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.
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.
