Simplifying a fraction means writing it in lowest terms. The value stays the same, but the numerator
and denominator become as small as possible.
1. Fraction form
A fraction has the form:
\[
\begin{aligned}
\frac{a}{b},
\qquad b\ne0.
\end{aligned}
\]
The numerator is \(a\), and the denominator is \(b\). The denominator cannot be zero.
2. Equivalent fractions
Multiplying or dividing 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:
\[
\begin{aligned}
\frac{2}{3}
&=
\frac{4}{6}
=
\frac{84}{126}.
\end{aligned}
\]
3. Lowest terms
A fraction is in lowest terms if the numerator and denominator have no common factor greater than 1:
\[
\begin{aligned}
\gcd(|a|,b)&=1.
\end{aligned}
\]
For example, \(2/3\) is in lowest terms because 2 and 3 have no common factor greater than 1.
4. Greatest common divisor method
The fastest way to simplify a fraction is to divide numerator and denominator by their greatest common divisor:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{a\div \gcd(|a|,b)}{b\div \gcd(|a|,b)}.
\end{aligned}
\]
This removes all common factors in one step.
5. Worked example: \(84/126\)
Find the greatest common divisor:
\[
\begin{aligned}
\gcd(84,126)&=42.
\end{aligned}
\]
Divide both numerator and denominator by 42:
\[
\begin{aligned}
\frac{84}{126}
&=
\frac{84\div42}{126\div42}\\
&=
\frac{2}{3}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\boxed{\frac{84}{126}=\frac{2}{3}}.
\end{aligned}
\]
6. Prime factorization method
Another way is to write numerator and denominator as products of prime factors.
\[
\begin{aligned}
84&=2^2\cdot3\cdot7,\\
126&=2\cdot3^2\cdot7.
\end{aligned}
\]
The common prime factors are \(2\), \(3\), and \(7\). Their product is:
\[
\begin{aligned}
2\cdot3\cdot7&=42.
\end{aligned}
\]
Canceling that common factor gives:
\[
\begin{aligned}
\frac{84}{126}
&=
\frac{2^2\cdot3\cdot7}{2\cdot3^2\cdot7}\\
&=
\frac{2}{3}.
\end{aligned}
\]
7. Why cancellation works
Cancellation works because a common factor divided by itself equals 1:
\[
\begin{aligned}
\frac{c\cdot m}{c\cdot n}
&=
\frac{c}{c}\cdot\frac{m}{n}\\
&=
1\cdot\frac{m}{n}\\
&=
\frac{m}{n}.
\end{aligned}
\]
This changes the form of the fraction, not the value.
8. Negative fractions
The negative sign can be placed in the numerator, denominator, or in front of the fraction:
\[
\begin{aligned}
\frac{-a}{b}
=
\frac{a}{-b}
=
-\frac{a}{b}.
\end{aligned}
\]
Standard form keeps the denominator positive:
\[
\begin{aligned}
\frac{45}{-60}
&=
-\frac{45}{60}
=
-\frac{3}{4}.
\end{aligned}
\]
9. Zero numerator
If the numerator is zero and the denominator is nonzero, the fraction equals zero:
\[
\begin{aligned}
\frac{0}{b}
&=0,
\qquad b\ne0.
\end{aligned}
\]
For example:
\[
\begin{aligned}
\frac{0}{18}
&=0.
\end{aligned}
\]
10. Improper fractions
Simplifying can also apply to improper fractions:
\[
\begin{aligned}
\frac{150}{90}
&=
\frac{150\div30}{90\div30}\\
&=
\frac{5}{3}.
\end{aligned}
\]
The simplified fraction may still be improper. Simplifying is different from converting to a mixed number.
11. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
12. Common mistakes
- Dividing only the numerator and forgetting to divide the denominator by the same number.
- Stopping too early before the fraction is fully reduced.
- Forgetting that the denominator cannot be zero.
- Leaving a negative sign in the denominator instead of moving it to the numerator or in front of the fraction.
- Confusing simplifying a fraction with converting it to a decimal or mixed number.
Key idea: divide numerator and denominator by their greatest common divisor to reduce the fraction to lowest terms.
