Percent quite literally means 'per' 'cent'. **Cent** means **one** **hundred**, and so percent means **per** **hundred**. We denote percentages using the **percentage** **symbol** %.

In a class of students, 62% have brown hair. This means that, if there were 100 students, 62 of them would have brown hair. We could also say that the remaining 38 of them would not have brown hair as they are not in the 62%.

Percentages are also great as they enable us to make **comparisons**. For example, if a student scored 51% on their maths exam, and 63% on their English exam, they can say they did better in English than in maths, despite the fact that both exams are very different in structure. So, let's now talk about how to actually **calculate** **percentages**.

## Percentage Formula

### Working out Percentages

Suppose you did a maths test and got 32 questions correct out of a total of 48. To determine how well you did in comparison to the total number of questions, you may wish to express this score as a percentage. Luckily, there is a handy percentage formula that enables us to do this.

To work out something as a percentage of a total amount, we do as follows:

$percentage=\frac{score}{total}\times 100\%$

Here, the 'score' is the amount and the 'total' is the total available.

### Percentage Formula with Example

Back to the original example, if you scored 32 out of 48 on a test, the **score** is 32 and the **total** is 48. In this case, the percentage is $\frac{32}{48}\times 100\%=67\%$. So you scored 67% on your test, which is pretty good. However, if you need 70% to pass, you may need to revise a little more...

### Percentages of Amounts

Sometimes, we are **given** a **percentage** and are required to do the **reverse** and find the **score**. This is called finding the **percentage** **of** **an** **amount**. For example, if we take a test, and we need 75% to pass, then we may wish to know how many marks we actually need to pass.

Below is a table that enables us to work out some common values. Using these facts, we can work out pretty much any percentage of anything.

Percentage | Method for Finding Percentage of Amount | Example |

50% | Half the amount (divide the amount by two). | Find 50% of 600:600 divided by 2 is 300. Thus, 50% of 600 is 300. |

25% | Half the amount, and then half again (divide the amount by four). | Find 25% of 600:We already know that half of 600 is 300, so one-quarter of 600 is 150 as half of 300 is 150. |

10% | Divide the amount by 10. | Find 10% of 600:600 divided by 10 is 60. Thus 10% of 600 is 60. |

5% | Half 10% | Find 5% of 60010% is 60, so 5% must be half of 60 which is 30. |

1% | Divide the amount by 100. | Find 1% of 600:600 divided by 100 is 6. Thus 1% of 600 is 6. |

Using this table, we can work out any percentage by combining percentages. For example, if we wanted to work out 28%, we could do$2\times 10\%+5\%+3\times 1\%$.

**Work out 36% of 120**

**Solution:**

Firstly, we can say that$36\%=30\%+6\%$.

__Finding 30%:__

We know that $30\%=10\%\times 3$and$10\%of120=120\xf710=12$.

Thus,$30\%of120=12\times 3=36$.

__Finding 6%__

Now, $6\%=5\%+1\%$

$5\%of120=12\xf72=6$.

$1\%of120=120\xf7100=1.2$

Thus,$6\%of120=6+1.2=7.2$.

__Finding 36%__

Thus,$36\%of120=36+7.2=43.2$.

### Fractions, Decimals and Percentages

Percentages are simply a way of representing information. We can also represent the same information using fractions and decimals. Consequently, percentages can be converted to fractions and decimals and vice versa. We follow the following rules:

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

To convert a fraction to a percentage, multiply the fraction by 100.

To convert a percentage to a decimal, divide the percentage by 100.

To convert a percentage to a fraction, write the percentage over 100 and simplify.

We will now look at some examples to put these rules into practice.

**Convert 34% to a fraction and decimal. **

**Solution:**

To convert a percentage to a fraction, we simply put it over one hundred. Recall that percent means per hundred, so 34% is 34 out of 100, or $\frac{34}{100}$. Now, this fraction can be simplified by dividing both the numerator and denominator by two. Thus, $\frac{34}{100}=\frac{17}{50}$.

Now, to convert to a decimal, we take the percentage and divide it by 100. So, in this case, we take 34 and divide it by 100. We can do this by shifting the decimal place two spaces to the left. So, it becomes 0.34. Thus,$34\%=0.34$.

**Convert$\frac{3}{10}$ to a percentage.**

**Solution:**

To convert a fraction to a percentage, we multiply the fraction by 100.

In this case, we get $\frac{3}{10}\times 100=\frac{300}{10}=30\%$. Thus, $\frac{3}{10}=30\%$.

**Convert 0.07 to a percentage**

**Solution**

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

In this case, we get $0.07\times 100=7\%$. Thus, $0.07=7\%$

## Percentage Change Formula

Suppose we were looking to book a holiday to France. We look at flights on Tuesday, and they cost £150. However, by Friday, the price has increased to £180. The cost has changed by a percentage of the original amount. To work out this percentage change, we can use the percentage change formula. It is as follows:

Percentage change = $\frac{difference}{originalamount}\times 100\%$.

