A rational number (or fraction) is represented as a ratio between two integers, a and b, and has the form \({a \over b}\) where a is the numerator and b is the denominator. An improper fraction (\({5 \over 3} \)) has a numerator with a greater absolute value than the denominator and can be converted into a mixed number (\(1 {2 \over 3} \)) which has a whole number part and a fractional part.

Average speed in miles per hour is the number of miles traveled divided by the number of hours:

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

\( \frac{4}{9} \) ÷ \( \frac{4}{5} \) = \( \frac{4}{9} \) x \( \frac{5}{4} \)

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

\( \frac{4}{9} \) x \( \frac{5}{4} \) = \( \frac{4 x 5}{9 x 4} \) = \( \frac{20}{36} \) = \(\frac{5}{9}\)

The number of students taking German or Spanish is 14 + 5 = 19. Of that group of 19, 4 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 19 - 4 = 15 who are taking at least one language. 24 - 15 = 9 students who are not taking either language.

The least common multiple (LCM) is the smallest positive integer that is a multiple of two or more integers.