Introduction to Fractions | Dividing Fractions |

Adding Fractions | Comparing Fractions |

Simplifying Fractions | Fractions Used in Cooking |

Subtracting Fractions | More Practice |

Multiplying Fractions |

## Introduction to Fractions

A lot of people have trouble with fractions, but they are really not that complicated. Like decimals, **fractions**** **can be thought as numbers that are in between “normal” numbers (integers), or when **less than $ 1$**, are **part of** something. Here’s a number line:

The numbers with the marks on them are called **integers. **The integer numbers that are positive (to the right of $ 0$, including $ 0$) are called **whole numbers** and the integer numbers to the right of $ 0$ (not including $ 0$) are called **counting numbers**, or **natural numbers**. We’ll talk about the numbers to the left of $ 0$, **negative numbers**, later, in the **Types of Numbers and Algebraic Properties** section.

Let’s first talk about fractions and we’ll use pizza pies to explain them. Everyone likes pizza, right?

Let’s say we are going to our friend’s birthday party and her parents order two large pizzas, which we’ll eat before the cake:

For each **whole** pizza, we can divide it into parts. A fraction is written where the **top **(**numerator**) is the number of parts we’re interested in, and the **bottom** (**denominator**) is the total number of parts.

For example, let’s say each pizza has $ 6$ pieces total, and your friend’s little brother sneaks a piece of pizza from the first pie. Since this pizza (and the other one) are divided into $ 6$ pieces, the piece that is missing represents $ \displaystyle \frac{1}{6}$ of the total first pizza. This is because there are $ 6$ pieces total (the bottom), and this is only $ 1$ piece (the top). It’s as easy as that! Also remember that $ \displaystyle \frac{1}{6}$ is the same thing as “$ 1$ divided by $ 6$”, or we can say “$ 1$ out of $ 6$” or “the ratio of $ 1$** **to $ 6$ or $ 1:6$”, which we’ll talk about more in the **Percentages, Ratios and Proportions** section.

Here are what the pizza pies look like now:

## Adding Fractions

Let’s try to add fractions now, which really isn’t too bad! Here is an example:

Now let’s say little brother eats **another** piece of pizza from the same pie, and let’s add the two fractions:

$ \displaystyle \frac{1}{6}+\frac{1}{6}\,\,=\,\,\frac{{1\text{ }+\text{ }1}}{6}\,\,=\,\,\frac{2}{6}$

Notice that when we **add** the two fractions, we **add across the top** (the numerators) and just keep** the denominator the same**. We always have to have the **same **denominator in order to add or subtract; if we don’t, we have to change the fractions to have the **same denominator**, which we’ll talk about below. **Memorize this!**

Now, $ \displaystyle \frac{2}{6}$ of the first pizza is gone, and we have $ \displaystyle \frac{4}{6}$ left:

Notice that the two pizzas above have the same amount eaten, but we can represent the amounts in two different ways: $ 2$ out of $ 6$ pieces gone, or $ 1$ out of $ 3$ pieces gone. What we have done is “build down” or reduce the fraction $ \displaystyle \frac{2}{6}$ since we have the same factor ($ 2$) that goes in the top of the fraction and the bottom of the fraction; this is also called **simplifying**. Thus, $ \displaystyle \frac{2}{6}$ is the same as $ \displaystyle \frac{1}{3}$. Similarly, $ \displaystyle \frac{4}{6}$ is the same as $ \displaystyle \frac{2}{3}$.

## Simplifying Fractions

To show this more mathematically, to get from $ \displaystyle \frac{2}{6}$ to $ \displaystyle \frac{1}{3}$, notice that $ \displaystyle \require{cancel} \frac{2}{6}=\frac{{1\times \cancel{2}}}{{3\,\times \cancel{2}}}$; we can cross out the $ 2$ on the top and $ 2$ on the bottom (since they divide to equal $ 1$) to get $ \displaystyle \frac{1}{3}$ (also notice that $ \displaystyle \frac{{2\div 2}}{{6\div 2}}=\frac{1}{3}$). To get from $ \displaystyle \frac{4}{6}$ to $ \displaystyle \frac{2}{3}$, notice that $ \displaystyle \require{cancel} \frac{4}{6}=\frac{{2\,\times \cancel{2}}}{{3\,\times \cancel{2}}}$; we can cross out the $ 2$ on the top and $ 2$ on the bottom to get $ \displaystyle \frac{2}{3}$. **We can only do this crossing out if we are multiplying the numbers on the top and the bottom — not adding them.**

The largest number that we can cross out on the top and the bottom of the fraction to reduce it is the **Greatest Common Factor** (**GCF**) from the **Multiplying and Dividing **section. We can also cross out these numbers in phases; for example, we can first cross out $ 2$’s on the top and bottom, then the $ 3$’s, if that’s another factor that goes into both, and so on. This process of simplifying fractions is called “**reducing fractions**” or “**simplifying fractions**”.

