Equivalent fractions are fractions that look different but represent the same value.
They describe the same point on a number line and the same portion of a whole.
1. Basic idea
A fraction has the form:
\[
\begin{aligned}
\frac{a}{b},
\qquad b\ne0.
\end{aligned}
\]
To generate an equivalent fraction, multiply both the numerator and the denominator by the same nonzero number:
\[
\begin{aligned}
\frac{a}{b}
&=
\frac{a\cdot k}{b\cdot k},
\qquad k\ne0.
\end{aligned}
\]
This works because \(k/k=1\), and multiplying by 1 does not change the value.
2. Worked example: \(3/5\)
Starting with \(3/5\), multiply numerator and denominator by 2:
\[
\begin{aligned}
\frac{3}{5}
&=
\frac{3\cdot2}{5\cdot2}\\
&=
\frac{6}{10}.
\end{aligned}
\]
Multiply by 3:
\[
\begin{aligned}
\frac{3}{5}
&=
\frac{3\cdot3}{5\cdot3}\\
&=
\frac{9}{15}.
\end{aligned}
\]
Multiply by 4:
\[
\begin{aligned}
\frac{3}{5}
&=
\frac{3\cdot4}{5\cdot4}\\
&=
\frac{12}{20}.
\end{aligned}
\]
Therefore:
\[
\begin{aligned}
\frac{3}{5}
=
\frac{6}{10}
=
\frac{9}{15}
=
\frac{12}{20}.
\end{aligned}
\]
3. Why the value stays the same
Multiplying by \(k/k\) means multiplying by 1:
\[
\begin{aligned}
\frac{a}{b}\cdot\frac{k}{k}
&=
\frac{a}{b}\cdot1\\
&=
\frac{a}{b}.
\end{aligned}
\]
The number of pieces changes, but the total amount represented stays the same.
4. Fraction wall interpretation
In a fraction wall, \(3/5\) and \(6/10\) can fill the same length.
The denominator 10 divides the whole into smaller pieces than denominator 5, but the filled portion is unchanged.
\[
\begin{aligned}
\frac{3}{5}
=
\frac{6}{10}.
\end{aligned}
\]
5. Simplified fraction and equivalent fractions
A simplified fraction is the lowest-terms representative of all equivalent fractions.
For example:
\[
\begin{aligned}
\frac{6}{8}
&=
\frac{3}{4}.
\end{aligned}
\]
From \(3/4\), we can generate equivalent fractions:
\[
\begin{aligned}
\frac{3}{4}
=
\frac{6}{8}
=
\frac{9}{12}
=
\frac{12}{16}.
\end{aligned}
\]
6. Cross-product check
Two fractions \(a/b\) and \(c/d\) are equivalent if their cross products are equal:
\[
\begin{aligned}
\frac{a}{b}
=
\frac{c}{d}
\quad\Longleftrightarrow\quad
ad=bc.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{3}{5}
&=
\frac{12}{20}
\end{aligned}
\]
because:
\[
\begin{aligned}
3\cdot20
&=
12\cdot5\\
60&=60.
\end{aligned}
\]
7. Negative equivalent fractions
Negative fractions also have equivalent forms:
\[
\begin{aligned}
-\frac{4}{7}
=
-\frac{8}{14}
=
-\frac{12}{21}.
\end{aligned}
\]
The negative sign stays with the whole value. The denominator is usually kept positive.
8. Zero numerator
If the numerator is zero and the denominator is nonzero, the value is zero:
\[
\begin{aligned}
\frac{0}{9}=0.
\end{aligned}
\]
Equivalent fractions can be generated by multiplying numerator and denominator by the same nonzero number:
\[
\begin{aligned}
\frac{0}{9}
=
\frac{0}{18}
=
\frac{0}{27}.
\end{aligned}
\]
9. Why multiplier zero is not allowed
The multiplier \(k\) must not be zero:
\[
\begin{aligned}
k\ne0.
\end{aligned}
\]
If \(k=0\), the new denominator would become zero:
\[
\begin{aligned}
b\cdot0=0,
\end{aligned}
\]
and fractions with denominator zero are undefined.
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Multiplying only the numerator and forgetting the denominator.
- Multiplying the numerator and denominator by different numbers.
- Using multiplier zero, which creates denominator zero.
- Thinking that equivalent fractions are different values instead of different forms of the same value.
- Forgetting to keep the denominator positive in standard form.
Key idea: equivalent fractions are made by multiplying or dividing the numerator and denominator by the same nonzero number.
