = \( \frac{14}{5} \) = 2\(\frac{4}{5}\)

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

\( \frac{3}{5} \) x \( \frac{1}{9} \) = \( \frac{3 x 1}{5 x 9} \) = \( \frac{3}{45} \) = \(\frac{1}{15}\)

To subtract 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:

\( \frac{6 x 2}{4 x 2} \) - \( \frac{9 x 1}{8 x 1} \)

\( \frac{12}{8} \) - \( \frac{9}{8} \)

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

\( \frac{12 - 9}{8} \) = \( \frac{3}{8} \) = \(\frac{3}{8}\)

To simplify this fraction, first find the greatest common factor between them. The factors of 16 are [1, 2, 4, 8, 16] and the factors of 60 are [1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60]. They share 3 factors [1, 2, 4] making 4 their greatest common factor (GCF).

Next, divide both numerator and denominator by the GCF:

\( \frac{16}{60} \) = \( \frac{\frac{16}{4}}{\frac{60}{4}} \) = \( \frac{4}{15} \)