##### Percentage

A percentage is a __fractions__ with a denominator of 100. The symbol used to represent percent is %. To find 40% of 25, you simply need to find $ \displaystyle \frac{{40}}{{100}}$ of 25. Using what you know about multiplying __fractions__:

$ \displaystyle \frac{{40}}{{100}}\times 25=\frac{{\cancel{{40}}}}{{\cancel{{100}}}}\times \frac{{25}}{1}=$

$ \displaystyle \frac{2}{{\cancel{5}}}\times \frac{{\cancel{{25}}}}{1}=\frac{2}{1}\times \frac{5}{1}=10$

So, 40% of 25 = 10

### Equivalent forms

A percentage can be converted into a decimal by dividing by 100 (notice that the digits move to places to the right). So, 45%=$\displaystyle \frac{{45}}{{100}}$=0.45 and $\displaystyle \frac{{3.1}}{{100}}$.

A decimal can be converted to a percentage by multiplying by 100(notice that the digits move to places to the left). So, $ \displaystyle 0.65=\frac{{65}}{{100}}=$65% and $ \displaystyle 0.7\times 100=$70%.

Converting percentages to vulgar fractions (and vice versa) involves a few more stages.

Example 1: Convert each of the following percentages to fractions in their simple form.

a) 25%

Write as __fraction__ with a denominator of 100, then simplify.

25%$ \displaystyle =\frac{{25}}{{100}}=\frac{1}{4}$

b) 30%

Write as __fraction__ with a denominator of 100, then simplify.

30%$ \displaystyle =\frac{{30}}{{100}}=\frac{3}{{10}}$

c) 3.5%

Write as __fraction__ with a denominator of 100, then simplify.

3.5%$ \displaystyle =\frac{{3.5}}{{100}}=\frac{{35}}{{1000}}=\frac{7}{{200}}$

**Note! **Remember that a fraction that contains a decimal is not in its simplest form.

Example 2: Convert each of the following fractions into percentages.

a) $ \displaystyle \frac{1}{{20}}$

Find the equivalent fraction with a denominator of 100. (Remember to do the same thing to both the numerator and denominator).

$ \displaystyle \frac{1}{{20}}=\frac{{1\times 5}}{{20\times 5}}=\frac{5}{{100}}=$5%

$ \displaystyle \frac{5}{{100}}=0.05$

$ \displaystyle 0.05\times 100$=5%

b) $ \displaystyle \frac{1}{8}$

Find the equivalent fraction with a denominator of 100. (Remember to do the same thing to both the numerator and denominator).

$ \displaystyle \frac{1}{8}=\frac{{1\times 12.5}}{{8\times 12.5}}=\frac{{12.5}}{{100}}$=12.5%

$ \displaystyle \frac{{12.5}}{{100}}=0.125$

$ \displaystyle 0.125\times 100$=12.5%

Although it is not always easy to find an equivalent fraction with a denominator of 100, any fraction can be converted into a percentage by multiplying by 100 and cancelling.

Example 3: Convert the following fractions into percentage:

a) $ \displaystyle \frac{3}{{40}}$

$ \displaystyle \frac{3}{{40}}\times \frac{{100}}{1}=\frac{{30}}{4}=\frac{{15}}{2}=7.5$

So, $ \displaystyle \frac{3}{{40}}$=7.5%

b) $ \displaystyle \frac{8}{{15}}$

$ \displaystyle \frac{8}{{15}}\times \frac{{100}}{1}=\frac{{160}}{3}=53.3$

So, $ \displaystyle \frac{8}{{15}}$=53.3%

### Finding one number as a percentage of another

To write one number as a percentage of another number, you start by writing the first number as a fraction of the second number then multiply by $ \displaystyle {100}$.

Example 4

a) Express 16 as a percentage of 48.

First write 16 as a fraction of 48 then multiply by 100.

This may be easier if you write the fraction in its simplest form first.

$ \displaystyle \frac{{16}}{{48}}=\frac{{16}}{{48}}\times 100$=33.3%

$ \displaystyle \frac{{16}}{{48}}=\frac{1}{3}\times 100$=33.3%

b) Express 15 as a percentage of 75$.

