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Percentages


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Numeracy Basics

Percentages

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This topic will cover how to:

  • calculate percentages of given quantities;
  • calculate the result of a percentage increases or decrease;
  • determine how much has been added or subtracted after a percentage increase or decrease;
  • express quantities as percentages of other quantities;
  • determine the percentage increase or decrease between two values;
  • convert between percentages and decimal numbers;
  • convert between percentages and fractions or mixed numbers.

Percentage

A percentage refers to the number of something 'out of 100', and is shown by the symbol %. For example a tax rate of 30% indicates that you need to pay $30 out of every $100 you earn in tax. Some other facts about percentages are as follows:

Percentages are commonly used to express sales discounts, increase or decrease in price over time, profit, loss, commission and rates of interest, as well as when analysing and comparing data

You can have percentages that are more than 100%, such as 150% or 300%

You can calculate a percentage of a given quantity (including determining percentage increase and decreases), and you can also express one quantity as a percentage of another; how to do all this will be explained in this topic...

Percentages are commonly used to express sales discounts, increase or decrease in price over time, profit, loss, commission and rates of interest, as well as when analysing and comparing data

You can have percentages that are more than 100%, such as 150% or 300%

You can calculate a percentage of a given quantity (including determining percentage increase and decreases), and you can also express one quantity as a percentage of another; how to do all this will be explained in this topic...

Calculating a Percentage of a Given Quantity

As an example of when you may need to calculate a percentage of a given quantity, consider the Goods and Services Tax (GST) in Australia. The rate of the GST is 10%, which means that an additional $10 needs to be paid for every $100 of the original price of the good or service.

In order to calculate a percentage of a given quantity, as per our GST example, you need to divide the percentage in question by 100, and then multiply by the quantity.

For example, to calculate 10% of $500, you would do:

10 ÷ 100 x 500 = $50

10÷
100x
500

=$50

Examples: Calculating a Percentage
of a Given Quantity

1. A product which was originally priced at $50 is reduced by 25%. What is the dollar value of this reduction?

25 ÷ 100 x $50 = $12.50 price reduction

2. A price reduction of a particular item means that the number of sales is expected to increase by 30%. Given that the number of sales was originally 20 000, how many extra sales can be expected?

30 ÷ 100 x 20 000 = 6 000 extra sales

25 ÷ 100

x $50

= $12.50 price reduction

30 ÷ 100

x 20 000

= 6 000 extra sales

Activity 1: Practice Questions

Calculating Percentage Increase or Decrease

Calculating the result of a percentage increase or decrease requires just a bit more work than calculating a percentage of an amount. It can be done in one of two ways:

1. Calculate the specified percentage of the initial amount, as described earlier, and then either add (for an increase) or subtract (for a decrease) this value from the initial amount

For example to determine the cost of an item originally priced at $500, after 10% GST is added, you can:

Determine that the amount of the increase is 10 ÷ 100 x $500 = $50

And therefore that the new price of the item is $500 + $50 = $550

2. Add (for an increase) or subtract (for a decrease) the specified percentage from 100%, and then calculate this new percentage of the initial amount (as described earlier)

For example to determine the cost of an item originally priced at $500, after 10% GST is added, you can:

Determine that the new price is 100% + 10% = 110% of the old price

And therefore that the new price of the item is 110 ÷ 100 x 500 = $550

1. Calculate the specified percentage of the initial amount, as described earlier, and then either add (for an increase) or subtract (for a decrease) this value from the initial amount

For example to determine the cost of an item originally priced at $500, after 10% GST is added, you can:

Determine that the amount of the increase is 10 ÷ 100 x $500 = $50

And therefore that the new price of the item is $500 + $50 = $550

2. Add (for an increase) or subtract (for a decrease) the specified percentage from 100%, and then calculate this new percentage of the initial amount (as described earlier)

For example to determine the cost of an item originally priced at $500, after 10% GST is added, you can:

Determine that the new price is 100% + 10% = 110% of the old price

And therefore that the new price of the item is 110 ÷ 100 x 500 = $550

Examples: Percentage Increase or Decrease

1. Sales reps at a company are paid 8% of the sale price of an item. If a rep sells an item for $500, how much will the company receive for the item after the commission has been paid?

Method 1: 8 ÷ 100 x $500 = $40 commission

So company receives $500 - $40 = $460

Method 2: 100% - 8% = 92%

So company receives 92 ÷ 100 x $500 = $460

2. Suppose that an asset which was originally worth $20 000 is determined to have depreciated (i.e. lost value) by 38%. What is the value of the asset now?

Method 1: 38 ÷ 100 x $20 000 = $7 600 reduction

So new value is $20 000 - $7 600 = $12 400

Method 2: 100% - 38% = 62%

So new price is 62 ÷ 100 x $20 000 = $12 400

Method 1: 8 ÷ 100 x $500 = $40 commission

So company receives $500 - $40 = $460

Method 2: 100% - 8% = 92%

So company receives 92 ÷ 100 x $500 = $460

Method 1: 38 ÷ 100 x $20 000 = $7 600 reduction

So new value is $20 000 - $7 600 = $12 400

Method 2: 100% - 38% = 62%

So new price is 62 ÷ 100 x $20 000 = $12 400

How much has been Added or Subtracted?

