Working with Percentages

Working with percentages

Writing percentages as fractions

Writing percentages as decimals

Changing fractions to percentages

Changing decimals to percentages

Applications of percentages

Percentages of amounts

Estimation and quick calculations

Expressing an amount as a percentage of another amount

Using diagrams to calculate percentage

Percentage discounts

Commissions in sales

GST

Answers to activities

Answers to Check your progress 4

Writing percentages as fractions

A per cent (%) is just a common fraction, which has a denominator of 100. Since you know the bottom line is always 100, you don’t need to write it, so you just put the % sign.

‘Per cent’ actually means ‘out of 100’ in Latin.

So, 50% just means a fraction with a top line of 50, and bottom line of 100:

50% =

You can of course cancel this to.

Examples

125% = = after cancelling.

To enter 25% just enter 25100 and the display is .

22% = =

2100 and the answer is .

337% = (we can’t cancel).

37100 and the answer is .

100% means or 1!

So, any percentage larger than 100 represents a number larger than 1.

4130% = = 1

130100 and the answer 1.

5280% = = 2. Use your calculator to cancel down.

6112.5% = = 1 (using the calculator).

Writing percentages as decimals

Percentages can be easily turned into decimals:

50% = = 0.5

Remember—when dividing by 100, you just move the decimal point two places to the left.

Examples

123% = = 0.23

On the calculator, 23100 gives 0.23

Actually, it’s easier to do 23100 which also gives 0.23

2140% = = 1.4

140100 and the answer is 1.4

317.25% = 17 = 0.1725

/ Activity 1

1Change these percentages to fractions in their simplest form (cancelled down):

(a)80%______(b)74%______

(e)387.5%______(f)0.16%______

2Change these percentages to decimals.

(a)60%______(b)37.5%______

Check your answers at the end of the section.

Changing fractions to percentages

You can also easily change fractions to percentages. All you need to do is to use equivalent fractions to turn the denominator to 100.

Examples

1To change to a percentage, multiply top and bottom by 20:

s4_eqn17

There is a simpler way to get the same answer. Just multiply the fraction by 100 and add a % sign. We use the calculator for this:

= × 100% = 80%

45100 and the display is 80.

2Change to a percentage.

= × 100% = 75%

34100 which displays 75.

3Change to a percentage.

= × 100% = 8.%

112100 and the display is 8.333333333333

4Change 3 to a percentage.

3 = 3× 100% = 362 %

358100 and the display is 36212.

Changing decimals to percentages

To change decimals to a percentage, you also just need to multiply by 100 and add a % sign:

0.4 = 0.4 × 100% = 40%

Examples

10.23 = 0.23 × 100% = 23%

22.07 = 2.07 × 100% = 207%

31.0671 = 1.0671 × 100% = 106.71%

/ Activity 2

Convert the following to percentages.

1______2______

3______43______

50.7______61.8______

72.075______85.6775______

Check your answers at the end of the section.

Applications of percentages

Percentages of amounts

Percentages are most often used to represent a fraction of a given amount; for example, a store may offer 20% off all items.

You will now look at how to find the percentage of an amount.

Example

1Find 20% of $55.

First, we convert the percent to a fraction, then replace the ‘of’ with ‘×’:

20% of $55 = × 55

2010055 Answer is $11.

or 2010055 Answer is $11.

2Find 18% of 350 mm.

18% of 350 = × 350

18100350 Answer is 63 mm.

3Find 135% of $20.

135% of 20 = × 20

13510020 Answer $27.

4Find 12.5% of $512.

12.5% of 512 = × 512

12.5100512 [=] Answer is $64.

Estimation and quick calculations

Often we can estimate or quickly calculate a percentage.

Examples

1Find 10% of $700.

Remember 10% means or . One-tenth of an amount is found by dividing it by 10 (just move the decimal place one place to the left).

So of $700 is $70.

2Find 20% of 450 m.

10% is found by dividing by 10, so10% is 45 m.

Therefore 20% is just twice that, that is 90 m.

3Find 5% of $68.

10% is $6.80 and so 5% will be half of that i.e. $3.40

4Find an approximate value for 22% of $600.

10% is $60, so 20% is $120, which is approximately the 22% wanted.

Expressing an amount as a percentage of another amount

You may want to know what percentage one value is of another.

Examples

1What percentage is 30 cm of 150 cm?

Here, you can write the first value as a fraction of the second, then convert to a percentage by multiplying by 100.

The percentage 30 cm is of 150cm is × 100%.

30150100 and the answer is 20%.

2What percentage is $15 of $300?

Remember, you write the first value as a fraction of the second, then convert to a percentage.

The percentage $15 is of $300 is × 100%.

15300100 and the answer is 5%.

/ Activity 3

1Find:

(a)50% of $300______(b)25% of $180______

(e)140% of 8 litres______(f)12.5% of $80______

2What percentage is:

(a)15cm of 120 cm? ______(b)$20 of $25? ______

(d)$21.25 of $85?______

Check your answers at the end of the section.

Using diagrams to calculate percentage

Often percentages are given as a part of a diagram.

Examples

1What percentage is shaded in the diagram?

Well, one part out of four, or , is shaded. So 25% is shaded.

2What percentage of the following diagram is shaded?

Here, is shaded, so 60% is shaded.

3This pie chart (or sector graph) shows the proportion of concrete components. What percentage of concrete is the blue metal?

