# Percentages

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## Percentages

Percentage means “number of parts per one hundred” and is denoted by the $\bf{\%}$ sign. For example, $50\%$ of a number means $50$ parts of it out of a total of $100$, and since $50$ is one half of $100$, $50\%$ means half of the total amount.

Make sure you are happy with the following topics before continuing.

This will help with this topic where we will look at how to find a percentage of an amount as well as percentage increase and decrease.

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## Skill 1: Percentage of an Amount

To calculate the percentage of an amount, we convert the percentage to a decimal or fraction and then multiply this by the amount.

Example: Calculate $\textcolor{blue}{16 \%}$ of $\textcolor{blue}{60}$.

With a calculator:

We need to multiply $\textcolor{blue}{16\%}$ as a decimal by $60$

$0.16 \times 60 = \dfrac{16}{100} \times 60 = 9.6$

Without a calculator it is better to split $\textcolor{blue}{16\%}$ into amounts that are easier to work out such as $\textcolor{blue}{10\%}$, $\textcolor{blue}{5\%}$,  and  $\textcolor{blue}{1\%}$

$\textcolor{blue}{10\%} \text{ of } 60 = 6, \quad \textcolor{blue}{5\%} \text{ of } 60 = 3, \quad \textcolor{blue}{1\%} \text{ of } 60 = 0.6$

\begin{aligned} \textcolor{blue}{16\%} & = 10\% + 5\% + 1\% \\ &= 6 +3+ 0.6 = 9.6 \end{aligned}

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## Skill 2: Percentage Increase

For a percentage increase, the decimal or fraction that you multiply the amount by will be greater than $\bf{1}$.

Example: Jane deposits $\textcolor{blue}{£1,360}$ into her bank account which has an interest rate of $\textcolor{blue}{2.2\%}$ per year. Assuming that she does not deposit or withdraw any money, how much money will she have in a year’s time?

The new total value of Jane’s account will be equal the original total plus $\textcolor{blue}{2.2\%}$ of the original total. To find this total we multiply $\textcolor{blue}{£1,360}$ by $\textcolor{blue}{1 + 0.022} = \textcolor{blue}{1.022}$.

Therefore, the total value is,

$\textcolor{blue}{1,360} \times \textcolor{blue}{1.022} = \pounds 1,389.92$

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## Skill 3: Percentage Decrease

For a percentage decrease, the decimal or fraction that you multiply the amount by will be less than $\bf{1}$.

Example:

If Jane decides to withdraw $\textcolor{blue}{25\%}$ of the total $\textcolor{blue}{£1,389.92}$, we find the decimal equivalent as  $1 - 0.25 = \textcolor{blue}{0.75}$

Therefore after the withdrawal, the value of the account is,

$\textcolor{blue}{1,389.9} \times \textcolor{blue}{0.75} = \pounds 1,042.44$

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## Skill 4: Percentage Change

Percentage change is used to find the change in a value as a percentage.

$\text{\textcolor{Black}{Percentage `Change'}} = \dfrac{\text{\textcolor{Red}{Change}}}{\text{\textcolor{blue}{Original}}} \times \textcolor{black}{100}$

Example: Calculate the percentage change when a car goes down in value from $\textcolor{blue}{£8,500}$ to $\textcolor{black}{£7,000}$.

Using the equation above we can calculate the percentage change by first calculating the difference, which is,

$\textcolor{black}{£8,500} - \textcolor{black}{£7,000} = \textcolor{Red}{£1,500}$

We then need to divide this difference by the original amount and multiply by $\textcolor{black}{100}$ to get the percentage change:

$\text{\textcolor{black}{Percentage Change}} = \dfrac{\textcolor{Red}{£1,500}}{\textcolor{blue}{£8,500}} \times \textcolor{black}{100} = \textcolor{black}{17.65\%}$

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## Percentages Example Questions

Question 1: Without using a calculator, work out $33\%$ of $180$.

[2 marks]

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In order to solve this question, we will need to break down the $33\%$ into easier, and more manageable, chunks.  $33\%$ can be broken down as follows:

$3 \times 10\% + 3 \times 1\%$

$10\%$ is a very easy amount to calculate since all we need to do is divide by $10$:

$10\%\text{ of }180 = 180\div 10 = 18$

Therefore, $30\%$ of $180$ is $3\times 18 = 54$.

Next, we will find $3\%$ by first finding $1\%$ and multiplying the answer by $3$. To find $1\%$ of $180$, we need to divide the total by $100$:

$1\%\text{ of }180 = 180\div 10 = 1.8$

Therefore, $3\%$ of $180$ is $3\times 1.8 = 5.4$.

Finally, we need to add together the $30\%$ amount and the $3\%$ amount, so $33\%$ is:

$54+5.4=59.4$

Gold Standard Education

Question 2: Matteo scored $99$ out $150$ on an exam. What is his score as a percentage?

[2 marks]

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To convert anything into a percentage, it is a lot easier to write the amount as a fraction first.  If Matteo scored “$99$ out of $150$”, then we should write this as:

$\dfrac{99}{150}$

To convert a fraction into a percentage, you need to divide the top by the bottom (it helps if you remember that the line in a fraction means ‘divide’) and then multiply by $100$.

So, Matteo’s score as a percentage can be calculated as follows:

$(99 \div 150) \times 100 = 66\%$

Gold Standard Education

Question 3: The value of an antique increases by $6\%$ from $£520$

What is the value of the antique now?

[2 marks]

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The value of the antique now will be the original value plus $6\%$ of the original value.

The multiplier is:

$1 + 0.06 = 1.06$

which corresponds to a percentage increase of $6\%$

So, the value of the antique now is:

$£520 \times 1.06 = £551.20$

Gold Standard Education

Question 4:  Mildred’s salary has increased from $£24,600$ to $£25,338$.  By what percentage has her salary increased?

[2 marks]

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In this question the difference between the two salaries is

$\pounds25,338 - \pounds24,600 = \pounds738$

The original amount (the amount before it was increased) was $£24,600$, so the percentage increase can be calculated as follows:

$\dfrac{\pounds738}{\pounds24,600} \times 100 = 3\%$

Gold Standard Education

Question 5: The price of a motorbike is reduced by $10\%$.  In a sale, the new price is reduced further by $10\%$.  By what percentage has the original price of the scooter been reduced in this sale?

[3 marks]

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To most people, this would appear a very easy question with an answer of $20\%$, but this answer is, sadly, incorrect!

The easiest thing to do to solve this question is to invent a price for the motorbike.  You can invent any price you want, but it would be advisable to make the price a nice, easy number and, since this question concerns percentages, giving the motorbike a price of $£100$ makes life extremely easy.

If the motorbike costs $£100$, when if it is reduced by $10\%$, then its new value is $£90$.

If the motorbike now costs $£90$ and is further reduced by $10\%$, then we need to deduct $10\%$ from this $£90$ value (and not the previous $£100$ value).

$10\%$ of $£90 = £9$

So the new value of the motorbike is $£81$.

So the motorbike has decreased in value from $£100$ to $£81$.  Since we set the motorbike’s original price as $£100$, the percentage decrease here should be relatively obvious.  If not, remember that to calculate a percentage decrease (or increase), you need to divide the difference between the two values by the original value and multiply by $100$.

The original value of the motorbike was $£100$, and its new value is $£81$, so the percentage decrease can be calculated as follows:

$\dfrac{\pounds100 - \pounds81}{\pounds 100} \times 100 = 19\%$

Therefore, the motorbike has decreased in value by $19\%$ and not $20\%$.

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