Here, the difference is the difference between the **initial** **value** and the **new** **value** and can be calculated by **subtracting** the **bigger** value from the **smaller** value (this will change depending on whether the amount has increased or decreased).

For example, if the cost of a tablet was £500, and then increased to £550, the difference is$550-500=\pounds 50$. However, if the cost instead decreased from £500 to £480, the difference would be$500-480=\pounds 20$. Simple right? Let's do some examples just to be sure.

On Monday, Sam took a test and scored 56 out of 82. On Wednesday, he retook the same test and scored 78. What was his percentage change?

**Solution:**

The difference is$78-56=22$. Thus, the percentage change is $\frac{22}{82}\times 100\%=26.8\%$. Therefore, Sam scored **26.8% higher** on Wednesday than he did on Monday.

Dave brought a house for £296,000. He sold the house for £400,000. Calculate the percentage increase in price.

**Solution:**

The difference is$400000-296000=\pounds 104000$. Thus, the percentage change is$\frac{104000}{296000}\times 100\%=35.1\%$. Therefore, Dave has made a profit of **35.1%** of what he originally paid.

A company sold 31,250 televisions in 2020. In 2021, they sold 29,876 televisions. Calculate the percentage decrease in the number of televisions sold.

**Solution:**

The difference is$31250-29876=1374$. Therefore, the percentage change is$\frac{1374}{31250}\times 100\%=4.4\%$. Thus, the company sold 4.4% less televisions in 2021 than 2020.

## Percentage increase

Suppose flights to Doha usually cost £500, however, a sports event has caused the flights to increase in price by 50% in July. We may wish to find the new cost of the flights. In this section, we will talk about the percentage increase.

To work out the new price after the percentage increase, we calculate the percentage of the amount that it has gone up by. So in this case, we would work out 50% of £500 to work out how much the price has increased. Then, we add it to the original amount to find the new price.

Kevin buys a house for £250,000. After refurbishing, the house is now worth 10% more. He decides to sell the house. How much does the house cost now?

**Solution:**

We first need to work out 10% of £250,000 to determine how much the price has increased by.

$10\%of250,000=250,000\xf710=\pounds 25,000$. Thus, the cost of the house has increased by £25,000.

Thus, the new price of the house is$\pounds 250,000+\pounds 25,000=\pounds 275,000$.

## Percentage Decrease

Similar to a percentage increase, we can also have a percentage decrease. This is the exact same idea, but rather than increasing something by an amount, we a decreasing something by an amount. We follow the same method, except rather than adding the amount to the original amount, we take it away.

**In a shop, all items are reduced by 30%. Sam wants to buy a t-shirt that is marked as £30 before the discount. He also wants to buy a pair of jeans that are labelled as £45 before the reduction. Sam has £52. Does he have enough money to buy both items? **

**Solution:**

The t-shirt is £30 and we need to decrease this by 30% to work out the new price. To do so, we work out 30% of £30. We can say that 10% of £30 is £3 since $30\xf710=3$. Now, to work out 30%, we need to multiply 10% by 3. So, 30% of 30 is £9.

We, therefore, need to reduce £30 by £9 to get the new price. Thus, the new price of the t-shirt is $\pounds 30-\pounds 9=\pounds 21$.

The pair of jeans is £45 so we need to reduce this by 30% to work out the new price of the jeans. We know that 10% of £45 is £4.50 since $45\xf710=4.5$. Thus, 30% of £45 must be £13.50 since$4.5\times 3=13.5$. Therefore, we need to decrease £45 by £13.50 to get the new price. Since $45-13.50=31.50$we know that the new price of the jeans is £31.50.

Therefore, the cost of the t-shirt and the jeans together is $31.50+21=52.50$. So, Sam needs £52.50. Since he only has £52, he does **not** have enough money to buy both items.

## Percentage - Key takeaways

- Percent quite literally means 'per' 'cent', with 'cent' meaning one hundred.
- To work out a percentage, we take the amount, divide it by the total and then multiply this by 100.
- To work out a percentage difference, we find the difference between the two values, divide this by the original, and then multiply by 100.
- Things can increase and decrease by percentages, so it is also useful to be able to work out percentage increases and decreases.

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##### Frequently Asked Questions about Percentage

What is the percentage?

The percentage is the amount per hundred.

How to calculate the percentage?

Divide the score by the total and then multiply by 100.

What is the purpose of percentage?

It enables us to compare data.

How to find a number as a percentage?

Divide the number by the total and then multiply by 100.

How to work out percentage?

Divide the score by the total and then multiply by 100.

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