MENUFraction Rules
All the rules for working with fractions on one page, each with an example. The letters stand for numbers, and no denominator may be 0: every rule below assumes that. For step-by-step explanations, follow the links to the lessons.
Jump to: What a fraction means · Equivalent fractions · Signs · Adding and subtracting · Multiplying and dividing · Complex fractions · Mixed numbers · Powers and roots · Comparing · Rules that are not true
What a fraction means
| Rule | Formula | Example |
|---|---|---|
| A fraction is a division | $\dfrac{a}{b}=a\div b$ | $\dfrac{3}{4}=3\div 4=0.75$ |
| A fraction counts equal parts | $\dfrac{a}{b}=a\cdot\dfrac{1}{b}$ | $\dfrac{3}{4}=\dfrac{1}{4}+\dfrac{1}{4}+\dfrac{1}{4}$ |
| Denominator 1 | $\dfrac{a}{1}=a$ | $\dfrac{9}{1}=9$ |
| A whole number as a fraction | $a=\dfrac{a\cdot b}{b}$ | $5=\dfrac{10}{2}=\dfrac{15}{3}$ |
| Equal numerator and denominator | $\dfrac{a}{a}=1$ | $\dfrac{7}{7}=1$ |
| Zero numerator | $\dfrac{0}{a}=0$ | $\dfrac{0}{5}=0$ |
| Zero denominator | $\dfrac{a}{0}$ is undefined | $\dfrac{5}{0}$ has no value: no number times 0 gives 5 |
| Proper and improper fractions | For positive $a$ and $b$: $\dfrac{a}{b}\lt 1$ if $a\lt b$, and $\dfrac{a}{b}\ge 1$ if $a\ge b$ | $\dfrac{3}{5}\lt 1$ and $\dfrac{7}{5}\gt 1$ |
Equivalent fractions
| Rule | Formula | Example |
|---|---|---|
| Multiply the top and the bottom | $\dfrac{a}{b}=\dfrac{a\cdot k}{b\cdot k}$, $k\ne 0$ | $\dfrac{2}{3}=\dfrac{2\cdot 4}{3\cdot 4}=\dfrac{8}{12}$ |
| Divide the top and the bottom (reduce) | $\dfrac{a}{b}=\dfrac{a\div k}{b\div k}$, $k\ne 0$ | $\dfrac{18}{24}=\dfrac{18\div 6}{24\div 6}=\dfrac{3}{4}$ |
| Cancel a common factor | $\dfrac{a\cdot k}{b\cdot k}=\dfrac{a}{b}$ | $\dfrac{6x}{9x}=\dfrac{2\cdot 3x}{3\cdot 3x}=\dfrac{2}{3}$ |
| Cross products | $\dfrac{a}{b}=\dfrac{c}{d}$ exactly when $ad=bc$ | $\dfrac{2}{3}=\dfrac{8}{12}$ because $2\cdot 12=3\cdot 8$ |
Lesson: reduction of fractions
Signs
A fraction has three signs: in front of it, in the numerator and in the denominator. Changing one of them changes the sign of the value. Changing two of them leaves the value the same.
