A fraction represents a part-to-whole ratio or a quotient of two integers.
The general form is
\[
\begin{aligned}
\frac{a}{b},
\qquad b\ne0.
\end{aligned}
\]
The numerator \(a\) tells how many parts are being counted, and the denominator \(b\) tells how many equal parts make one whole.
1. Equivalent fractions
Multiplying or dividing the numerator and denominator by the same nonzero number gives an equivalent fraction:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{ka}{kb},
\qquad k\ne0.
\end{aligned}
\]
For example:
\[
\begin{aligned}
\frac{1}{2}
&=
\frac{2}{4}
=
\frac{3}{6}.
\end{aligned}
\]
2. Simplifying fractions
A fraction is simplified by dividing numerator and denominator by their greatest common divisor:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{a/\gcd(a,b)}{b/\gcd(a,b)}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{12}{18}
&=
\frac{12\div6}{18\div6}\\
&=
\frac{2}{3}.
\end{aligned}
\]
3. Adding fractions
To add fractions with different denominators, use a common denominator:
\[
\begin{aligned}
\frac{a}{b}+\frac{c}{d}
&=
\frac{ad+bc}{bd}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{3}{4}+\frac{5}{6}
&=
\frac{3\cdot6+5\cdot4}{4\cdot6}\\
&=
\frac{18+20}{24}\\
&=
\frac{38}{24}\\
&=
\frac{19}{12}.
\end{aligned}
\]
4. Subtracting fractions
Subtraction also uses a common denominator:
\[
\begin{aligned}
\frac{a}{b}-\frac{c}{d}
&=
\frac{ad-bc}{bd}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{7}{8}-\frac{5}{12}
&=
\frac{7\cdot12-5\cdot8}{8\cdot12}\\
&=
\frac{84-40}{96}\\
&=
\frac{44}{96}\\
&=
\frac{11}{24}.
\end{aligned}
\]
5. Multiplying fractions
To multiply fractions, multiply numerators together and denominators together:
\[
\begin{aligned}
\frac{a}{b}\times\frac{c}{d}
&=
\frac{ac}{bd}.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{5}{6}\times\frac{2}{3}
&=
\frac{5\cdot2}{6\cdot3}\\
&=
\frac{10}{18}\\
&=
\frac{5}{9}.
\end{aligned}
\]
6. Dividing fractions
To divide by a nonzero fraction, multiply by its reciprocal:
\[
\begin{aligned}
\frac{a}{b}\div\frac{c}{d}
&=
\frac{a}{b}\times\frac{d}{c}\\
&=
\frac{ad}{bc},
\qquad c\ne0.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{4}{9}\div\frac{2}{3}
&=
\frac{4}{9}\times\frac{3}{2}\\
&=
\frac{12}{18}\\
&=
\frac{2}{3}.
\end{aligned}
\]
7. Powers of fractions
For a positive integer exponent, raise both numerator and denominator to the power:
\[
\begin{aligned}
\left(\frac{a}{b}\right)^n
&=
\frac{a^n}{b^n},
\qquad n>0.
\end{aligned}
\]
For a negative exponent, use the reciprocal first:
\[
\begin{aligned}
\left(\frac{a}{b}\right)^{-n}
&=
\left(\frac{b}{a}\right)^n,
\qquad a\ne0.
\end{aligned}
\]
8. Mixed numbers
A mixed number combines a whole number and a fraction, such as \(1\frac{2}{3}\).
To calculate with it, first convert it to an improper fraction:
\[
\begin{aligned}
1\frac{2}{3}
&=
\frac{1\cdot3+2}{3}\\
&=
\frac{5}{3}.
\end{aligned}
\]
For negative mixed numbers, the sign applies to the whole value:
\[
\begin{aligned}
-1\frac{2}{3}
&=
-\frac{5}{3}.
\end{aligned}
\]
9. Order of operations
Fraction expressions follow the same order of operations as other arithmetic expressions:
- Parentheses
- Powers
- Multiplication and division from left to right
- Addition and subtraction from left to right
Example:
\[
\begin{aligned}
\frac{3}{4}+\frac{5}{6}\times\frac{2}{3}-\frac{1}{2}
&=
\frac{3}{4}+\frac{5}{9}-\frac{1}{2}\\
&=
\frac{27}{36}+\frac{20}{36}-\frac{18}{36}\\
&=
\frac{29}{36}.
\end{aligned}
\]
10. Decimal and fraction connection
A terminating decimal can be converted into a fraction by using a power of ten as the denominator:
\[
\begin{aligned}
2.5
&=
\frac{25}{10}\\
&=
\frac{5}{2}.
\end{aligned}
\]
This is why the calculator can evaluate expressions that mix decimals and fractions.
11. Formula summary
The table below uses plain text formulas in cells to avoid raw LaTeX rendering problems.
12. Common mistakes
- Adding denominators when adding fractions. Denominators are not added.
- Forgetting to use a common denominator before addition or subtraction.
- Dividing fractions without taking the reciprocal of the second fraction.
- Forgetting that a denominator cannot be zero.
- Not simplifying the final answer.
- Ignoring order of operations in longer expressions.
Key idea: keep fractions exact, follow order of operations, then simplify the final fraction using the greatest common divisor.