Here is another example: If we had $ 6$ people at the party and each one had exactly one piece of pizza, how would we write the fraction that represents all of the pizza eaten, including the two pieces eaten by little brother (still $ 6$ pieces of pizza per pie)?

Each person had one piece, so all $ 6$ ate one whole pizza ($ \displaystyle \frac{6}{6}$ gone). Little brother’s pizza is $ \displaystyle \frac{2}{6}$ or $ \displaystyle \frac{1}{3}$ of the first pizza. Here are what the two pizzas look like:

How much of the total pizza (two pies) is gone? We figure this out by adding the two fractions: $ \displaystyle \frac{6}{6}+\frac{2}{6}=\frac{{6\text{ }+\text{ }2}}{6}=\frac{8}{6}$, which is $ \displaystyle \require{cancel} \frac{{4\,\times \cancel{2}}}{{3\,\times \cancel{2}}}$, which is $ \displaystyle \frac{4}{3}$. Remember again, that we added across with the numbers on top (the numerators) and kept the number on the bottom (the denominator). $ \displaystyle \frac{4}{3}$ is called an **improper fraction**, since the top is bigger than the bottom; improper fractions are greater than $ 1$.

To turn this into what we call a **mixed fraction**** **(a fraction with one “regular” number and one fraction), we would notice that $ 3$ goes into $ 4$ one time, and there is $ 1$ left over (which is a fractional part of $ 3$); the mixed fraction is

$ \displaystyle 1\frac{1}{3}$:

$ \displaystyle \begin{array}{l}3\overset{{1\,\,\text{R1}}}{\overline{\left){{4\,\,\,\,\,}}\right.}}\text{ }\,\,\,\,\,\text{The remainder of 1 is the same as }\displaystyle \frac{1}{3}\\\,\,\,\underline{{\,\,3\,\,\,\,\,\,}}\\\,\,\,\,\,1\end{array}$

Another way to see how this improper fraction $ \displaystyle \frac{4}{3}$ turns into the mixed fraction is to separate the fractions as we did below. The reason we separated $ 4$ into $ 3$ and $ 1$ is because $ 3$ is the highest number that goes into $ 3$, so we can make $ \displaystyle \frac{3}{3}$ into a whole number.

$ \displaystyle \frac{4}{3}\,\,=\,\,\frac{{3+1}}{3}\,\,=\,\,\frac{3}{3}+\frac{1}{3}\,\,=\,\,1+\frac{1}{3}\,\,=\,\,1\frac{1}{3}$

We have $ \displaystyle 1\frac{1}{3}$ of the pizzas gone, or **one** pizza gone, and **one third **of another pizza gone. When we add or subtract mixed fractions, we often do this vertically, and sometimes we have to carry over if what’s on the numerator turns out to be more than the denominator. We work from the right to the left, adding the fractions first. For example, let’s add $ \displaystyle 1\frac{5}{6}+2\frac{3}{6}$. Note that the last mixed fraction that was added ($ \displaystyle 2\frac{3}{6}$) could have been reduced to $ \displaystyle 2\frac{1}{2}$, but we’ll keep it as is, so we can do the addition:

$ \displaystyle \begin{array}{l}\,\,\,\,\,\,\,1\,\,\displaystyle \frac{5}{6}\\\underline{{+\,\,\,2\,\,\displaystyle \frac{3}{6}}}\\\,\,\,\,\,3\,\,\,\displaystyle \frac{8}{6}\,=\,\,\,3+\,\,\displaystyle \frac{8}{6}\,\,=\,\,\,3\,\,+\,\displaystyle \frac{{6+2}}{6}\,\,\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,3\,\,+\displaystyle \frac{6}{6}\,\,+\displaystyle \frac{2}{6}\,\,=\,\,\,3\,\,+1\,\,+\,\displaystyle \frac{2}{6}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,4\,\,+\,\displaystyle \frac{2}{6}\,=\,\,\,4\,\,+\displaystyle \frac{{1\,\times \,\cancel{2}}}{{3\times \,\cancel{2}}}\,\,\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,\,4\,\,+\displaystyle \frac{1}{3}\,\,=\,\,\,4\displaystyle \frac{1}{3}\,\end{array}$