The total weight of concrete is 12 + 10 + 8 + 20 = 50 kg.

So the blue metal is 10 kg out of 50 kg.

1050100 which is 20%.

4The diagram shows Australia’s exports. If our total exports are worth $86 billion per annum, what is the contribution from tourism?

Farming
22% / Mining
28% / Tourism
16% / Ed
8% / Other
26%

Tourism is 16% of $86 billion.

1610086 and the answer is $13.76 billion.

Percentage discounts

There are many times when we come across percentages in real life. Shops often advertise a sale with a given percentage off the regular price. The amount taken off the regular price is called the discount.

Examples

1A department store has a ‘25% off all clothing’ sale. What is the discount on a shirt that regularly sells for $24? What is its sale price?

The discount is 25% of $24

× 24.

2510024 Answer is $6.

The price I pay at the sale is therefore $24 – $6 = $18.

2A hardware store offers 12.5% off all drills. I bought a drill, which normally retails for $160. How much did I pay at the sale?

The usual full price is 100%.

If I get a 12.5% off, it means I only pay (100 – 12.5)% = 87.5%.

So I pay 87.5% of $160.

87.5100160 The answer is $140.

Commissions in sales

Another place you often see percentages used is where a commission is paid to a salesperson. A commission is the percentage of the total sale paid to the salesperson as part or all of their wage.

Examples

1A used car saleswoman receives 7.5% commission on all the cars she sells.

On Tuesday, she sold a Camry for $13 500 and a Prado for $24 000. What was her total commission?

Total sales = $13 500 + $24 000 = $37 500.

The commission is 7.5% of $37 500.

7.510037 500 and the answer is $2812.50

2A real estate agent charges 2% commission on a house sale. What is the commission on a house which sold for $840 000?

The commission is 2% of $840 000.

2100840 000 which gives an answer of $16 800.

GST

Another situation where we often come across percentages is in the Australian Goods andServices Tax (GST). The Australian Government charges a 10% tax on all sales and services.

Examples

The total cost of a car service was $311.15. What GST would be charged?

GST = 10 % of $311.15

10100311.15 and the display is 31.115

Rounded off, this is $31.12.

/ Activity 4

1A store offers a ‘20% off’ sale on all items. What is the discount on a $315 TV and a $42 picture frame?

______

2At a ‘15% off’ sale, Dale bought an electric chainsaw, which normally retails for $140. What discount did he get, and how much did he have to pay for the saw?

______

3A store has 12.5% off all women’s clothes. Saffron bought a top (normal price $45) and a skirt (normal price $35). What was the total price she paid?

______

4Charlotte bought jeans, normally priced at $85, on sale for just $63.75. What was the percentage discount?

______

5A used caravan salesman receives 5% commission on all the vans he sells. He sold one van for $12 500, and another for $14 000. What was his total commission?

______

6A real estate agent charges 1.8% commission on a house sale. What is the commission on a house which sold for $650 000?

______

7What GST is charged on a refrigerator priced at $800?

______

8I bought items costing for $45, $67, and $11. What total GST did I pay?

______

Check your answers at the end of the section.

/ Check your progress 4

1Change the following percentages to fractions and simplify:

(a)74%______(b)22.5%______

2Change these percentages to decimals:

(a)35%______(b)18.3%______

3Convert the following to percentages:

(a)2______(b)1______

4Find:

(a)33% of $3.39______(b)18% of 550 mm______

5What percentage is:

(a)$14 of $90? ______(b)12.8 km of 32 km?______

6At a ‘20% off’ sale, Susan bought a pair of slippers that normally retail for $25. What discount did she get, and how much did she have to pay for the slippers?

______

7A store has a ‘45% off clearance’ sale on all electrical items. Jason bought a juicer (normal price $29) and a microwave oven (normal price $161). What was the total price he paid at the sale?

______

8A used car salesman receives 2.5% commission on all the cars he sells. He sold one car for $8 500, another for $11 000, and another for $23 500. What was his total commission?

______

9An art auction house retains 8% of all money gained at auction. A ‘Dobell’ was sold for $240 000. What was the commission?

______

10A washing machine man came to fix my washer. The cost was $112 labour, and $28 parts. How much GST must be added?

______

Answers to activities

Activity 1

1(a)(b)(c)(d)1

(e)3(f)

2(a)0.6(b)0.375(c)2.13(d)0.0045

Activity 2

125%220%387.5%4375%

570%6180%7207.5%8567.75%

Activity 3

1(a)$150(b)$45(c)90 mm(d)16.2 kg

(e)11.2 litres(f)$10

2(a)12%(b)80%(c)20%(d)25%

Activity 4

1$63 on TV; $8.40 on the picture frame.

2$21 discount, paid $119

3$70

4she paid 75%, so discount was 25%

5$1325

6$11 700

7$80

8$12.30

Answers to Check your progress 4

1(a) (b) (c)1

2(a) 0.35(b) 0.183(c)5.673

3(a) 280%(b) 133%(c)45.6%(d)275%

4(a) $1.13(b) 99 mm(c)18.75 min or 18 mins 45 secs

5(a) 15.%(b) 40%(c)112.5%

6$5 discount, paid $20

7$104.50

8$1075

9$19 200

10$14

4930N: 4 Arithmetic and the Calculator1

 CLI 2004/055/06/2004