Write 15 as a fraction of 75, then simplify and multiply by 100.

You know that 100 divided by 5 is 20, so you don’t need a calculator.

$ \displaystyle \frac{{15}}{{75}}\times 100=\frac{1}{5}\times 100$=20%

c) Express 18 as a percentage of 23.

You need to calculate $ \displaystyle \frac{{18}}{{23}}\times 100$, but this is not easy using basic fractions because you cannot simplify it further, and 23 does not divide neatly into 100. Fortunately, you can use your calculator.

Simply type: $ \displaystyle 18\div 23\times 100$=78.26%

### Percentage increases and decreases

Suppose the cost of a book increases from 12 to 15. The actual increase is 3. As a fraction of the original value, the increase is $ \displaystyle \frac{3}{{12}}=\frac{1}{4}$. This is the fractional change and you can write this fraction as 25% of the original value. This is called the percentage increase. If the value had reduced (for example if something was on sale in a shop) then it would have been a percentage-decrease.

Be careful! Whenever increases or decreases are stated as percentages, they are stated as percentages of the original value.

Example 5: The value of the house increases from 120000 to 124800 between August and December. What percentage increase is this?

First calculate the increase.

124800-120000=4800

Write the increase as a fraction of the original and multiply by 100.

%increase=$\displaystyle \frac{{increase}}{{original}}\times $ 100%

Then do the calculation (either in your head or using a calculator).

$\displaystyle \frac{{4800}}{{120000}}\times $100%=4%

### Increasing and decreasing by a given percentage

If you know what percentage you want to increase or decrease an amount by, you can find the actual increase or decrease by finding a percentage of the original. If you want to know the new value you either add the increase to or subtract the decrease from the original value.

Example 6: Increase 56 by:

a) 10%

First of all, you need to calculate 10% of 56 to work out the size of the increase.

10% of 56$\displaystyle =\frac{{10}}{{100}}\times 56$

$ \displaystyle \frac{1}{{10}}\times 56=5.6$

To increase the original 10% you need to add this to 56.

56+5.6=61.6

If you don’t need to know the actual increase but just the final value, you can use this method.

If you consider the original to be 100% then adding 10 to this will give you 110% of the original. So multiply 56 by $ \displaystyle \frac{{110}}{{100}}$, which gives 61.6.

b) 15%

A 15% increase will lead to 115% of the original.

$ \displaystyle \frac{{115}}{{100}}\times 56=64.4$

c) 4%

A 4% increase will lead to 104% of the original.

$ \displaystyle \frac{{104}}{{100}}\times 56=58.24$

Example 7: In a sale all items are reduced 15%. If the normal selling price for a bicycle is $120. Calculate the sale price.

Note that reducing a number by 15% leaves you with 85% of the original.

100-15=85.

So you simply find 85% of the original value.

$ \displaystyle \frac{{85}}{{100}}\times \$120=\$102$

### Reverse Percentages

Sometimes you are given the value or amount of an item after a percentage increase or decrease has been applied to it and you need to know what the original value was. To solve this type of reverse percentage question it is important to remember that you are always dealing with percentages of the original values. The method used in example 6 (b) and (c) is used to help us solve these types of problems.

Example 8: A store is holding a sale in which every item is reduced by 10%. A jacket in sale is sold for $108. How can you find the original price of the Jacked?

If an item is reduced by 10%, the new coast is 90% of the original (100-10). If $ \displaystyle x$ is the original value of the jacked then you can write a formula using the new price.

$ \displaystyle \frac{{90}}{{100}}\times x=108$

$ \displaystyle x=\frac{{100}}{{90}}\times 108$

Original price $120.

Notice that when the $ \displaystyle \times \frac{{90}}{{100}}$ was moved to the other side of the $ \displaystyle =$ sign it became its reciprocal, $ \displaystyle \frac{{100}}{{90}}$.

Careful! Undoing 10% decrease is not the same as increasing the reduced value by 10%. If you increase the sale price of $108 by 10% you will get $ \displaystyle \frac{{110}}{{100}}\times 108=118.80$ which is different (and incorrect) answer.