We could have also just turned these into improper fractions first and added:

$ \displaystyle \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\,1\displaystyle \frac{5}{6}\,\,=\displaystyle \frac{{(1 \times 6)+5}}{6}\,\,=\,\,\displaystyle \frac{{11}}{6}\,\,\,\\\,\,\,\,\,\,\,\,\,\,\,2\displaystyle \frac{3}{6}\,\,=\,\,\displaystyle \frac{{(2\,\times \,6)+3}}{6}\,\,=\,\,\displaystyle \frac{{15}}{6}\,\\\\\displaystyle \frac{{11}}{6}+\displaystyle \frac{{15}}{6}\,\,=\,\,\displaystyle \frac{{26}}{6}\,\,=\,\,\displaystyle \frac{{24+2}}{6}\,\,=\,\,\displaystyle \frac{{24}}{6}+\displaystyle \frac{2}{6}\,\,\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\,\,4+\displaystyle \frac{2}{6}\,\,=\,\,4\displaystyle \frac{2}{6}\,\,=\,\,4\displaystyle \frac{1}{3}\end{array}$

Note that after we got the answer $ \displaystyle \frac{{26}}{6}$, we separated the $ 26$ above into $ 24$ and $ 2$, since $ 6$ goes into $ 24$ exactly.

Sometimes with fractions (actually, most of the time!), we won’t have the **same denominator**, so you can’t just add or subtract them across. To add or subtract them, it’s easiest to use we call the **Lowest Common Denominator**, which is the **Least Common Multiple** (**LCM**), from the the Multiplying and Dividing section. Then we’ll have to “build up” our fractions (top and bottom) so we can add the numerators on the top and keep the one denominator on the bottom.

Let’s say we are baking and the recipe calls for $ \displaystyle \frac{2}{3}$ of a cup of sugar and $ \displaystyle \frac{3}{4}$ of a cup of flour. We want to know if our $ \displaystyle 1\frac{1}{2}$ cup measuring cup is large enough to use for both ingredients. We add:

$ \displaystyle \frac{2}{3}+\frac{3}{4}\,\,\,=\,\,\,?$

To find the **Least Common Multiple** (LCM) of the denominators $ 3$ and $ 4$**,** find the smallest number that they both go into:

**MULTIPLES of **$ \boldsymbol {3}$: $ \color{black}{3,6,9},\color{#800000}{12},\color{black}{15},18,21,\color{#800000}{24},\color{black}{27}…$

**MULTIPLES of **$ \boldsymbol {4}$: $ \color{black}{4,8},\color{#800000}{12},\color{black}{16},20,\color{#800000}{24},\color{black}{28,32}…$

(We can **never** have a denominator of $ 0$, so we have to ignore that multiple. Remember, if you have a fraction with a denominator of $ 0$, it is undefined!)

The lowest common denominator is $ 12$. Note in this case that we could have gotten the least common denominator by multiplying the two numbers, since they had no common factors – this is usually a clue that you have to multiply the numbers together to get the lowest common denominator.

Now we have to “build up” the fractions by multiplying each by $ 1$ (or the same number on the top and bottom of the fraction) to get the common denominator:

$ \displaystyle \begin{array}{l}\displaystyle \frac{2}{3}+\displaystyle \frac{3}{4}=\displaystyle \frac{{2\times \color{#800000}{4}}}{{3\times \color{#800000}{4}}}+\displaystyle \frac{{3\times \color{blue}{3}}}{{4\,\times \color{blue}{3}}}\,\,\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,=\displaystyle \frac{8}{{12}}+\displaystyle \frac{9}{{12}}=\displaystyle \frac{{17}}{{12}}=1\displaystyle \frac{5}{{12}}\end{array}$

Since $ \displaystyle 1\frac{5}{{12}}\,\,<\,\,1\frac{1}{2}$, which is $ \displaystyle 1\frac{6}{{12}}$), we can use our $ \displaystyle 1\frac{1}{2}$ measuring cup!

