3.3 Percentages: Finding Percentages and Expressing Quantities as Percentages

The term per cent ($\%$) means "per hundred" or "out of $100$". Every percentage $p\%$ is simply another way of writing the fraction $\frac{p}{100}$ or the decimal $p \div 100$. In Australian schools, percentages are used to report assessment scores, attendance benchmarks, demographic breakdowns, and budget allocations.

Fluency Across the Fraction–Decimal–Percentage Triangle

Moving rapidly between fractions, decimals, and percentages requires two core operations—multiplying by $100\%$ and dividing by $100\%$:

• Percentage $\rightarrow$ Decimal: Divide by $100$ (shift the decimal point $2$ places left) and remove the $\%$ sign. For example, $37.5\% = 0.375$.
• Decimal $\rightarrow$ Percentage: Multiply by $100\%$ (shift the decimal point $2$ places right) and attach the $\%$ sign. For example, $0.085 = 8.5\%$.
• Percentage $\rightarrow$ Fraction: Write the percentage over $100$ and simplify to lowest terms. If the percentage contains a decimal, multiply numerator and denominator by $10$ or $100$ first: $12.5\% = \frac{12.5}{100} = \frac{125}{1000} = \frac{1}{8}$.
• Fraction $\rightarrow$ Percentage: Either scale the denominator to $100$ (for example, $\frac{17}{25} = \frac{68}{100} = 68\%$) or multiply the fraction by $100\%$ (for example, $\frac{3}{8} \times 100\% = \frac{300}{8}\% = 37.5\%$).

Percentages Greater Than $100\%$ and Less Than $1\%$

Two boundary cases frequently catch candidates off guard:

1. Percentages greater than $100\%$: A percentage above $100\%$ indicates a quantity larger than the baseline whole. For instance, if a rapidly growing outer-suburban school records current enrolments at $135\%$ of its five-year-ago baseline of $600$ students, its current enrolment is $1.35 \times 600 = 810$ students.
2. Percentages less than $1\%$: When a percentage is already less than $1\%$, candidates often forget to divide by $100$ because they see a decimal point and assume it is already in decimal form. Remember that $1\% = 0.01$, so a sub-$1\%$ rate must be smaller than $0.01$:

$$0.25\% = \frac{0.25}{100} = 0.0025 = \frac{25}{10000} = \frac{1}{400}$$

Never write $0.25\%$ as $0.25$ (which is $25\%$) or $0.025$ (which is $2.5\%$). Similarly, $0.4\% = 0.004$.

Finding $p\%$ of a Quantity

Section 1 (Calculator): Decimal Multipliers

When a calculator is available, convert $p\%$ to its decimal multiplier $\frac{p}{100}$ and multiply directly by the quantity. To find $18.5\%$ of a $14,600 curriculum budget (in dollars):

$$0.185 \times 14{,}600 = 2{,}701$$

Section 2 (Non-Calculator): Benchmark Decomposition

Without a calculator, break the target percentage into simple mental building blocks:

• $50\% = \frac{1}{2}$: Divide the quantity by $2$.
• $25\% = \frac{1}{4}$: Divide the quantity by $4$ (halve twice).
• $12.5\% = \frac{1}{8}$: Divide the quantity by $8$ (halve three times).
• $10\% = \frac{1}{10}$: Divide by $10$ (shift the decimal point one place left).
• $5\% = \frac{1}{20}$: Take half of $10\%$.
• $1\% = \frac{1}{100}$: Divide by $100$ (shift the decimal point two places left).

Worked Example: Suppose $35\%$ of $240$ Year 8 students participate in the school choir or band. To find $35\%$ of $240$ mentally:

• $10\%$ of $240 = 240 \div 10 = 24$
• $30\%$ of $240 = 3 \times 24 = 72$
• $5\%$ of $240 = 24 \div 2 = 12$
• $35\%$ of $240 = 72 + 12 = 84\text{ students}$

Similarly, to find $17.5\%$ of $160$, add $10\%$ ($16$), $5\%$ ($8$), and $2.5\%$ ($4$) to get $16 + 8 + 4 = 28$. To find $98\%$ of $350$, subtract $2\%$ from $100\%$: since $1\%$ of $350 = 3.5$, $2\%$ is $7$, giving $350 - 7 = 343$.

Expressing One Quantity as a Percentage of Another

To express a part as a percentage of a whole, form the fraction $\frac{\text{Part}}{\text{Whole}}$ and multiply by $100\%$:

$$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%$$

The Same-Units Rule

Before writing the fraction $\frac{\text{Part}}{\text{Whole}}$, verify that both quantities are measured in the exact same unit. Mixing minutes with hours, grams with kilograms, millilitres with litres, or cents with dollars produces an invalid ratio.

Worked Example 1 (Time): A primary school teacher spends $45\text{ minutes}$ of a $2.5\text{-hour}$ morning teaching block on explicit phonics instruction. What percentage of the morning block is spent on phonics?

1. Convert both quantities to the same unit (minutes): $2.5\text{ hours} = 2.5 \times 60 = 150\text{ minutes}$. (Never compute $\frac{45}{2.5}$!)
2. Form the fraction and convert to a percentage:

$$\frac{45\text{ minutes}}{150\text{ minutes}} \times 100\% = \frac{3}{10} \times 100\% = 30\%$$

Worked Example 2 (Volume and Currency): Expressing $350\text{ mL}$ as a percentage of a $1.4\text{ L}$ jug requires converting $1.4\text{ L}$ to $1400\text{ mL}$, giving $\frac{350}{1400} \times 100\% = \frac{1}{4} \times 100\% = 25\%$. Likewise, a $75\text{-cent}$ fundraising levy out of a $3.00 ($300\text{-cent}$) ticket price is $\frac{75}{300} \times 100\% = 25\%$.

Nested Percentages: "Percentage of a Percentage"

Many LANTITE questions describe a subgroup within a subgroup—a percentage of a percentage. To find the subgroup's share of the entire population, multiply the decimal equivalents:

$$\text{Overall Share} = \left(\frac{p_1}{100}\right) \times \left(\frac{p_2}{100}\right)$$

Worked Example: At a secondary school with $850$ students, $60\%$ of the students catch the school bus. Of those bus riders, $15\%$ live more than $10\text{ km}$ from the school.

• Percentage of all school students who catch the bus and live over $10\text{ km}$ away:

$$0.60 \times 0.15 = 0.09 = 9\%$$

Notice that $60\%$ of $15\%$ is $9\%$, not $75\%$ (adding) or $45\%$ (subtracting). If instead $20\%$ of the bus riders live over $10\text{ km}$ away, that is $0.60 \times 0.20 = 0.12 = 12\%$ of the school, or $0.12 \times 850 = 102$ students.