Fractions and Percentages as Operators
Fractions and Percentages as Operators
Both fractions and percentages are operators — they act on a quantity to produce a result. When we say "three-fifths of 240" or "35% of £480", we are applying an operator to a value. This benchmark covers the full toolkit: finding fractions and percentages of quantities, percentage increase and decrease, expressing one quantity as a percentage of another, and reversing a percentage change to find an original value.
Fractions as Operators
The word "of" in mathematics means multiply. To find a fraction of a quantity, multiply the quantity by the fraction.
\[ \frac{3}{5} \text{ of } 240 = \frac{3}{5} \times 240 = \frac{720}{5} = 144 \]For a mixed number, convert to an improper fraction first. For a unit fraction (\( \frac{1}{n} \)), divide by \( n \); for other fractions, divide by the denominator then multiply by the numerator.
Percentages as Operators
Convert the percentage to a decimal (divide by 100), then multiply by the quantity. This makes percentage calculations consistent with fraction calculations.
\[ 35\% \text{ of } £480 = 0.35 \times 480 = £168 \]To express one quantity as a percentage of another: divide the part by the whole, then multiply by 100.
\[ \frac{20}{32} \times 100 = 62.5\% \]Percentage Multipliers
The most efficient method for percentage change is to use a single percentage multiplier. Instead of finding the increase/decrease separately and then adding/subtracting, one multiplication does everything:
| Percentage change | Calculation | Multiplier |
|---|---|---|
| 20% increase | 100% + 20% = 120% | \( \times 1.20 \) |
| 7% increase | 100% + 7% = 107% | \( \times 1.07 \) |
| 15% decrease | 100% − 15% = 85% | \( \times 0.85 \) |
| 40% decrease | 100% − 40% = 60% | \( \times 0.60 \) |
For a reverse percentage: divide the known (final) value by the multiplier to find the original. Never make the common error of finding a percentage of the final value — the percentage always applies to the original.
Worked Examples
Find \( \frac{5}{8} \) of 720 g, and find what percentage 45 is of 360.
\[ \frac{5}{8} \times 720 = \frac{3600}{8} = 450 \text{ g} \]Express 45 as a percentage of 360:
\[ \frac{45}{360} \times 100 = 12.5\% \]A TV costs £640. Its price increases by 12.5%. Find the new price. Then find the price if instead it decreased by 35%.
Increase 12.5%: multiplier = \( 1 + 0.125 = 1.125 \)
\[ £640 \times 1.125 = £720 \]Decrease 35%: multiplier = \( 1 - 0.35 = 0.65 \)
\[ £640 \times 0.65 = £416 \]After a 20% increase, a price is £156. Find the original price. After a 25% decrease, a price is £120. Find the original price.
Reverse 20% increase: multiplier = 1.20
\[ \text{Original} = £156 \div 1.20 = £130 \]Reverse 25% decrease: multiplier = 0.75
\[ \text{Original} = £120 \div 0.75 = £160 \]Key Takeaways
- "Of" means multiply: \( \frac{a}{b} \) of a quantity = \( \frac{a}{b} \times ext{quantity} \).
- Convert a percentage to a decimal (÷ 100) before multiplying; to express as a percentage, divide then multiply by 100.
- Percentage multiplier for an increase of \( n\% \): multiply by \( 1 + \frac{n}{100} \). For a decrease: multiply by \( 1 - \frac{n}{100} \).
- To reverse a percentage change: divide the final value by the multiplier — never take a percentage of the final value.
- Two successive percentage changes are combined by multiplying their multipliers: +10% then −10% is \( 1.10 \times 0.90 = 0.99 \) (a net 1% decrease, not zero).