Fractions | Adriamics Study

Prerequisites

Arithmetic | 9 of 17

Prerequisites: Division. Enables: Algebra.

A fraction represents part of a whole, a ratio, or division.

\frac{a}{b}

where:

Equivalent fractions

Multiplying numerator and denominator by the same nonzero number gives an equivalent fraction.

\frac{a}{b} = \frac{ak}{bk}

where

k \ne 0

Example:

\frac{2}{3} = \frac{2\cdot4}{3\cdot4} = \frac{8}{12}

Divide numerator and denominator by their greatest common factor.

Example:

\frac{18}{24}

Since

$$ GCF(18,24)=6 $$

then:

\frac{18}{24}=\frac{18\div6}{24\div6}=\frac{3}{4}

Multiply numerators and multiply denominators.

\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}

where

b \ne 0,\quad d \ne 0

Example:

\frac{2}{5}\cdot\frac{3}{4}=\frac{6}{20}=\frac{3}{10}

To divide by a fraction, multiply by its reciprocal.

\frac{a}{b}\div\frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}

where

b \ne 0,\quad c \ne 0,\quad d \ne 0

Example:

\frac{3}{5}\div\frac{2}{7}=\frac{3}{5}\cdot\frac{7}{2}=\frac{21}{10}

Fractions need a common denominator before addition or subtraction.

\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}

\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}

For unlike denominators:

\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}

Example:

\frac{2}{3}+\frac{1}{4} = \frac{8}{12}+\frac{3}{12} = \frac{11}{12}

A mixed number contains a whole number and a fraction.

Example:

3\frac{2}{5}

Convert to an improper fraction:

3\frac{2}{5}=\frac{3\cdot5+2}{5}=\frac{17}{5}

Convert an improper fraction to a mixed number:

\frac{17}{5}=3\frac{2}{5}

because

$$ 17 = 5(3) + 2 $$