| Rule | Formula | Example |
|---|---|---|
| The minus sign can go in three places | $-\dfrac{a}{b}=\dfrac{-a}{b}=\dfrac{a}{-b}$ | $-\dfrac{3}{4}=\dfrac{-3}{4}=\dfrac{3}{-4}$ |
| Two minus signs cancel | $\dfrac{-a}{-b}=\dfrac{a}{b}$ and $-\dfrac{-a}{b}=\dfrac{a}{b}$ | $\dfrac{-6}{-2}=\dfrac{6}{2}=3$ |
| Minus in front of a difference | $-\dfrac{a-b}{c}=\dfrac{b-a}{c}$ | $-\dfrac{x-1}{2}=\dfrac{1-x}{2}$ |
| Swapping a difference in the denominator | $\dfrac{a}{b-c}=-\dfrac{a}{c-b}$ | $\dfrac{1}{2-x}=-\dfrac{1}{x-2}$ |
Adding and subtracting
| Rule | Formula | Example |
|---|---|---|
| Same denominator | $\dfrac{a}{c}+\dfrac{b}{c}=\dfrac{a+b}{c}$ $\dfrac{a}{c}-\dfrac{b}{c}=\dfrac{a-b}{c}$ |
$\dfrac{1}{5}+\dfrac{2}{5}=\dfrac{3}{5}$ $\dfrac{5}{8}-\dfrac{2}{8}=\dfrac{3}{8}$ |
| Different denominators | $\dfrac{a}{b}+\dfrac{c}{d}=\dfrac{ad+bc}{bd}$ $\dfrac{a}{b}-\dfrac{c}{d}=\dfrac{ad-bc}{bd}$ |
$\dfrac{1}{2}+\dfrac{1}{3}=\dfrac{3+2}{6}=\dfrac{5}{6}$ $\dfrac{3}{4}-\dfrac{1}{6}=\dfrac{18-4}{24}=\dfrac{7}{12}$ |
| A whole number and a fraction | $a+\dfrac{b}{c}=\dfrac{ac+b}{c}$ | $2+\dfrac{3}{4}=\dfrac{8+3}{4}=\dfrac{11}{4}$ |
| Splitting a numerator | $\dfrac{a+b}{c}=\dfrac{a}{c}+\dfrac{b}{c}$ | $\dfrac{x+6}{3}=\dfrac{x}{3}+2$ |
Lessons: addition of fractions, subtraction of fractions
Multiplying and dividing
| Rule | Formula | Example |
|---|---|---|
| Multiply | $\dfrac{a}{b}\cdot\dfrac{c}{d}=\dfrac{ac}{bd}$ | $\dfrac{2}{3}\cdot\dfrac{3}{4}=\dfrac{6}{12}=\dfrac{1}{2}$ |
| Multiply by a whole number | $k\cdot\dfrac{a}{b}=\dfrac{ka}{b}$ | $3\cdot\dfrac{2}{5}=\dfrac{6}{5}$ |
| A fraction of a number | $\dfrac{a}{b}$ of $n$ is $\dfrac{a}{b}\cdot n$ | $\dfrac{2}{3}$ of 12 is $\dfrac{2}{3}\cdot 12=8$ |
| Reciprocal | The reciprocal of $\dfrac{a}{b}$ is $\dfrac{b}{a}$, and $\dfrac{a}{b}\cdot\dfrac{b}{a}=1$ ($a\ne 0$) | $\dfrac{3}{5}\cdot\dfrac{5}{3}=1$ |
| Divide | $\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\cdot\dfrac{d}{c}=\dfrac{ad}{bc}$ ($c\ne 0$) | $\dfrac{3}{4}\div\dfrac{2}{5}=\dfrac{3}{4}\cdot\dfrac{5}{2}=\dfrac{15}{8}$ |
| Divide by a whole number | $\dfrac{a}{b}\div k=\dfrac{a}{bk}$ | $\dfrac{6}{7}\div 3=\dfrac{6}{21}=\dfrac{2}{7}$ |
Lessons: multiplication of fractions, division of fractions
Complex fractions
| Rule | Formula | Example |
|---|---|---|
| A fraction over a fraction | $\dfrac{\dfrac{a}{b}}{\dfrac{c}{d}}=\dfrac{ad}{bc}$ | $\dfrac{\dfrac{3}{4}}{\dfrac{5}{8}}=\dfrac{3\cdot 8}{4\cdot 5}=\dfrac{24}{20}=\dfrac{6}{5}$ |
| A number over a fraction | $\dfrac{a}{\dfrac{b}{c}}=\dfrac{ac}{b}$ | $\dfrac{2}{\dfrac{3}{5}}=\dfrac{10}{3}$ |
| A fraction over a number | $\dfrac{\dfrac{a}{b}}{c}=\dfrac{a}{bc}$ | $\dfrac{\dfrac{2}{3}}{4}=\dfrac{2}{12}=\dfrac{1}{6}$ |
