Fractions

A fraction is a part of a whole number. Fractions appear in our lives everyday: eating a slice of a whole pizza that was cut into eight, separating a class of \(20\) students into groups of \(5\), \(5\), and \(10\).

A fraction is written with a numerator on top and denominator on the bottom separated by a line such as \(\frac{1}{8}, \frac{5}{20} \). The denominator tells you how many pieces the whole was separated into (like a pizza cut into 8 equal slices). The numerator tells you how many are of interest (like eating 1 of the 8 slices). If you eat \(1\) of \(8\) slices, than \( \frac{7}{8} \) are left.

You can visually represent fractions by taking an object, dividing it into equal parts and shading in the parts of interest.

	\(\cfrac{1}{4}\)
	\(\cfrac{2}{3}\)

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Equivalent Fractions

Equivalent fractions are fractions that are written differently but are actually equal. For example, if you cut up a pizza into \(4\) slices and eat \(2\) of them you have eaten half the pizza (\(\frac{2}{4}\)) which is the same thing as cutting up the same pizza into 6 slices and eating 3 slices (\(\frac{3}{6}\)). Both of these are equivalent to \(\frac{1}{2}\). All three of these fractions are equal or equivalent:

\(\cfrac{2}{4} = \cfrac{3}{6} = \cfrac{1}{2}\)

You can make an equivalent fraction by multiplying or divided both the numerator and denominator of a fraction by the same number. Let's take the fraction \( \frac{6}{14} \) and create an equivalent fraction by multiply the numerator and denominator by \(3\) and another equivalent fraction by dividing both the numerator and denominator by \(2\):

\( \cfrac{6}{14} = \cfrac{6*3}{14*3} = \cfrac{18}{42}\)

\( \cfrac{6}{14} = \cfrac{6/2}{14/2} = \cfrac{3}{7}\)

All of these fractions are equivalent:

\(\cfrac{6}{14} = \cfrac{18}{42} = \cfrac{3}{7}\)

Why does this work? Well, when you multiply/divide by the same number to the numerator and denominator, you are effectively multiplying the fraction by \(1\). When you multiply something by \(1\), it does not change! This is a math trick to manipulate numbers (or expressions) you will see throughout your math classes.

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Reducing Fractions

Reducing a fraction means to find an equivalent fraction with the lowest, simpliest numbers such that you cannot reduce it anymore. To reduce a fraction, we will divide the numerator and denominator by the same number. You need to use factors of the numerator/denominator so that when you divide it, you end up with a whole number.

Example

Reduce the fraction \( \cfrac{20}{100}\).

We want to find a number to divde both the top and bottom so that we get whole numbers. We can divide both by \(2\):

\(\cfrac{20}{100} = \cfrac{20/2}{100/2} = \cfrac{10}{50}\)

But notice, we can continue reducing! There are other numbers we can divde by to simplify this fraction. Let's divide them both by \(10\):

\(\cfrac{10}{50} = \cfrac{10/10}{50/10} = \cfrac{1}{5}\)

We cannot reduce any further because the numerator is 1. You cannot divde it by anything because it will give a fraction. Therefore, we reduced the fraction to its lowest terms:

\(\cfrac{20}{100} = \cfrac{1}{5}\)

Now, there are a lot of different ways we could have ended up with this answer. The fastest way is to find the greatest common factor (GCF) between the numerator and denominator:

\(20: 20, 1, 10, 2, 5, 4\)

\(100: 100, 1, 50, 2, 25, 4, 5, 20, 10\)

The greatest common factor is \(20\) so dividing both top and bottom by \(20\) would give us the reduced fraction:

\(\cfrac{20}{100} = \cfrac{20/20}{100/20} = \cfrac{1}{5}\)

NOTE: try to divde both top and bottom by the numerator. That will give you a \(1\) in the numerator and reduce the fraction (only if you get a whole number in the denominator!).

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Converting Fractions to Decimals

Since fractions are part of a whole, we can represent them with decimal numbers. For example, \(\cfrac{1}{2} = 0.5\).

To convert a fraction to decimal, you can divide the numbers in your calculate. Give it a try, what is \(\cfrac{3}{8} \) as a decimal? Since dividing numbers by \(100\) is nice and easy, one strategy is to write an equivalent fraction with a denominator of \(100\). Then you can conert to decimal without a calculator.

Example

Convert \(\cfrac{6}{25}\) to a decimal number.

We want to write an equivalent fraction with a bottom of \(100\):

\(\cfrac{6*?}{25*?} = \cfrac{?}{100}\)

In order to turn the \(25\) to \(100\) we have to multiply by \(4\). We will also need to multiply the top by \(4\) to keep the fractions equal:

\(\cfrac{6*4}{25*4} = \cfrac{24}{100}\)

Since a fraction represents division, dividing \(24\) by \(100\) involves moving the decimal point two places to the left:

\(\cfrac{24}{100} = 0.24\)

Therefore, \(\cfrac{6}{25}\) is \(\boldsymbol{0.24}\) as a decimal.

Not all numbers will work out this nicely. You can try to write an equivalent fraction using a base with tens like \(10, 100, 1000\).

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Converting Fractions to Percents

We might also want to express a fraction as a percent. This could be helpful to see how much of a whole the fraction occupies. To convert a fraction to a percent, first convert to a decimal. Then multiply by \(100%\):

\( = \cfrac{24}{100} = 0.24 \)

\(0.24 * 100\% = 24\% \)

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