Fractions, decimals, and percentages are three different ways to write the same value.
For example, \(3/4\), \(0.75\), and \(75\%\) all represent the same amount.
1. Fraction form
A fraction is a ratio:
\[
\begin{aligned}
\frac{a}{b},
\qquad b\ne0.
\end{aligned}
\]
The numerator \(a\) tells how many parts are selected, and the denominator \(b\) tells how many equal parts make one whole.
2. Decimal form
A decimal writes a number using place value. To convert a fraction to a decimal, divide the numerator by the denominator:
\[
\begin{aligned}
\frac{a}{b}
&=
a\div b.
\end{aligned}
\]
Example:
\[
\begin{aligned}
\frac{3}{4}
&=
3\div4\\
&=
0.75.
\end{aligned}
\]
3. Percentage form
A percentage means “out of 100.” To convert a decimal to a percentage, multiply by 100:
\[
\begin{aligned}
\text{percent}
&=
\text{decimal}\times100\%.
\end{aligned}
\]
Example:
\[
\begin{aligned}
0.75
&=
75\%.
\end{aligned}
\]
4. Converting a decimal to a fraction
To convert a terminating decimal to a fraction, place the decimal number over a power of 10 and simplify.
\[
\begin{aligned}
0.75
&=
\frac{75}{100}\\
&=
\frac{3}{4}.
\end{aligned}
\]
The number of decimal places determines the power of 10.
Two decimal places means denominator \(100\).
5. Converting a percentage to a fraction
Since percent means “per 100,” divide the percentage number by 100:
\[
\begin{aligned}
75\%
&=
\frac{75}{100}\\
&=
\frac{3}{4}.
\end{aligned}
\]
Another example:
\[
\begin{aligned}
12.5\%
&=
\frac{12.5}{100}\\
&=
\frac{125}{1000}\\
&=
\frac{1}{8}.
\end{aligned}
\]
6. Converting a fraction to a percentage
To convert a fraction to a percentage, first divide to get a decimal, then multiply by 100:
\[
\begin{aligned}
\frac{3}{4}
&=
0.75,\\
0.75\times100\%
&=
75\%.
\end{aligned}
\]
Directly:
\[
\begin{aligned}
\frac{3}{4}\times100\%
&=
75\%.
\end{aligned}
\]
7. Terminating and recurring decimals
Some fractions produce terminating decimals:
\[
\begin{aligned}
\frac{1}{4}
&=
0.25.
\end{aligned}
\]
Some fractions produce recurring decimals:
\[
\begin{aligned}
\frac{1}{3}
&=
0.333\ldots
\end{aligned}
\]
A reduced fraction has a terminating decimal only when its denominator has no prime factors except 2 and 5.
8. Mixed numbers
A mixed number such as \(1\frac{1}{2}\) can be converted to an improper fraction first:
\[
\begin{aligned}
1\frac{1}{2}
&=
\frac{1\cdot2+1}{2}\\
&=
\frac{3}{2}.
\end{aligned}
\]
Then:
\[
\begin{aligned}
\frac{3}{2}
&=
1.5
=
150\%.
\end{aligned}
\]
9. Real-life examples
Fractions, decimals, and percentages are common in everyday contexts:
- Grades: \(75\%\) means 75 points out of 100.
- Discounts: \(25\%\) off means \(1/4\) of the price is removed.
- Recipes: \(0.5\) of a cup means \(1/2\) cup.
- Probability: \(0.2\) means \(1/5\), or \(20\%\).
10. Formula summary
The table below uses plain text formulas in table cells to avoid raw LaTeX rendering problems.
11. Common mistakes
- Forgetting that percent means “out of 100.”
- Writing \(75\%\) as \(75\) instead of \(0.75\).
- Forgetting to simplify a fraction after converting from a decimal or percentage.
- Assuming every fraction has a terminating decimal.
- Rounding too early when doing a multi-step calculation.
- Using a denominator of zero, which is undefined.
Key idea: a fraction, decimal, and percentage can represent the same value; only the format changes.
