What are percentages?
Percent or % means 'out of one hundred'.
In this diagram, \({30}\) out of \({100}\) squares have been shaded. So 30% has been shaded.
Examples of percentages
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Writing one number as a percentage of another
Key point
To write one number as a percentage of another, divide the first number by the second number, then multiply by 100.
Question
a) A dress which originally cost \(\pounds{80}\) is reduced by \(\pounds{10}\) in the sale. What percentage reduction is this?
b) A swimming team has \({20}\) members and \({12}\) of these are boys.What percentage of the swimming team are boys?
Answer
a) \(10 \div 80 = 0.125\)
\(0.125 \times 100\% = 12.5\%\)
So the dress has been reduced by \(12.5\%\).
b) \(\frac{12}{20} \times 100 = 60\)
So \(60\%\) of the team are boys.
Finding the percentage of a quantity
It's often useful to be able to find a percentage of a quantity.
For example, you might be told that bus fares are going up by \(5\%\) and you need to know how much more you will need to pay each week.
Here are a couple of ways you could do it:
Method 1
Find \(1\%\) of the quantity by dividing it by \({100}\).
Find \(5\%\) of a quantity by multiplying \(1\%\) of the quantity by \(5\).
To find \(x\%\) of a quantity you need to multiply \(1\%\) of the quantity by \(x\).
Question
Rakesh has a box containing \({60}\) pens. \(20\%\) of the pens are red.
How many red pens does the box contain?
Answer
You need to find \(20\%\) of \({60}\).
\(1\%\) of \({60}\) is \(60 \div 100 = 0.6\)
So \(20\%\) of \({60}\) is \(0.6 \times 20 = 12\)
Method 2
Find the percentage of a quantity by multiplying that quantity by the percentage expressed as a fraction.
This combines the division and multiplication of Method 1 into one sum.
Questions
Q1. In a class of \({25}\) pupils, \(24\%\) live in a flat. How many pupils live in a flat?
Q2. Last year Sam earned \(\pounds{20,000}\). This year her pay has increased by \(8\%\).
a) By how much money has Sam's pay increased?
b) What is her new salary?
Answers
A1. \(24\%\) is equivalent to \(\frac{24}{100}\)
\(24\%\) of \(25 = \frac{24}{100} \times 25 = 6\)
Therefore \(6\) pupils live in a flat.
A2.
a) Method 1:
\(1\%\) of \(20,000 = 20,000 \div 100 = 200\)
\(8\%\) of \(20,000 = 8 \times 200 = 1,600\)
b) Method 2:
\(\frac{8}{100} \times 20,000 = 1,600\)
So Sam's new salary is:
\(\pounds{20,000} + \pounds{1,600} = \pounds{21,600}\)
How to work out the percentage of an amount
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Work out percentages slideshow
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A bargain for Gran: Calculating percentages
Test section
Question 1
\({40}\) squares are shaded in a \({100}\)-square grid.
What percentage of the grid is shaded?
Answer
\(\frac{40}{100}\), which is \({40}\%\).
Question 2
There are \({35}\) squares shaded in a \({50}\)-square grid.
What percentage of the grid is shaded?
Answer
\(\frac{35}{50}=\frac{70}{100}\), which is \({70}\%\).
Question 3
What is \({40}\%\) of \({90}\) people?
Answer
The method of calculating a percentage of a quantity is to find \({1}\%\) then multiply it to get the percentage you need.
\({90}\div{100}={0.9}\)
\({0.9}\times{40}={36}\).
So the answer is \({36}\) people.
Question 4
What is \({35}\%\) of \({210}~{g}\)?
Answer
The method of calculating a percentage of a quantity is to find \({1}\%\) then multiply it to get the percentage you need.
\({210}~{g}\div{100}={2.1}~{g}\)
\({2.1}~{g}\times{35}={73.5}~{g}\).
Question 5
There is \({15}\%\) off the price of clothes in a shop.
A sweater's original price is \(\pounds{36}\).
What is its price after the reduction?
Answer
Remember that one way of calculating this answer is to find \({1}\%\), then multiplying it to get the percentage you need.
Then, you need to subtract this number from the original sum.
\(\pounds{36}\div{100}=\pounds{0.36}\)
\(\pounds{0.36}\times{15}=\pounds{5.40}\)
\(\pounds{36}-\pounds{5.40}=\pounds{30.60}\)
Question 6
The number of participants in a cycling competition has increased by \({20}\%\) since last year.
If there were \({1,240}\) competitors last year, how many are competing this year?
Answer
Remember that one way of calculating the answer is to find \({1}\%\), then multiply it to get the percentage you need.
Then, you need to add this number to the original sum.
\({1,240}\div{100}={12.4}\)
\({12.4}\times{20}={248}\)
\({1,240}+{248}={1,488}\)
Question 7
Ben gets \(\frac{14}{20}\) in his Geography test.
What percentage is this?
Answer
Remember that one way of calculating a percentage is to convert the fraction into an equivalent fraction over \({100}\).
\(\frac{14}{20}=\frac{70}{100}={70}\%\)
Question 8
In a class of \({30}\) pupils, \({12}\) are boys.
What percentage of the class are boys?
Answer
Remember that one way of calculating a percentage is to convert the fraction into an equivalent fraction over \({100}\).
\(\frac{12}{30}=\frac{4}{10}=\frac{40}{100}={40}\%\)
Question 9
Diane eats \({4}\) pieces of a \({16}\) piece chocolate bar.
What percentage of the bar has she eaten?
Answer
Remember that you have to divide to get the decimal, then multiply by \({100}\) to get the percentage.
\({4}\div{16}={0.25}={25}\%\)
Question 10
If \({27}\) out of \({60}\) cars are red, what is the percentage of non-red cars?
Answer
After calculating how many cars are not red, remember to divide to get the decimal, then multiply by \({100}\) to get the percentage.
\({60}-{27}={33}\)
\({33}\div{60}={0.55}={55}\%\)
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