2. Percentages

Australian Curriculum 11 Essential Mathematics - 2020 Edition

2.07 Solve problems with percentages

Lesson

Sometimes businesses want to attract customers by lowering the price and giving a **discount**, which is usually expressed in percentages. We've just seen in this chapter how to calculate percentage increases and decreases, usually the hardest part is interpreting the word problems and what they mean mathematically.

Lisa wants to buy a new phone for $\$360$$360, and was delighted to find out that she gets a $20%$20% discount. Calculate** **the value of the discount and hence the price Lisa pays for the phone.

**Think **about which numbers are the ones you need, and the meaning of 'value'. Value means how much the discount actually is in terms of money.

**Do: ** That means we need to calculate $20%$20% of $\$360$$360.

$360\times20%$360×20% | $=$= | $72$72 |

Therefore the discount is $\$72$$72.

This would mean the final price Lisa pays would be $\$360-\$72=\$288$$360−$72=$288

Watch out!

Remember to always attach the correct unit to your final answer, whether they're dollars, pounds or in other questions maybe even kilograms or metres.

Steph is going to buy a hat that is marked as $25%$25% off. The original price was $\$36$$36.

What is the value of the discount in dollars?

What is the price that Steph will pay for the hat?

The full price is $\$300$$300. Yvonne receives a discount of $55%$55%.

How much would a $10%$10% discount be?

How much would a $5%$5% discount be?

Hence find the discount that Yvonne receives.

1. **Discount: **how much the cost of something is reduced by. Eg. a $10%$10% discount on $\$200$$200 means taking $10%\times\$200=\$20$10%×$200=$20 off the full price of $\$200$$200

2. **Commission: **how much money a salesperson earns. It is usually a percentage of how much they manage to sell. Eg. $50%$50% commission on $\$10$$10 worth of sales is $50%\times\$10=\$5$50%×$10=$5, which is how much the salesperson gets to keep.

3. **Profit: **if an item is sold for more than what it cost to make or obtain, you make a profit. So it is the difference between the selling price and the cost price. Eg. If I bought a bike for $\$10$$10 and sold it for $\$15$$15, my profit is $\$5$$5

4. **Loss: **if an item is sold for less than what it cost to make or obtain, you make a loss. So it is the difference between a product's sale price and buy price when the sale price is lower. Eg. If I bought a bike for $\$20$$20 and sold it for $\$18$$18, my loss is $\$2$$2

5. **Stocks: **a type of 'virtual' product which represents ownership of part of a company

6. **Percentage profit/loss: **the profit/loss as a percentage of the buying price. Eg. I bought a bike for $\$20$$20 and sold it for $\$22$$22, so my percentage profit is $\frac{22-20}{20}\times100%=\frac{2}{20}\times100%$22−2020×100%=220×100% = $10%$10%. A profit of $\$100$$100 sounds good, but is it really that great if the buying price was $\$100000$$100000?

7. **Interest: **if you have savings in a bank account, or owe money in the form of loans, interest is the amount by which the savings or loans grow. Eg. If I have $\$500$$500 in the bank and the interest rate offered is $10%$10%, I will earn an extra $10%\times\$500=\$50$10%×$500=$50 worth of interest.

A sales assistant is paid a commission of $15%$15% on her weekly sales. Find her commission for a week in which she sells products to the value of $\$2000$$2000.

Luke deposits $\$8635$$8635 in a bank for one year. If the interest rate is $7%$7% p.a:

How much interest, in dollars, did Luke earn for the year? Write your answer to the nearest cent.

Find the total amount (to the nearest cent) in his bank account at the end of the year.

The Howard Government reformed the then state and federal run sales tax system in 2000 by introducing the Goods and Services Tax, or GST. It is a value-added tax, usually of $10%$10%, on most goods and services transactions. In other words, the prices 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$$10, the GST on this item $\$1$$1 since $10%$10% of $\$10$$10 is $\$1$$1. The price you would be charged would be $\$11$$11 since $10+1=11$10+1=11. The GST is given to the government by the business therefore they 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.

Calculating price after and before GST

To calculate the price of an item after GST is added we **multiply** by $1.1$1.1

To calculate the original price of an item before GST if we know the final price including GST we **divide** by $1.1$1.1

Aaron received a bill of $\$68$$68 for consultation and $\$10$$10 for herbal supplements from a medical professional. If GST is only applied to the consultation fee, what is the total bill that Aaron must pay? Give your answer correct to two decimal places.

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

**Do: **Consultation increased by $10$10% GST: $68\times1.1=\$74.80$68×1.1=$74.80

Herbal supplements: $\$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 also work this out in one step:

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

Calculate the GST-included sales price for an item with an original price of $\$20$$20.

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.

determine one amount expressed as a percentage of another

apply percentage increases and decreases in situations; for example, mark-ups, discounts and GST