Comparing fractions means deciding which fractions are smaller, larger, or equal.
Ordering fractions means arranging them from least to greatest or from greatest to least.
1. What a fraction represents
A fraction has the form
\[
\begin{aligned}
\frac{a}{b},
\qquad b\ne0.
\end{aligned}
\]
The numerator \(a\) tells how many parts are counted, and the denominator \(b\) tells how many equal parts make one whole.
2. Simplifying before comparing
Simplifying a fraction does not change its value. It only writes the same number in a clearer form:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{a/\gcd(a,b)}{b/\gcd(a,b)}.
\end{aligned}
\]
For example:
\[
\begin{aligned}
\frac{6}{8}
&=
\frac{3}{4}.
\end{aligned}
\]
This matters because equivalent fractions such as \(2/4\), \(3/6\), and \(1/2\) should be recognized as equal.
3. Comparing fractions with the same denominator
If two fractions have the same denominator, compare their numerators:
\[
\begin{aligned}
\frac{a}{d}<\frac{b}{d}
\quad\Longleftrightarrow\quad
a
Example:
\[
\begin{aligned}
\frac{3}{8}<\frac{5}{8},
\end{aligned}
\]
because \(3<5\).
4. Common denominator method
Fractions with different denominators can be rewritten using a common denominator.
A good choice is the least common multiple of the denominators:
\[
\begin{aligned}
L&=\operatorname{LCM}(b_1,b_2,\ldots,b_n).
\end{aligned}
\]
Each fraction is rewritten with denominator \(L\):
\[
\begin{aligned}
\frac{a_i}{b_i}
&=
\frac{a_i\left(L/b_i\right)}{L}.
\end{aligned}
\]
Then all fractions have the same denominator, so the rewritten numerators determine the order.
5. Worked example: \(5/8\), \(7/11\), \(3/5\)
The denominators are \(8\), \(11\), and \(5\). Their least common multiple is:
\[
\begin{aligned}
\operatorname{LCM}(8,11,5)&=440.
\end{aligned}
\]
Rewrite the fractions:
\[
\begin{aligned}
\frac{5}{8}
&=
\frac{275}{440},\\
\frac{7}{11}
&=
\frac{280}{440},\\
\frac{3}{5}
&=
\frac{264}{440}.
\end{aligned}
\]
Now compare the numerators:
\[
\begin{aligned}
264<275<280.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\frac{3}{5}<\frac{5}{8}<\frac{7}{11}.
\end{aligned}
\]
6. Cross-multiplication method
To compare two fractions \(a/b\) and \(c/d\), compare \(ad\) and \(bc\):
\[
\begin{aligned}
\frac{a}{b}<\frac{c}{d}
\quad\Longleftrightarrow\quad
ad0,\ d>0.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{5}{8}\ ?\ \frac{7}{11}
\end{aligned}
\]
Cross multiply:
\[
\begin{aligned}
5\cdot11&=55,\\
7\cdot8&=56.
\end{aligned}
\]
Since \(55<56\), we get:
\[
\begin{aligned}
\frac{5}{8}<\frac{7}{11}.
\end{aligned}
\]
7. Why rounded decimals can be risky
Decimals are useful for checking intuition, but rounded decimals can hide the exact order.
For example, two fractions may both round to \(0.67\) even if one is slightly larger.
Exact fraction comparison avoids this problem because it uses integer arithmetic rather than rounded decimal approximations.
8. Negative fractions
Negative fractions follow the number line:
\[
\begin{aligned}
-\frac{3}{4}<-\frac{1}{2}<0<\frac{1}{3}.
\end{aligned}
\]
More negative values are smaller. A fraction closer to zero is greater than a fraction farther to the left.
9. Equivalent fractions
Two fractions are equivalent if they represent the same value:
\[
\begin{aligned}
\frac{a}{b}=\frac{c}{d}
\quad\Longleftrightarrow\quad
ad=bc.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{2}{4}
=
\frac{3}{6}
=
\frac{1}{2}.
\end{aligned}
\]
10. Mixed numbers
A mixed number should be converted to an improper fraction before comparing:
\[
\begin{aligned}
1\frac{2}{3}
&=
\frac{1\cdot3+2}{3}\\
&=
\frac{5}{3}.
\end{aligned}
\]
After conversion, it can be compared like any other fraction.
11. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
12. Common mistakes
- Comparing only numerators when denominators are different.
- Comparing only denominators without considering the numerator.
- Relying on rounded decimals when fractions are close together.
- Forgetting that negative fractions farther left on the number line are smaller.
- Not recognizing equivalent fractions such as \(2/4\) and \(1/2\).
- Using a denominator of zero, which is undefined.
Key idea: use common denominators or cross multiplication to compare exact fraction values without rounding.
