Lesson

Percentages are used in everyday life to represent proportions and to show rates of increases and decreases. Some examples of percentages include:

- Discounts e.g. $30%$30% off
- Interest rates e.g. $2%$2% p.a.
- The amount of battery charge on a device e.g. $64%$64% remaining

The words "per cent" mean "per $100$100 parts". For example, $2%$2% means $2$2 out of every $100$100 parts. Percentages are equivalent to fractions out of $100$100 and can be written with a $%$% sign, or as a fraction or decimal. For example

$35%=0.35=\frac{35}{100}$35%=0.35=35100

When using percentages in the real world, we often deal with changes to amounts using percentages, whether they're increases or decreases. In both cases, we can go through some shortcuts to find our final amount.

When calculating with percentages, it is usually a good idea to convert the percentage to a fraction or a decimal first.

Let's say we wanted to increase $40$40 by $2%$2% and find the end amount.

We could first find $2%$2% of $40$40, which we can work out here (expressing the percentage as the fraction $\frac{2}{100}$2100):

$\frac{2}{100}\times40=0.8$2100×40=0.8

The final amount would then be the original amount $40$40 plus this extra $2%$2% of $40$40. That is, the final amount would be

$40+0.8=40.8$40+0.8=40.8

There is a quicker way to do this, however, if we think completely in percentages.

To find the total amount after the increase, we are essentially finding $100%+2%=102%$100%+2%=102% of $40$40.

Therefore we can just work out the increase this way:

$\frac{102}{100}\times40=40.8$102100×40=40.8

This gives us the same amount as before, but involved less steps.

Increase by a percentage

To increase $x$`x` by $y%$`y`%, we can calculate

$x\times\left(100+y\right)%$`x`×(100+`y`)%

Decreasing by a percentage has a similar shortcut.

For example, if we want to calculate a $25%$25% discount off $\$60$$60 we could find $25%$25% of $60$60 first, which is $60\times0.25=15$60×0.25=15, and subtract this from $60$60, giving $60-15=45$60−15=45.

But discounting a price by $25%$25% is the same as just paying $75%$75% of the price. So the easier way to find this amount is to calculate

$0.75\times\$60=\$45$0.75×$60=$45,

which is the same as the answer we got before.

So to decrease an amount by a percentage, we can just multiply the amount by ($100%$100% - percentage).

Decrease by a percentage

To decrease $x$`x` by $y%$`y`%, we can calculate

$x\times\left(100-y\right)%$`x`×(100−`y`)%

We want to increase $1300$1300 by $40%$40% by following the steps outlined below.

First find $40%$40% of $1300$1300.

Add the percentage increase to the original amount to find the amount after the increase.

Calculate $140%$140% of $1300$1300.

Is increasing an amount by $40%$40% equivalent to finding $140%$140% of that amount?

Yes

ANo

BYes

ANo

B

We want to decrease $1500$1500 by $15%$15% by following the steps outlined below.

First find $15%$15% of $1500$1500

Subtract the percentage decrease from the original amount to find the amount after the decrease.

Calculate $85%$85% of $1500$1500

Is decreasing an amount by $15%$15% equivalent to finding $85%$85% of that amount?

Yes

ANo

BYes

ANo

B

A bag of rice weighs $110$110kg. If the weight of the bag decreases by $40%$40% find the new weight of the bag.

Repeated percentage change can seem a bit misleading at first. If a TV was initially priced at $\$1000$$1000 but had been reduced by $20%$20% and then later by $50%$50%, it would be easy to think that the overall discount was $70%$70% of $1000$1000 (i.e. $\$700$$700). But this isn't actually correct!

Amounts can change more than once. If you have ever been sale shopping, you may well have seen that a product had its price reduced and then reduced again. This is sometimes called **successive discounts**. A sale that offers $20%$20% off and then $50%$50% off sounds great, right? Is it any different to offering $50%$50% then $20%$20% off?

The reduction happens successively (one after the other). Using the TV example from above, the first discount reduces the price from $\$1000$$1000 to $\$800$$800, so the initial savings is $\$200$$200. Applying the second discount next, the reduced price of $\$800$$800 is reduced again by $50%$50%, making the final price $\$400$$400 and corresponding to a saving of $\$400$$400. In total, this is a saving of $\$600$$600.

In summary, to calculate repeated percentage changes we simply calculate each percentage change in succession (one after the other).

A $\$70$$70 pair of shoes is on sale for $30%$30% off and later it is discounted by a further $10%$10% off. Calculate the final sale price.

