Percents, fractions & decimals

Examples

• If we have 100 pizzas and 35 of them are pepperoni, 10 are cheese, and 55 are veggie: 35% pepperoni, 10% cheese, 55% veggie = 100% of all pizzas.
• If we have 20 hats (${\displaystyle 20*5=100}$) and 7 are blue, 9 are green, and 4 are red: 35% blue (${\displaystyle 7*5=35}$), 45% green (${\displaystyle 9*5=45}$), 20% red (${\displaystyle 4*5=20}$) = 100% of all hats.

Simplifying fractions

Fractions can be simplified down to their lowest term by dividing both the numerator and denominator by the greatest common factor.

A greatest common factor is the highest factor that both the numerator and denominator share. For more information on factors: Factors and multiples. Dividing the numerator and denominator by the same number will ensure that the fraction retains the same value, but is represented in a simpler form.

For example, if my fraction is ${\displaystyle {\frac {4}{10}}}$, I would determine my factors for both 4 and 10. I know that 4's factors are 1, 2, 4 while 10's factors are 1, 2, 5, 10. Between the two of them, the highest factor is 2. Dividing both the numerator and denominator by 2 will result in the fraction: ${\displaystyle {\frac {2}{5}}}$.

We can check if we simplified correctly using multiple methods. The simplest one is converting both the unsimplified and simplified forms to decimals by dividing the numerator by the denominator. If both are the same, then the simplification was done correctly.

${\displaystyle {\frac {4}{10}}\ =0.4}$

${\displaystyle {\frac {2}{5}}\ =0.4}$

Another way is to set the two fractions equal to each other and cross-multiply.

${\displaystyle {\frac {4}{10}}\ ={\frac {2}{5}}\ }$

Cross-multiplying means multiplying fraction 1's numerator with fraction 2's denominator and vice versa. Therefore, we would do ${\displaystyle 4*5=20}$ and ${\displaystyle 10*2=20}$. Since both are the same, the simplification is correct, and both fractions are equivalent.

Examples

• If we have 50 fruits and 43 of them are grapes and the other 7 are raspberries: ${\displaystyle {\frac {43}{50}}}$ is the fraction for grapes and ${\displaystyle {\frac {7}{50}}}$ is the fraction for raspberries.
• If we have 37 bunnies and 16 of them are white and the other 21 are black: ${\displaystyle {\frac {16}{37}}}$ is the fraction for white bunnies and ${\displaystyle {\frac {21}{37}}}$ is the fraction for black bunnies.

Examples

• 0.25
• 3.75
• 9.9
• 10.16
• 100.44

Percents to fractions

Converting percents to fractions is as simple as putting the percent over 100.

For example, 17% is the same as ${\displaystyle {\frac {17}{100}}\ }$.

If we have 26%, we can write it as ${\displaystyle {\frac {26}{100}}\ }$ and then simplify it using the simplification method we learned before! We will find the factors of 26 (1, 2, 13, 26) and 100 (1, 2, 4, 5, 10, 20, 25, 50, 100). The greatest common factor is 2. Divide both the numerator and denominator by 2 to get ${\displaystyle {\frac {13}{50}}\ }$.

Fractions to percents

If we want to convert fractions to percents, we would have to find a way to convert the denominator to 100.

For example, ${\displaystyle {\frac {4}{5}}\ }$: we know that ${\displaystyle 5*20=100}$. So, we will multiply both the top and bottom by 20, giving us ${\displaystyle {\frac {60}{100}}\ }$, which is the same as 60%.

This method becomes harder and more advanced for denominators that do not evenly multiply to 100.

Percents to decimals

Converting percents to decimals follows the same initial process as converting percents to fractions. First, we must put the percent over 100. But after, we will divide the numerator by the denominator.

For example, 3% is the same as ${\displaystyle {\frac {3}{100}}\ }$. ${\displaystyle 3\div 100=0.03}$, which is three-hundredths.

Percents to decimals follows the path of percents → fractions → decimals.

Decimals to percents

In order to convert decimals to percents, we must be able to represent the decimal in fraction format first with a denominator of 100. Only then can we convert it to a percent.

For example, 0.29 is the same as ${\displaystyle {\frac {29}{100}}\ }$, which then can be written as 29%.

Fractions serve as the intermediate for both conversions between percents and decimals or vice versa.

Fractions to decimals

In order to make a fraction a decimal, divide the numerator by the denominator.

For example, ${\displaystyle {\frac {4}{10}}\ }$ can be written as ${\displaystyle 4\div 10=0.4}$.

Decimals to fractions

To convert decimals to fractions, the best way is to place the decimal over 100 and then simplify.

For example, 0.19 can be written as ${\displaystyle {\frac {19}{100}}\ }$.

However, if we had 0.22, we can be write it as ${\displaystyle {\frac {22}{100}}\ }$ and then simplify using the greatest common factor of 2 to ${\displaystyle {\frac {11}{50}}\ }$.

For more: Return to Pre-Algebra