Calculating Markups, Discounts, and Unit Rates

Hello, and welcome to the first lesson in your exam-revision course. Over the next short lessons, you will build the General Mathematics skills needed for your assessment, alongside PE and Biology revision. This first module focuses on practical consumer maths: the kind used in shopping, sport-program budgets, equipment orders, and everyday financial decisions.

Today’s goal is to calculate percentage markups, discounts, and unit rates, including questions where more than one change is made. By the end, you should be able to set up a clear calculator expression rather than guessing which amounts to add or subtract.

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Percentages are multipliers, not just numbers to subtract

A percentage change is always based on a starting amount, sometimes called the base amount. Before calculating, identify what the percentage is acting on.

A markup is a percentage increase. A discount is a percentage decrease.

The most reliable exam method is to turn each percentage into a multiplier:

Change	Multiplier
$p\%$ markup or increase	$1+\frac{p}{100}$
$p\%$ discount or decrease	$1-\frac{p}{100}$
$10\%$ GST added	$1.10$

For example, a sports store marks up a volleyball costing AUD 84 by $25\%$.

The $25\%$ markup means the new price is $125\%$ of the original price. As a multiplier, $125\%$ is $1.25$.

$$84\times 1.25=105$$

So the selling price is AUD 105.

You could work out the markup separately:

$$0.25\times 84=21$$ $$84+21=105$$

That method is correct. However, using the multiplier $1.25$ is faster and becomes much safer when a question has several steps.

For a discount, use the percentage left after the discount. If knee pads cost AUD 68 and are discounted by $30\%$, the customer pays $70\%$ of the original price.

$$68\times 0.70=47.60$$

The sale price is AUD 47.60.

A quick reasonableness check helps:

• After a markup, the final price should be higher than the original price.
• After a discount, the final price should be lower than the original price.
• A large discount should produce a noticeably smaller answer.

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Multistep questions: apply each change to the current amount

In consumer arithmetic, an item may be marked up, then discounted, then have GST added. Each new percentage change acts on the current price, not always on the original price.

Suppose a retailer buys a volleyball for AUD 18. It applies a $35\%$ markup, offers a $20\%$ sale discount, and then adds GST. Assume all prices in the question are stated as excluding GST.

Write one multiplier for each change:

• $35\%$ markup: $1.35$
• $20\%$ discount: $0.80$
• $10\%$ GST: $1.10$

$$\text{Final price}=18\times 1.35\times 0.80\times 1.10$$ $$\text{Final price}=21.384$$

Money is rounded to the nearest cent at the end:

$$\text{Final price}=\text{AUD }21.38$$

In an exam response, do not just write the calculator answer. Show the factors so the marker can see what each percentage means.

A strong working layout is:

$$\text{Price after markup}=18\times 1.35=24.30$$ $$\text{Price after discount}=24.30\times 0.80=19.44$$ $$\text{Price including GST}=19.44\times 1.10=21.384$$ $$\therefore \text{Final price}=\text{AUD }21.38$$

You can use either the step-by-step layout or one calculator line. For an exam, the step-by-step method is easier to check and less likely to go wrong.

Why equal percentage changes do not cancel

A common trap is thinking that a $10\%$ markup followed by a $10\%$ discount returns an item to its original price. It does not.

$$\text{Final price}=\text{Original price}\times 1.10\times 0.90$$ $$1.10\times 0.90=0.99$$

So the final price is $99\%$ of the original price: it is actually $1\%$ lower.

The reason is that the discount is calculated from the higher, marked-up price, not from the original price.

GST in Australian questions

GST is a $10\%$ increase, so:

$$\text{Price including GST}=\text{Price excluding GST}\times 1.10$$

If a question says a price is exclusive of GST, multiply by $1.10$. If it says a price includes GST, GST has already been added, so do not add it again.

If you need to find a pre-GST price from a GST-inclusive price, divide by $1.10$, rather than subtracting $10\%$. For example:

$$\text{Pre-GST price}=\frac{55}{1.10}=50$$

So a GST-inclusive price of AUD 55 has a pre-GST price of AUD 50.

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Unit rates: comparing fairly

A unit rate describes an amount for one unit of something. Common consumer unit rates include:

• dollars per item
• dollars per kilogram
• cents per $100\text{ mL}$
• kilometres per litre
• dollars per hour

To compare two deals, both must be converted to the same unit. The lower unit price is usually the better value, assuming you will use the full amount.

The general calculation is:

$$\text{Unit rate}=\frac{\text{total cost}}{\text{number of units}}$$

For example, a school is buying drinks for a sports carnival:

• Option A: $750\text{ mL}$ bottle for AUD 3.60
• Option B: $1.25\text{ L}$ bottle for AUD 5.25

Use cents per $100\text{ mL}$. First convert Option B to millilitres:

$$1.25\text{ L}=1250\text{ mL}$$

For Option A:

$$\frac{3.60}{750}\times 100=0.48$$

Option A costs 48 cents per $100\text{ mL}$.

For Option B:

$$\frac{5.25}{1250}\times 100=0.42$$

Option B costs 42 cents per $100\text{ mL}$.

Because $42$ cents per $100\text{ mL}$ is lower than $48$ cents per $100\text{ mL}$, Option B is the better value.

Combining a discount with a unit rate

Some questions require you to apply a percentage change before finding the unit rate.

For a sports day, a shop offers:

• Option A: 24 ice blocks for AUD 18.00, discounted by $15\%$
• Option B: 30 ice blocks for AUD 21.60

First calculate the discounted price for Option A:

$$18.00\times 0.85=15.30$$

Then calculate the price per ice block:

$$\frac{15.30}{24}=0.6375$$

Option A costs 63.75 cents per ice block.

For Option B:

$$\frac{21.60}{30}=0.72$$

Option B costs 72 cents per ice block.

Therefore, Option A is the better-value option, because $63.75$ cents per ice block is less than $72$ cents per ice block.

The important sequence is:

1. Apply any markup, discount, or GST that changes the price.
2. Make units consistent, such as converting litres to millilitres.
3. Divide to find the unit rate.
4. Compare rates with their units written clearly.

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A quick exam checklist

Before you commit to an answer, pause and check these points:

1. What is the starting amount?
   A percentage is always a percentage of something.

2. Is the price increasing or decreasing?
   Use a multiplier greater than $1$ for a markup and less than $1$ for a discount.

3. Are there several changes?
   Use one multiplier for every change.

4. Have you used the correct order?
   Apply a discount to the price that exists at that point in the question.

5. Are the units consistent?
   Do not compare dollars per litre with dollars per millilitre, or a pack price with a single-item price.

6. Have you rounded sensibly?
   Keep calculator digits during working, then round a money answer to two decimal places at the end unless the question instructs otherwise.

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Key takeaways

A percentage markup or discount is best handled with a multiplier:

$$\text{New amount}=\text{Original amount}\times \text{multiplier}$$

Use $1.25$ for a $25\%$ markup, $0.70$ for a $30\%$ discount, and $1.10$ when adding GST. For multistep questions, multiply by every relevant factor; do not add percentage changes together. For unit-rate questions, first adjust the price if necessary, convert quantities to matching units, and compare the cost per one unit.

Next, you will move from prices of items to gross earnings: calculating pay involving hourly rates, overtime, and allowances.