Find a percentage of a quantity, express one quantity as a percentage of another, calculate percentage change, and solve money problems including discount, profit and simple interest

What this dot point is asking

SEAB wants you to find a percentage of a quantity, express one quantity as a percentage of another, calculate percentage increases and decreases, and apply these to money problems such as discounts, profit and loss, and simple interest. Percentages are everywhere in real life, so this is one of the most useful topics in the syllabus.

The answer

What percentage means

"Per cent" means "per hundred", so $25\%$ means $\dfrac{25}{100} = 0.25$. To use a percentage in a calculation, convert it to a fraction or a decimal first.

Finding a percentage of a quantity

Multiply the quantity by the percentage written as a fraction or decimal:

Expressing one quantity as a percentage of another

Write the two quantities as a fraction, then multiply by $100$. For example, a student who scores $18$ out of $20$ achieves:

Percentage change

Percentage change compares the change with the original amount:

If a price rises from \40$to$\$50$, the change is \10$and the percentage increase is$\dfrac{10}{40} \times 100 = 25%$.

Money problems

• Discount. A reduction in price. New price equals original minus discount.
• Profit and loss. Profit is selling price minus cost price; a loss is a negative profit. Profit percentage is usually based on the cost price.
• Simple interest. Interest paid on the original amount (the principal) only. For each year the interest is the same, so total interest equals the one-year interest multiplied by the number of years.

Examples in context

Example 1. A restaurant service charge. A meal costs \60$and a$10%$service charge is added. The charge is$\dfrac{10}{100} \times 60 = \$6$, so the total bill is \66$. The same answer comes from finding$110%$of$\$60$ in one step, which is a handy shortcut for increases.

Example 2. Comparing two test scores. A student scores $24$ out of $40$ in one test and $30$ out of $50$ in another. As percentages these are $60\%$ and $60\%$, so the performances are equal even though the raw marks differ. Converting to percentages makes scores from different totals comparable.

Try this

• Cue. Find $15\%$ of \120$. Compute$\dfrac{15}{100} \times 120 = \$18$.
• Cue. A price falls from \50$to$\$45$. The change is \5$, so the percentage decrease is$\dfrac{5}{50} \times 100 = 10%$.
• Cue. Find the simple interest on \800$at$5%$per year for$2$years. One year is$\$40$, so two years give \80$.