Suppose you know that the total price of a TV is $880, but that this includes 10% GST. How can you determine how much of this total is the GST?

To do this, consider that the new total amount is 110% of the initial price, as the 10% GST has already been added (i.e. 100% + 10% = 110%). Hence rather than dividing our percentage (10) by 100, as we have done previously, we divide it by 110. Finally, we multiply by our total amount to get the value of our GST:

10 ÷ 110 x 880 = $80

Alternatively, we can determine that the original price of the TV is:

In general:

AMOUNT OF INCREASE = PERCENTAGE ÷ (100 + PERCENTAGE) x NEW VALUE

AMOUNT OF DECREASE = PERCENTAGE ÷ (100 - PERCENTAGE) x NEW VALUE

10 ÷ 110

x 880

= $80

100 ÷ 110

x 880

= $800

AMOUNT OF INCREASE

= PERCENTAGE ÷ (100 + PERCENTAGE)

x NEW VALUE

AMOUNT OF DECREASE

= PERCENTAGE ÷ (100 + PERCENTAGE)

x NEW VALUE

Examples: How much has been
Added or Subtracted?

1. The price of a fridge, inclusive of GST, is $1 320. What is the value of the GST component of this price?

GST component = 10 ÷ (100 + 10) x $1 320

= 10 ÷ 110 x $1 320

= $120

2. A dress has been reduced in price by 20%, and the new price is $90. What is the dollar value of this price reduction?

Price reduction = 20 ÷ (100 - 20) x $90

= 20 ÷ 80 x $90

= $22.50

GST component = 10 ÷ (100 + 10) x $1 320

= 10 ÷ 110 x $1 320

= $120

Price reduction = 20 ÷ (100 - 20) x $90

= 20 ÷ 80 x $90

= $22.50

Activity 3: Practice Questions

Expressing One Quantity
as a Percentage of Another

Another way you might need to use percentages is to express one quantity as a percentage of another. For example, consider that a company sells an item for $80 but that, after expenses are taken into account, the profit they make on the item is $60. If the company wishes to know what percentage of the sale price is profit, then they need to determine what percentage $60 is of $80.

In order to express one quantity as a percentage of another, as in this example, you need to divide the first quantity by the second, and then multiply by 100.

For example, to calculate what percentage $60 is of $80, we would do:

60 ÷ 80 x 100 = 75%

60÷

80

x100

=75%

Examples: Expressing One Quantity as a Percentage of Another

1. A telephone bill totals $80 for a given period, and the charge for the service and equipment is $20 of this total. What percentage of the bill is in payment of service and equipment?

20 ÷ 80 x 100 = 25% of the bill is to pay for service and equipment

2. If a company sells an item for $240, of which $180 is profit, what percentage of the sale price is profit?

180 ÷ 240 x 100 = 75% of the sale price is profit

20 ÷ 80

x 100

= 25%

of the bill is to pay for service and equipment

180 ÷ 240

x 100

= 75%

of the sale price is profit

Activity 4: Practice Questions

What is the Percentage Increase or Decrease?

Something else you may need to calculate is the percentage increase or decrease from one amount to another. You can do this by following these steps:

1. Subtract the old amount from the new amount

For example to calculate the percentage change from $100 to $150 you would determine:

150 (new amount) - 100 (old amount) = $50

2. Divide this difference by the old amount, and then multiply by 100

For example to calculate the percentage change from $100 to $150 you would determine:

50 ÷ 100 (old amount) x 100 = 0.5 x 100

= 50%

3. If this value is positive you have a percentage increase, while if it is negative you have a percentage decrease

For example the percentage change from $100 to $150 is a 50% increase, since the value is positive

1. Subtract the old amount from the new amount

For example to calculate the percentage change from $100 to $150 you would determine:

150 (new amount) - 100 (old amount) = $50

2. Divide this difference by the old amount, and then multiply by 100

For example to calculate the percentage change from $100 to $150 you would determine:

50 ÷ 100 (old amount) x 100 = 0.5 x 100

= 50%

3. If this value is positive you have a percentage increase, while if it is negative you have a percentage decrease

For example the percentage change from $100 to $150 is a 50% increase, since the value is positive

Examples: What is the Percentage
Increase or Decrease?

1. A patient's weight initially was 50kg, and after treatment it was recorded as 65kg. How much of a percentage increase or decrease is this?

65 - 50 = 15

15 ÷ 50 x 100 = 30%

Since this is positive, we have a percentage increase of 30% in the weight of the patient

2. The number of customers frequenting a particular business in the previous financial year was 53 000, while this financial year it was 45 000. How much of a percentage increase or decrease is this (rounded to two decimal places)?