If you are adding two fractions and one of the denominators goes into the other one perfectly (without any remainders), the largest one is the lowest common denominator. In this example, the lowest common denominator is $ 10$:

$ \displaystyle \frac{2}{5}+\frac{3}{{10}}=\frac{{2\times \color{#800000}{2}}}{{5\times \color{#800000}{2}}}+\frac{3}{{10}}=\frac{4}{{10}}+\frac{3}{{10}}=\frac{7}{{10}}$

A bit more complicated example:

$ \displaystyle \frac{5}{{12}}+\frac{5}{{18}}\,\,=\,\,?$

Find the least common denominator (don’t forget to try to use the **Prime Factor Tree** in the **Multiplying and Dividing** section to find the least common multiple, or lowest common denominator):

**MULTIPLES of **$ \boldsymbol {12}$: $ \color{black}{12,24},\color{#800000}{36},\color{black}{48}…$

**MULTIPLES of **$ \boldsymbol {18}$: $ \color{black}{18},\color{#800000}{36},\color{black}{54}…$

$ 36$ is the least common denominator, so turn each fraction to the same fraction with denominator $ 36$:

$ \displaystyle \frac{5}{{12}}+\frac{5}{{18}}=\frac{{5\times \,\color{#800000}{3}}}{{12\times \,\color{#800000}{3}}}+\frac{{5\times \color{#800000}{2}}}{{18\times \color{#800000}{2}}}=\frac{{15}}{{36}}+\frac{{10}}{{36}}=\frac{{25}}{{36}}$

## Subtracting Fractions

Subtracting fractions works the same way, but sometimes you have to **borrow** if you are working with mixed fractions, or you can turn all the fractions into improper fractions. In this example, we’ll work with fractions with the same denominator:

$ \displaystyle 4\frac{3}{8}-2\frac{5}{8}=\,\,\,?$

We could have also turned both of the fractions into improper fractions and done the subtracting this way:

$ \displaystyle \begin{align}4\frac{3}{8}-2\frac{5}{8}&=\frac{{(4\times 8)+3}}{8}-\frac{{(2\times 8)+5}}{8}\\&=\frac{{35}}{8}-\frac{{21}}{8}\,\,=\,\,\frac{{14}}{8}\\&=\frac{{8+6}}{8}=\frac{8}{8}+\frac{6}{8}\\&=1+\frac{6}{8}=1\frac{6}{8}=1\frac{3}{4}\end{align}$

## Multiplying Fractions

**Multiplying fractions** is actually easier than adding and subtracting fractions. Don’t forget this:

**OF = TIMES**

What does this mean? When we say something like “What is half of your age?” we are actually translating this into “What is one half **times **your age?”. Try it — it works, right? It’s weird, but it works!!

**Here’s an example:** If you’re $ 8$ years old, what is half your age? To get one half of your age, multiply the following:

$ \displaystyle \frac{1}{2} \,\times \,8\,\,=\,\,?$

To multiply fractions, put **both** fractions into fraction form (turn any mixed fractions into improper ones) and multiply across the top **and** across the bottom. It’s as easy as that! If you have a regular number like the $ 8$ above, just turn it into $ \displaystyle \frac{8}{1}$. Thus,

$ \displaystyle \begin{align}\frac{1}{2} \times \,8&=\,\frac{1}{2} \times \frac{8}{1}\,=\,\frac{{1\times 8}}{{2\times 1}}\,\\&=\,\frac{8}{2}\,=\,\frac{{4\times \cancel{2}}}{{1\times \cancel{2}}}\,=\,\frac{4}{1}\,=4\end{align}$

See — **much easier** than addition! Now you can multiply any two fractions. Remember to always turn the fractions into improper fractions before you multiply, and, depending on the problem, you may have to turn them back into mixed fractions if needed. For example, multiply two mixed fractions:

$ \displaystyle \begin{align}3\frac{2}{5}\times \,2\frac{1}{3}&=\frac{{(3\,\,\times \,\,5)+2}}{5}\times \frac{{(2\,\,\times \,\,3)+1}}{3}\\&=\frac{{17}}{5}\,\times \,\frac{7}{3}=\frac{{17\,\times \,7}}{{5\,\times \,3}}=\frac{{119}}{{15}}\\&=\frac{{(7\times 15)+14}}{{15}}=7\frac{{14}}{{15}}\end{align}$

