Understanding Percentages
What is a Percentage?
The word "percent" literally means "per hundred". When we talk about 45%, we are really talking about the fraction \(\frac{45}{100}\) or the decimal \(0.45\).
1. Decimal to Percentage (and back)
This is the easiest conversion because it only involves moving the decimal point.
Decimal \(\rightarrow\) Percentage: Multiply by 100 (move decimal 2 places right).
Example: \(0.34 \times 100 = 34\%\)
Example: \(1.2 \times 100 = 120\%\)
Percentage \(\rightarrow\) Decimal: Divide by 100 (move decimal 2 places left).
Example: \(85\% \div 100 = 0.85\)
Example: \(4.5\% \div 100 = 0.045\)
2. Fraction to Percentage
Method A: Change the denominator to 100.
\(\frac{3}{20} = \frac{15}{100} = 15\%\)
Method B: Divide the top by the bottom, then multiply by 100.
\(\frac{5}{8} = 5 \div 8 = 0.625 \rightarrow 62.5\%\)
3. Common Equivalence Table
These are the core values you should memorize for exams:
| Fraction |
Decimal |
Percentage |
| \(\frac{1}{2}\) | 0.5 | 50% |
| \(\frac{1}{4}\) | 0.25 | 25% |
| \(\frac{3}{4}\) | 0.75 | 75% |
| \(\frac{1}{5}\) | 0.2 | 20% |
| \(\frac{1}{10}\) | 0.1 | 10% |
| \(\frac{1}{3}\) | \(0.\dot{3}\) | \(33.\dot{3}\%\) |
| \(\frac{2}{3}\) | \(0.\dot{6}\) | \(66.\dot{6}\%\) |
| \(\frac{1}{8}\) | 0.125 | 12.5% |
4. Percentage to Fraction
Put the percentage value over 100 and simplify.
Example: \(35\% = \frac{35}{100}\)
Divide both by 5 \(\rightarrow \frac{7}{20}\)
Example: \(125\% = \frac{125}{100} = \frac{5}{4} = 1\frac{1}{4}\)