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

\( \frac{2}{9} \) x \( \frac{4}{5} \) = \( \frac{2 x 4}{9 x 5} \) = \( \frac{8}{45} \) = \(\frac{8}{45}\)

The distributive property for division helps in solving expressions like \({b + c \over a}\). It specifies that the result of dividing a fraction with multiple terms in the numerator and one term in the denominator can be obtained by dividing each term individually and then totaling the results: \({b + c \over a} = {b \over a} + {c \over a}\). For example, \({a^3 + 6a^2 \over a^2} = {a^3 \over a^2} + {6a^2 \over a^2} = a + 6\).

To simplify this fraction, first find the greatest common factor between them. The factors of 32 are [1, 2, 4, 8, 16, 32] 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:

\( \frac{32}{60} \) = \( \frac{\frac{32}{4}}{\frac{60}{4}} \) = \( \frac{8}{15} \)

By buying two shirts, Bob will save $27 x \( \frac{45}{100} \) = \( \frac{$27 x 45}{100} \) = \( \frac{$1215}{100} \) = $12.15 on the second shirt.

By buying two shirts, Frank will save $10 x \( \frac{5}{100} \) = \( \frac{$10 x 5}{100} \) = \( \frac{$50}{100} \) = $0.50 on the second shirt.

So, his total cost will be
$10.00 + ($10.00 - $0.50)
$10.00 + $9.50
$19.50