Rational numbers

Rational Numbers

A Rational Number is a real number that can be written as a ratio or fraction: $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

Examples

$5 = \frac{5}{1}$ → Rational

$1.75 = \frac{7}{4}$ → Rational

$0.001 = \frac{1}{1000}$ → Rational

$-0.1 = \frac{-1}{10}$ → Rational

$0.111\ldots = \frac{1}{9}$ → Rational

$\sqrt{2}$ → Not Rational

Operations on Rational Numbers

Addition of Rational Numbers

To add two or more rational numbers, make sure the denominators are the same. If not, find a common denominator and convert the fractions.

Examples:

1. $\frac{1}{3} + \frac{4}{3} = \frac{1+4}{3} = \frac{5}{3}$

2. $\frac{1}{3} + \frac{1}{5} = \frac{5}{15} + \frac{3}{15} = \frac{8}{15}$

3. $\frac{1}{4} + \frac{1}{6} = \frac{3 + 2}{12} = \frac{5}{12}$

4. $\frac{2}{3} + \frac{1}{2} = \frac{4 + 3}{6} = \frac{7}{6}$

5. $\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$

6. $\frac{5}{8} + \frac{3}{4} = \frac{5 + 6}{8} = \frac{11}{8}$

7. $\frac{7}{10} + \frac{2}{5} = \frac{7 + 4}{10} = \frac{11}{10}$

8. $\frac{1}{9} + \frac{5}{18} = \frac{2 + 5}{18} = \frac{7}{18}$

Subtraction of Rational Numbers

To subtract rational numbers, make the denominators the same if they are different, then subtract the numerators.

Examples:

1. $\frac{4}{3} - \frac{2}{3} = \frac{4 - 2}{3} = \frac{2}{3}$

2. $\frac{1}{3} - \frac{1}{5} = \frac{5}{15} - \frac{3}{15} = \frac{2}{15}$

3. $\frac{3}{4} - \frac{1}{2} = \frac{6 - 4}{8} = \frac{1}{4}$

4. $\frac{5}{6} - \frac{1}{3} = \frac{10 - 2}{12} = \frac{8}{12} = \frac{2}{3}$

5. $\frac{7}{8} - \frac{3}{8} = \frac{4}{8} = \frac{1}{2}$

6. $\frac{9}{10} - \frac{2}{5} = \frac{9 - 4}{10} = \frac{5}{10} = \frac{1}{2}$

7. $\frac{5}{9} - \frac{2}{3} = \frac{5 - 6}{9} = -\frac{1}{9}$

8. $\frac{11}{12} - \frac{1}{6} = \frac{11 - 2}{12} = \frac{9}{12} = \frac{3}{4}$

Multiplication of Rational Numbers

To multiply rational numbers, multiply the numerators together and the denominators together.

Examples:

1. $\frac{4}{3} \times \frac{2}{3} = \frac{4 \cdot 2}{3 \cdot 3} = \frac{8}{9}$

2. $\frac{3}{5} \times \frac{10}{7} = \frac{3 \cdot 10}{5 \cdot 7} = \frac{30}{35} = \frac{6}{7}$

3. $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$

4. $\frac{5}{6} \times \frac{2}{5} = \frac{10}{30} = \frac{1}{3}$

5. $\frac{7}{8} \times 2 = \frac{14}{8} = \frac{7}{4}$

6. $\frac{3}{10} \times \frac{5}{6} = \frac{15}{60} = \frac{1}{4}$

7. $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$

8. $\frac{3}{7} \times \frac{7}{3} = 1$

Division of Rational Numbers

To divide rational numbers, multiply the first by the reciprocal of the second.

Examples:

1. $\frac{4}{3} \div \frac{2}{5} = \frac{4}{3} \times \frac{5}{2} = \frac{20}{6} = \frac{10}{3}$

2. $\frac{5}{6} \div \frac{10}{9} = \frac{5}{6} \times \frac{9}{10} = \frac{45}{60} = \frac{3}{4}$

3. $\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$

4. $\frac{5}{6} \div \frac{1}{2} = \frac{5}{6} \times \frac{2}{1} = \frac{10}{6} = \frac{5}{3}$

5. $\frac{7}{8} \div 2 = \frac{7}{16}$

6. $\frac{2}{3} \div \frac{4}{9} = \frac{2}{3} \times \frac{9}{4} = \frac{18}{12} = \frac{3}{2}$

7. $1 \div \frac{3}{5} = \frac{5}{3}$

8. $\frac{9}{10} \div \frac{3}{2} = \frac{9}{10} \times \frac{2}{3} = \frac{18}{30} = \frac{3}{5}$