45 000 - 53 000 = -8 000

-8 000 ÷ 53 000 x 100 = -15.09%

Since this is negative we have a percentage decrease of 15.09% in customers

65 - 50 = 15

15 ÷ 50 x 100 = 30%

Since this is positive, we have a percentage increase of 30% in the weight of the patient

45 000 - 53 000 = -8 000

-8 000 ÷ 53 000 x 100 = -15.09%

Since this is negative we have a percentage decrease of 15.09% in customers

Activity 5: Practice Questions

Converting between
Percentages and Decimal Numbers

You may sometimes need to convert a percentage to a decimal number, or a decimal number to a percentage. This can be done as follows:

To convert a percentage to a decimal number, simply divide your percentage by 100.

For example 87.5% = 87.5 ÷ 100 = 0.875

To convert a decimal number to a percentage, simply multiply the decimal number by 100.

For example 0.875 = 0.875 x 100 = 87.5%

To convert a percentage to a decimal number, simply divide your percentage by 100.

For example 87.5% = 87.5 ÷ 100 = 0.875

To convert a decimal number to a percentage, simply multiply the decimal number by 100.

For example 0.875 = 0.875 x 100 = 87.5%

Examples: Converting between
Percentages and Decimal Numbers

1. Convert the following percentages into decimal numbers:

a.10%

10 ÷ 100 = 0.1

b.150%

150 ÷ 100 = 1.5

2. Convert the following decimal numbers into percentages:

a.0.4

0.4 x 100 = 40%

b.0.015

0.015 x 100 = 1.5%

10 ÷ 100 = 0.1

150 ÷ 100 = 1.5

0.4 x 100 = 40%

0.015 x 100 = 1.5%

Activity 6: Practice Questions

Converting Fractions or
Mixed Numbers to Percentages

You can convert a fraction or mixed number to a percentage by following these steps:

1. If your number is a mixed number, multiply the whole number part of the mixed number by 100

For example to convert 2 4/5 to a percentage do 2 x 100 = 200

2. Divide the numerator of your fraction by the denominator and multiply this result by 100, rounding as necessary. If your number is not a mixed number then this is your percentage equivalent

For example to convert 2 4/5 to a percentage do 4 ÷ 5 x 100 = 80

3. If your number is a mixed number, add the two values you have calculated previously together to determine the percentage equivalent

For example to convert 2 4/5 to a percentage do 200 + 80 = 280%

1. If your number is a mixed number, multiply the whole number part of the mixed number by 100

For example to convert 2 4/5 to a percentage do 2 x 100 = 200

2. Divide the numerator of your fraction by the denominator and multiply this result by 100, rounding as necessary. If your number is not a mixed number then this is your percentage equivalent

For example to convert 2 4/5 to a percentage do 4 ÷ 5 x 100 = 80

3. If your number is a mixed number, add the two values you have calculated previously together to determine the percentage equivalent

For example to convert 2 4/5 to a percentage do 200 + 80 = 280%

Converting Percentages to
Fractions or Mixed Numbers

Similarly, you may sometimes wish to convert between percentages and fractions or mixed numbers. You can convert a percentage to a fraction or mixed number by following these steps:

1.Rewrite the percentage as a fraction with a denominator of 100

For example to convert 112.5% to a fraction or mixed number write it as 112.5/100

2. If necessary, multiply the numerator and denominator of your fraction by appropriate multiples of ten to ensure the numerator is a whole number.

For example to convert 112.5% to a fraction or mixed number multiply the numerator and denominator by 10 to give 1125/1000

3. Simplify the fraction, converting improper fractions to mixed numbers if necessary

For example to convert 112.5% to a fraction or mixed number convert 1125/1000 to 1 1/8

1.Rewrite the percentage as a fraction with a denominator of 100

For example to convert 112.5% to a fraction or mixed number write it as 112.5/100

2. If necessary, multiply the numerator and denominator of your fraction by appropriate multiples of ten to ensure the numerator is a whole number.

For example to convert 112.5% to a fraction or mixed number multiply the numerator and denominator by 10 to give 1125/1000

3. Simplify the fraction, converting improper fractions to mixed numbers if necessary

For example to convert 112.5% to a fraction or mixed number convert 1125/1000 to 1 1/8

Examples: Converting between
Percentages and Fractions or Mixed Numbers

1. Convert the following percentages to fractions or mixed numbers:

a.25%

25/100 = 1/4

b.165%

165/100 = 1 13/20

2. Convert the following fractions or mixed numbers to percentages:

a.3/8

3 ÷ 8 x 100 = 37.5%

b.2 2/5

2 x 100 = 200

2 ÷ 5 x 100 = 40

200 + 40 = 240%

25/100 = 1/4

165/100 = 1 13/20

3 ÷ 8 x 100 = 37.5%

2 x 100 = 200

2 ÷ 5 x 100 = 40

200 + 40 = 240%

End of Topic

Congratulations, you have completed this topic.

You should now have a better understanding of Percentages.

 


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