8.01 Percentage increase and decrease

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.

 

Increasing by a percentage

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.

 

 

 

Decreasing by a percentage

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).

 

Practice questions

QUESTION 1

QUESTION 2

QUESTION 3

 

Repeated percentage change

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).

 

Worked example

Example 1

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

 

Practice question

question 4

 

GST

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%.

 

Worked example

Example 2

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

 

Practice question

Question 5