**Think:** We want to decrease $\$70$$70 by $30%$30%, and then decrease that amount by $10%$10%.

**Do:** Decreasing by $30%$30% is the same as finding $70%$70% of $\$70$$70: $0.7\times70=49$0.7×70=49

Decreasing by $10%$10% is the same as finding $90%$90% of the amount: $0.9\times49=44.1$0.9×49=44.1

Therefore the final sale price is $\$44.10$$44.10

The price of a phone was increased by $40%$40% and then again by $40%$40%.

What was the overall percentage increase?

The overall percentage increase when the price is increased by $40%$40% twice is $2\times40%$2×40%. True or false?

True

AFalse

BTrue

AFalse

B

For business owners and managers, it is important to know whether the business is making or losing money. In other words, it is important to know whether there is a profit or a loss.

In this section, we'll look at how to calculate profits and losses, as well as how to calculate the percentage profit or loss.

Terminology

**Cost price** (or **buying price**): The price that is paid for the item.

**Selling price:** The price that the item is sold for.

**Marked price**: The price labelled on the item.

**Profit:** If more money is earned than than is spent, the extra amount earned is a profit.

**Loss:** If more money is spent than is earned, the extra amount spent is a loss.

**Break-even point:** If exactly the same amount of money is earned and spent, then there is no profit or loss. This is called the break-even point.

**Revenue:** another name for income or money that is earned by a business.

To calculate whether a business will register a profit or a loss, first calculate the gross income for the business (i.e. the total amount earned), as well as the total expenses for the business. Then subtract the two amounts:

$\text{Gross income}-\text{Total expenses}=\text{Profit or Loss}$Gross income−Total expenses=Profit or Loss

Note that a loss can also be expressed as a negative profit, which is reflected in the formula above. For example, a profit of $-\$1200$−$1200 would actually mean a loss of $\$1200$$1200.

We can also use this formula to calculate the total expenses, if we know our revenue and our profit. We just substitute the values we know to work out the one that we don't.

It is common to express a profit or loss from a sale as a percentage of the cost price. To do so, we first write the profit or loss as a fraction of the cost price, then convert the fraction to a percentage:

$\text{Percentage Profit/Loss}=\frac{\text{Profit/Loss }}{\text{Cost Price}}\times100%$Percentage Profit/Loss=Profit/Loss Cost Price×100%

This can be helpful as it gives a good indication of the relative amount earned. For example, making $\$100$$100 profit sounds good in isolation, but is it really that good if you sold the item for $\$100000$$100000 and only made $\$100$$100 profit?

Amelia bought a truck for $\$17300$$17300 and sold it four years later, making a loss of $25%$25%. How much did she sell it for?

Xavier bought a property for $\$371000$$371000. In the first year, it increased in value by $12%$12%, but in the second year, it decreased in value by $6%$6%. If running costs during the two years amounted to $\$1500$$1500, find Xavier’s profit or loss at the end of two years.

First, calculate the amount Xavier earned from the changing value of the property. Give your answer correct to the nearest cent.

Now calculate Xavier's total profit. Give your answer correct to the nearest cent.

The Goods and Services Tax, or GST, was introduced and implemented in Australia in the year 2000. It is a value added tax, usually $10%$10%, on most goods and services transactions. In other words, the prices that businesses charge you include an additional $10%$10% of the original price as a GST amount.

For example, if the original price of an item was $\$10$$10, the GST on this item would be $\$1$$1 (since $10%$10% of $\$10$$10 is $\$1$$1). The price that is actually charged would then be the total of $\$11$$11.

For tax reasons, businesses in particular need to keep track of how much GST they pay and receive, so it is important to be able to calculate prices before and after GST, as well as the amount and rate of GST.

Other countries have a variation of GST, such as VAT (value added tax), and can be different rates. For example, in Germany the VAT is $19%$19%.

Aaron received a bill of $\$68$$68 for consultation and $\$10$$10 for medication from a medical professional. If GST is applied to the consultation fee only, what is the total bill that Aaron must pay?

**Think:** GST is an additional $10%$10% of the cost.

**Do:** Consultation including GST: $68+68\times0.1=\$74.80$68+68×0.1=$74.80

Medication: $\$10$$10 (we don't need to change this amount as the GST only applies to the consultation.

Total: $\$74.80+\$10=\$84.80$$74.80+$10=$84.80

**Reflect:** We could have performed this calculation in one step, as

$\$68\times1.1+\$10=\$84.80$$68×1.1+$10=$84.80

The sales price of an item, including GST, is $\$40$$40. Calculate the price of the item without GST.

Round your answer to the nearest cent.

models relevant financial situations using appropriate tools