Calculate percentages, percentage change and reverse percentages, and apply them to profit, loss, discount, taxation and simple and compound interest

What this dot point is asking

SEAB wants you to find percentages of quantities, calculate percentage increase and decrease, work backwards from a final amount in reverse-percentage problems, and apply all of this to money, profit and loss, discount, tax, and both simple and compound interest. Money questions reward careful identification of which amount the percentage is taken of.

The answer

Percentage of a quantity

A percentage is a fraction out of $100$. To find a percentage of an amount, convert to a decimal and multiply: $15\%$ of $240 = 0.15 \times 240 = 36$.

Percentage increase and decrease

The most efficient method uses a multiplier. To increase by $20\%$, multiply by $1.20$; to decrease by $20\%$, multiply by $0.80$. To express a change as a percentage:

The denominator is always the original value.

Reverse percentages

When you are given the amount after a change and must find the original, divide by the multiplier rather than applying the percentage to the new figure. If a price after a $30\%$ increase is $260$ dollars, the original is $\dfrac{260}{1.30} = 200$ dollars.

Profit, loss and discount

Profit and loss are measured against the cost price:

A discount is a percentage reduction off the marked price, so a $25\%$ discount means paying $75\%$ of the marked price.

Simple and compound interest

Simple interest is the same each year, calculated on the original principal:

Compound interest is added to the balance each period, so later interest is earned on earlier interest. The amount after $n$ years at rate $r$ percent is:

Examples in context

Example 1. Comparing savings accounts. An account paying compound interest grows faster than one paying the same rate as simple interest, because each year the compound balance is larger. Over many years the gap widens noticeably, which is why long-term saving favours compounding.

Example 2. A sale price with tax. A marked price reduced by a sale percentage and then increased by a service or sales tax involves two multipliers applied in turn. Keeping the multipliers separate and applying them to the correct base avoids the common error of combining them carelessly.

Try this

Q1. Increase $250$ dollars by $12\%$. [1 mark]

• Cue. Multiply by $1.12$: $250 \times 1.12 = 280$ dollars.

Q2. A price after a $20\%$ discount is $96$ dollars. Find the original price. [2 marks]

• Cue. Divide by the multiplier $0.80$: $\dfrac{96}{0.80} = 120$ dollars.

Q3. Find the simple interest on $1500$ dollars at $4\%$ per year for $3$ years. [2 marks]

• Cue. $I = \dfrac{1500 \times 4 \times 3}{100} = 180$ dollars.