**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.

- Fractions, decimals and percentage conversions. 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.

**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}

**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

**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

**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\%}

### Take an Online Exam

#### Percentage of Amount & Percentage Change Online Exam

### Example Questions

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

**[2 marks]**

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

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

**[2 marks]**

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\%

**Question 3:** Mildred’s salary has increased from £24,600 to £25,338. By what percentage has her salary increased?

**[2 marks]**

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\%

**Question 4:** 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]**

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\%.

### Worksheets and Exam Questions

#### (NEW) Percentages Exam Style Questions - MME

Level 4-5 New Official MME### Drill Questions

#### Ratio and Percentages

#### Fractions of amounts and percentages

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