The commutative property states that, when adding or multiplying numbers, the order in which they're added or multiplied does not matter. For example, 3 + 4 and 4 + 3 give the same result, as do 3 x 4 and 4 x 3.

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

\( \frac{4}{8} \) ÷ \( \frac{2}{8} \) = \( \frac{4}{8} \) x \( \frac{8}{2} \)

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

\( \frac{4}{8} \) x \( \frac{8}{2} \) = \( \frac{4 x 8}{8 x 2} \) = \( \frac{32}{16} \) = 2

The absolute value is the positive magnitude of a particular number or variable and is indicated by two vertical lines: \(\left|-5\right| = 5\). In the case of a variable absolute value (\(\left|a\right| = 5\)) the value of a can be either positive or negative (a = -5 or a = 5).

The number of students taking German or Spanish is 11 + 9 = 20. Of that group of 20, 5 are taking both languages so they've been counted twice (once in the German group and once in the Spanish group). Subtracting them out leaves 20 - 5 = 15 who are taking at least one language. 21 - 15 = 6 students who are not taking either language.

The amount of flour you need is (3\(\frac{1}{4}\) - 1\(\frac{5}{8}\)) cups. Rewrite the quantities so they share a common denominator and subtract:

(\( \frac{26}{8} \) - \( \frac{13}{8} \)) cups
\( \frac{13}{8} \) cups
1\(\frac{5}{8}\) cups