In many cases, we can simplify fractions **before **multiplying by crossing out factors in the numerator and denominator before we multiply across:

$ \displaystyle \begin{align}\frac{1}{2} \times \,8\,&=\,\frac{1}{2} \times \,\frac{8}{1}\,=\,\frac{1}{{{}_{1}\cancel{2}}} \times \frac{{{{{\cancel{8}}}^{4}}}}{1}\\\,&=\,\frac{{1\,\times \,4}}{{1\,\times \,1}}\,=\,\frac{4}{1}\,=\,4\end{align}$

## Dividing Fractions

Dividing fractions isn’t too difficult either; it’s a little strange, but it works! What you do is the same exact thing as multiplication, but take the **second** fraction and **flip it** (called the** reciprocal **of that fraction). and then multiply across. Here is an example:

$ \displaystyle \begin{align}\color{#800000}{{3\frac{2}{5}\div \,2\frac{1}{3}}}\,&=\,\frac{{(3\times 5)+2}}{5}\div \frac{{(2\times 3)+1}}{3}\,\\&=\,\frac{{17}}{5}\,\div \,\frac{7}{3}\,=\,\frac{{17}}{5}\,\times \frac{3}{7}\,\\&=\frac{{17\,\,\times \,\,3}}{{5\,\,\times \,\,7}}\,=\,\frac{{51}}{{35}}\,\\&=\,\frac{{(1\times 35)+16}}{{35}}\,=\,1\frac{{16}}{{35}}\end{align}$

Remember, when you see fractions to be divided: “**Don’t cry! Flip the second and multiply!**”

Be sure to go through each step over and over again until you understand it. These will get **much** easier as you do more of them.

## Comparing Fractions

Comparing fractions, or determining if they are the same, the first is smaller, or the first is larger, can always be achieved by building up fractions (if needed) to have the **same denominator** and comparing numerators. The larger (or smaller) the numerator, the larger (or smaller) the fraction.

Another way I like to compare fractions is to use the “butterfly”: you multiply up diagonally and put the product close to the fraction on the outside. If the two products are equal, the fractions are equal. Whichever fraction is larger has the larger “butterfly” product near it. It’s a great shortcut so you don’t have to find common denominators!

Remember that “$ >$” means **greater than**, and the “mouth” opens up to the larger number. “$ <$” means **less than**, and again the “mouth” opens up to the larger number. These are called “**Inequalities**” and we’ll talk more about them in later sections.

Here are some examples, where we’ll determine if the following fractions are equal, and, if they are not, determine which one is larger:

## Fractions Used in Cooking

Like we saw in an earlier example, you have probably experienced fractions when you cook and bake! An American measuring cup, since we don’t use the **Metric System**, subdivides (divides up) the amounts into half cups, quarter cups, and ounces. Similarly, measuring spoons measure teaspoons and tablespoons and fractions of both of them.

It can be a little confusing since **ounces** can either be an amount you measure in a cup, or also a weight of something (part of a pound). The following table may help you remember and understand some of the different types of measurements you’ve used:

Measurements | Fractional Measurements |

1 tablespoon (T or tbsp) = 3 teaspoons | 1 teaspoon = $ \displaystyle \frac{1}{3}$ tablespoon |

8 ounces (oz) = 1 cup | 1 ounce = $ \displaystyle \frac{1}{8}$ cup |

2 cups = 1 pint | 4 ounces = $ \displaystyle \frac{1}{2}$ cup |

4 cups = 1 quart | 1 cup = $ \displaystyle \frac{1}{2}$ pint |

2 pints = 1 quart | 1 cup = $ \displaystyle \frac{1}{4}$ quart |

4 quarts = 1 gallon | 1 quart = $ \displaystyle \frac{1}{4}$ gallon |

16 ounces (oz) = 1 pound (lb.) (Weight) | 1 ounce = $ \displaystyle \frac{1}{16}$ pound (Weight) |

When we get to the **Percentages, Ratios and Proportions** section, we will see how easy it is to convert back and forth among different measurements by comparing ratios.

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