In real life, numbers appear in many forms. Foundation Mathematics requires you to read, interpret and convert between whole numbers, decimals, fractions, percentages, and ratios — and apply them to practical situations.
KEY TAKEAWAY: The same quantity can be written as a fraction, decimal, or percentage. Being fluent in all three forms lets you choose the most useful one for each context.
Used for counting exact quantities: number of items, population, pages.
| Place | Value |
|---|---|
| Tenths | $0.1$ |
| Hundredths | $0.01$ |
| Thousandths | $0.001$ |
Common contexts: money ($\$4.75$), measurements ($2.35\text{ m}$), fuel ($1.689\text{ L}$)
Converting decimals to fractions:
$\$0.6 = \frac{6}{10} = \frac{3}{5}, \quad 0.25 = \frac{25}{100} = \frac{1}{4}$$
Key equivalences:
| Fraction | Decimal | Percentage |
|---|---|---|
| $\frac{1}{2}$ | $0.5$ | $50\%$ |
| $\frac{1}{4}$ | $0.25$ | $25\%$ |
| $\frac{3}{4}$ | $0.75$ | $75\%$ |
| $\frac{1}{5}$ | $0.2$ | $20\%$ |
| $\frac{1}{3}$ | $0.\overline{3}$ | $33.3\%$ |
Percentages mean per hundred: $35\% = \frac{35}{100} = 0.35$
Finding a percentage of an amount:
$$15\%\ \text{of}\ \$240 = \frac{15}{100} \times 240 = 0.15 \times 240 = \$36$$
Finding what percentage one number is of another:
$$\frac{18}{24} \times 100 = 75\%$$
Percentage increase/decrease:
$$\text{New value} = \text{Original} \times \left(1 + \frac{\text{rate}}{100}\right)$$
$$\text{e.g. } \$80 \text{ increased by } 20\% = 80 \times 1.20 = \$96$$
EXAM TIP: For percentage decrease (e.g. a $15\%$ discount), multiply by $(1 - 0.15) = 0.85$. This is faster than finding $15\%$ and subtracting.
A ratio compares two or more quantities in the same units.
$$a : b \quad \text{read as “} a \text{ to } b \text{“}$$
Simplifying ratios: Divide both parts by their HCF.
$\$12 : 8 = 3 : 2$$
Using ratios to share quantities:
Share $\$150$ in the ratio \$2 : 3$.
Step 1: Total parts = \$2 + 3 = 5$
Step 2: Each part $= \$150 \div 5 = \$30$
Step 3: Shares are \$2 \times \$30 = \$60$ and \$3 \times \$30 = \$90$
Ratio vs fraction vs percentage:
$\$2 : 3 \implies \frac{2}{5} = 40\% \text{ and } \frac{3}{5} = 60\%$$
COMMON MISTAKE: Confusing a ratio \$2:3$ with a fraction $\frac{2}{3}$. In the ratio \$2:3$, the first part is $\frac{2}{5}$ of the total — not $\frac{2}{3}$.
| Context | Number Form Used |
|---|---|
| Bank interest rate | Percentage (e.g. $3.5\%$ p.a.) |
| Paint mixing | Ratio (e.g. \$3 : 1$ paint to thinner) |
| Cooking measurement | Fractions/decimals (e.g. $\frac{3}{4}$ cup) |
| Price per kg | Decimal (e.g. $\$4.69$ per kg) |
| GST on invoice | Percentage ($10\%$) |
VCAA FOCUS: Tasks often give one form (e.g. a fraction) and expect you to convert and apply it (e.g. as a percentage). Practice all conversions fluently.
