Fractions, decimals and percentages

Fraction essentials

Simplify by dividing top and bottom by their highest common factor: $\frac{35}{42} = \frac{5}{6}$. Convert both ways between improper fractions and mixed numerals: $\frac{41}{4} = 10\frac{1}{4}$ and $5\frac{6}{11} = \frac{61}{11}$. The reciprocal of a fraction flips it — but write a mixed numeral as an improper fraction first, so the reciprocal of $3\frac{1}{2} = \frac{7}{2}$ is $\frac{2}{7}$.

Operating with fractions

• Add or subtract with a common denominator: $\frac{1}{3} + \frac{2}{9} = \frac{3}{9} + \frac{2}{9} = \frac{5}{9}$.
• Multiply: cancel common factors first, then multiply tops and bottoms.
• Divide: keep the first fraction, change $\div$ to $\times$, flip the second.

Decimal calculations

Add and subtract with the decimal points lined up. When multiplying, count decimal places: $0.2 \times 0.5 = 0.1$ (two places in the question, two in $0.10$). To divide by a decimal, shift both numbers until the divisor is whole: $96 \div 1.2 = 960 \div 12 = 80$.

Converting is routine exam territory: $\frac{35}{8} = 4.375$ by division; $\frac{2}{11} = 0.\dot{1}\dot{8}$ as a recurring decimal; $0.56 = \frac{56}{100} = \frac{14}{25}$ going the other way.

Percentage conversions

Per cent means out of 100, which links all three forms: $20\% = \frac{20}{100} = \frac{1}{5} = 0.2$. To write one amount as a percentage of another, make a fraction and multiply by 100: 20 out of 50 is $\frac{20}{50} \times 100 = 40\%$.

Percentage problems

• Percentage of a quantity: $13\%$ of 4 dollars is $0.13 \times 400$ cents $= 52$ cents.
• Increase or decrease: decreasing 360 by $20\%$ leaves $80\%$, so $360 \times 0.8 = 288$.
• Percentage change: $\frac{\text{change}}{\text{original}} \times 100$. Growing from 80 cm to 100 cm is $\frac{20}{80} \times 100 = 25\%$.
• Reverse problems: if $2\%$ of a number is 8, then $1\%$ is 4, so the number is 400.