Converting Percent
Fractions & DecimalsConverting percent means changing a percent to a decimal or fraction, or changing a decimal or fraction to a percent.
Formula
\text{percent to decimal: divide by 100; decimal to percent: multiply by 100}
Definition
Converting a percent means changing it into a fraction or a decimal. To change a percent to a decimal, divide by $100$ (move the decimal point two places to the left). To change a decimal to a percent, multiply by $100$ (move the decimal point two places to the right).
Example
$45\%$ to decimal: $45/100 = 0.45$. Decimal to percent: $0.72$ to percent $= 0.72 \times 100 = 72\%$. Fraction to percent: $3/4 = 0.75 = 75\%$.
Key Insight
Moving the decimal point two places is the quick shortcut, but understanding why helps: percent means "per hundred," so $45\% = 45/100$, and dividing by $100$ shifts the decimal point two places left.
Definition
Conversion rules: (1) Percent to decimal: $p\% = p/100$; shift decimal left $2$. (2) Decimal to percent: $d = d \times 100\%$; shift decimal right $2$. (3) Fraction to percent: $a/b \times 100\%$. (4) Percent to fraction: $p\% = p/100$, then simplify. Repeating decimal percents (like $33.333\ldots\%$) arise from fractions with prime factors other than $2$ and $5$ in the denominator.
Example
$37.5\%$ to fraction: $37.5/100 = 375/1000 = 3/8$. Fraction $5/6$ to percent: $5/6 \times 100 = 500/6 = 83.333\ldots\% = 83$ and $1/3\%$. Convert $0.008$ to percent: $0.008 \times 100 = 0.8\%$.
Key Insight
Percent-to-fraction conversion sometimes produces a non-terminating decimal in the percent. $1/3 = 33.333\ldots\%$ - this does not mean there is anything wrong with $1/3$; it means $1/3$ cannot be expressed as a nice round percent because $3$ does not evenly divide $100$.
Definition
Percent conversion is an isomorphism between $(\mathbb{Q}, \times)$ scaled by $100$ and its unscaled form: $f: \mathbb{Q} \to \mathbb{Q}$ by $f(x) = x/100$, with inverse $f^{-1}(x) = 100x$. In applied contexts, unit conversion between percent and decimal is essential in probability (converting between probability and percent chance), statistics (relative frequency), and finance (interest rates). The conversion is trivial but the choice of representation affects intuition and error rates.
Example
A probability $p = 0.034$ expressed as a percent is $3.4\%$. The percent form is more intuitive for communication but the decimal form is required for arithmetic: $P(A \text{ and } B) = p_A \times p_B = 0.034 \times 0.21 = 0.00714 = 0.714\%$. Mixing percent and decimal forms in calculations is a common error in science.
Key Insight
The convention of using percent rather than raw probability has cognitive roots: humans are better at reasoning about "$34$ out of $1000$" than "$0.034$." Gigerenzer's research on statistical literacy shows that natural frequency representations ($34$ in $1000$) produce far fewer reasoning errors than equivalent percent or probability statements - a practical argument for percent as a teaching tool.