Percentages in context

What you will learn

increase and decrease a quantity by a percentage in one step,
solve reverse problems (“find the original after a $20\%$ rise”),
work with profit, loss, mark-ups, discounts, and GST,
calculate percentage error between an estimate and an actual value,
recognise when percentages compound rather than simply adding.

Worked example 0 Real-world example: sale price with GST

A jacket is marked $120, on sale for $25\%$ off. After the discount, $10\%$ GST is added. What do you actually pay?

Discount: $120 \times 0.75 = 90$ dollars.
Add GST: $90 \times 1.10 = 99$ dollars.
You pay $99, not $120 with ” $25\% - 10\% = 15\%$ off.”

Key idea: the discount and GST apply to different bases ($120 then $90), so you cannot just subtract the percentages. Percentage changes compound — they must be applied one step at a time.

1. Percentage increase and decrease

Percentage change in one step

Increase by p percent

\text{new} = A \times \left(1 + \dfrac{p}{100}\right).

Decrease by p percent

\text{new} = A \times \left(1 - \dfrac{p}{100}\right).

Reverse problem: find the original

If the new value is given after a change of $p\%$ , divide by the bracket:

A = \dfrac{\text{new}}{1 \pm \dfrac{p}{100}}.

Use $+$ for an increase, $-$ for a decrease.

Worked example 1 Straightforward increase

A bike costs $320 and its price rises $5\%$ . Find the new price.

320 \times 1.05 = 336.

New price: $336.

Worked example 2 Reverse: find the original

A shirt is sold for $42 after a $20\%$ mark-up. Find the original cost.

A = \dfrac{42}{1.20} = 35.

Original cost: $35.

2. Profit, loss, mark-ups, discounts

Mark-up (profit): add a percentage to the cost price to get the selling price.
Discount: subtract a percentage from the marked price.
Profit = selling price − cost price. Loss = cost price − selling price when the cost exceeds the selling price.

Worked example 3 Profit as a percentage

A shopkeeper buys a chair for $80 and sells it for $100. Find the profit and express it as a percentage of the cost.

Profit: $100 - 80 = 20$ dollars.
Percentage profit: $\dfrac{20}{80} \times 100 = 25\%$ .

3. GST (Goods and Services Tax)

In Australia, GST is $10\%$ added to most sales. To find the GST-included price:

\text{GST-inc.} = \text{pre-GST} \times 1.10.

To pull GST out of a GST-included price:

\text{pre-GST} = \dfrac{\text{GST-inc.}}{1.10}, \qquad \text{GST} = \dfrac{\text{GST-inc.}}{11}.

Worked example 4 GST in reverse

A receipt shows a total of $66 (GST included). Find the GST.

\text{GST} = \dfrac{66}{11} = 6 \text{ dollars}.

(Pre-GST would be $60$ ; then $60 \times 1.10 = 66$ .)

4. Percentage error

When an estimate or a measurement is compared with an actual (or accepted) value, the percentage error describes the relative size of the gap.

Percentage error

\text{error }\% = \dfrac{|\text{estimate} - \text{actual}|}{\text{actual}} \times 100.

Absolute value so the error is always positive.

Worked example 5 Estimate vs actual

A student estimates a fence is $25$ m long. The true length is $28$ m. Find the percentage error.

\dfrac{|25 - 28|}{28} \times 100 = \dfrac{3}{28} \times 100 \approx 10.7\%.

Practice: Year 8 core

Fluency

Increase and decrease

Increase $80$ by $25\%$ .
Decrease $150 by $12\%$ .
Increase $85 by $20\%$ .
Decrease $250$ by $6\%$ .
A jacket is $85, discounted by $20\%$ . Find the sale price.
A subscription is $120 and rises $8\%$ . Find the new price.

Fluency

Reverse problems

After a $15\%$ increase, a price is $46. Find the original.
After a $25\%$ discount, a shirt costs $36. Find the original price.
A salary increased by $5\%$ to $63,000. Find the old salary.
A population fell by $8\%$ to $9200$ . Find the original population.

Fluency

Profit, loss, GST

Buy at $60, sell at $75. Percentage profit?
Buy at $200, sell at $170. Percentage loss?
A price is $49 before GST ( $10\%$ ). What is the GST-included price?
A total (GST-inc.) is $77. How much GST is in it?
A meal costs $55 GST-inc. How much is the pre-GST price?

Reasoning

Percentage error & explain

A scientist measures a rod as $12.3$ cm; the true length is $12.0$ cm. Find the percentage error.
A shop advertises “was $100, now $75”. What percentage discount is this?
Sam says a $30\%$ rise then a $30\%$ fall brings the price back to the start. Show with a worked example whether this is true.
Explain the difference between ” $20\%$ off then $10\%$ off” and a single ” $30\%$ off”.

Problem solving

Real contexts

A laptop’s sticker price is $1400. In a sale it is reduced $15\%$ . After the sale the shop also adds GST ( $10\%$ ). What does the customer pay?
Mira earns $800 per week. She gets a $4\%$ rise, then six months later a further $3\%$ rise. What does she earn per week now?
A town’s population rose from $12\,500$ to $14\,000$ over five years. What was the percentage increase?
A phone’s retail price includes $10\%$ GST and is $1089. How much GST is included in the price?
A supermarket sells a $500$ g jar for $5.50 and a $750$ g jar for $7.70. Which is better value, and by what percentage is the better-value jar cheaper per gram?

Challenge

Reasoning

Harder reasoning

A value rises by $r\%$ and then falls by $r\%$ . Show that the end value is always less than the starting value (for $r > 0$ ), and find a formula for the net percentage change.
A shop marks up cost by $25\%$ and then gives a $20\%$ discount from the marked price. Is the final price above or below cost? By how much?
The true area of a rectangle is $24$ cm². A student estimates the area as $27$ cm² by rounding the sides up. What is the percentage error?
A bank account earns $6\%$ interest per year, compounded yearly. If the starting balance is $500, find the balance after $3$ years.