| One over a fraction | $\dfrac{1}{\dfrac{a}{b}}=\dfrac{b}{a}$ | $\dfrac{1}{\dfrac{2}{7}}=\dfrac{7}{2}$ |
Mixed numbers
| Rule | Formula | Example |
|---|---|---|
| A mixed number is a sum | $w\dfrac{n}{d}=w+\dfrac{n}{d}$ | $2\dfrac{1}{3}=2+\dfrac{1}{3}$ |
| Mixed number to improper fraction | $w\dfrac{n}{d}=\dfrac{w\cdot d+n}{d}$ | $3\dfrac{2}{5}=\dfrac{3\cdot 5+2}{5}=\dfrac{17}{5}$ |
| Improper fraction to mixed number | Divide: the quotient is the whole part, the remainder is the new numerator | $\dfrac{29}{6}=4\dfrac{5}{6}$ because $29=4\cdot 6+5$ |
| Negative mixed number | $-w\dfrac{n}{d}=-\left(w+\dfrac{n}{d}\right)$ | $-2\dfrac{1}{3}=-\dfrac{7}{3}$, not $-2+\dfrac{1}{3}$ |
Practice: mixed fractions worksheet
Powers and roots
| Rule | Formula | Example |
|---|---|---|
| Power of a fraction | $\left(\dfrac{a}{b}\right)^n=\dfrac{a^n}{b^n}$ | $\left(\dfrac{2}{3}\right)^3=\dfrac{8}{27}$ |
| Negative exponent | $a^{-n}=\dfrac{1}{a^n}$ ($a\ne 0$) | $2^{-3}=\dfrac{1}{8}$ |
| Fraction to the power $-1$ | $\left(\dfrac{a}{b}\right)^{-1}=\dfrac{b}{a}$ ($a\ne 0$) | $\left(\dfrac{3}{4}\right)^{-1}=\dfrac{4}{3}$ |
| Fraction to a negative power | $\left(\dfrac{a}{b}\right)^{-n}=\left(\dfrac{b}{a}\right)^{n}$ ($a\ne 0$) | $\left(\dfrac{2}{5}\right)^{-2}=\left(\dfrac{5}{2}\right)^2=\dfrac{25}{4}$ |
| Root of a fraction | $\sqrt{\dfrac{a}{b}}=\dfrac{\sqrt{a}}{\sqrt{b}}$ ($a\ge 0$, $b\gt 0$) | $\sqrt{\dfrac{9}{16}}=\dfrac{3}{4}$ |
Comparing fractions
| Rule | Formula | Example |
|---|---|---|
| Same denominator: the bigger numerator wins | $\dfrac{a}{c}\lt\dfrac{b}{c}$ if $a\lt b$ ($c\gt 0$) | $\dfrac{3}{7}\lt\dfrac{5}{7}$ |
| Same numerator: the bigger denominator loses | $\dfrac{a}{b}\gt\dfrac{a}{c}$ if $0\lt b\lt c$ ($a\gt 0$) | $\dfrac{1}{3}\gt\dfrac{1}{4}$ |
| Cross-multiply | $\dfrac{a}{b}\lt\dfrac{c}{d}$ exactly when $ad\lt bc$ ($b,d\gt 0$) | $\dfrac{5}{6}\lt\dfrac{7}{8}$ because $5\cdot 8=40$ and $6\cdot 7=42$ |
Practice: comparing fractions worksheet
Rules that are not true
These look right but are wrong. One counterexample is enough to show it.
| Mistake | Not true | Counterexample |
|---|---|---|
| Adding the tops and the bottoms | $\dfrac{a}{b}+\dfrac{c}{d}\ne\dfrac{a+c}{b+d}$ | $\dfrac{1}{2}+\dfrac{1}{2}=1$, but $\dfrac{1+1}{2+2}=\dfrac{1}{2}$ |
| Splitting a denominator | $\dfrac{a}{b+c}\ne\dfrac{a}{b}+\dfrac{a}{c}$ | $\dfrac{6}{2+1}=2$, but $\dfrac{6}{2}+\dfrac{6}{1}=9$ |
| Cancelling a term | $\dfrac{a+b}{a}\ne b$ | $\dfrac{2+5}{2}=\dfrac{7}{2}$, not 5 |
| Cancelling a common term | $\dfrac{a+b}{a+c}\ne\dfrac{b}{c}$ | $\dfrac{1+2}{1+3}=\dfrac{3}{4}$, not $\dfrac{2}{3}$ |
| Subtracting from the top and the bottom | $\dfrac{a}{b}\ne\dfrac{a-k}{b-k}$ | $\dfrac{5}{7}\ne\dfrac{5-4}{7-4}=\dfrac{1}{3}$ |
Fraction Calculators with Steps