# Fractions

This section covers:

# Introduction to Fractions

OK, now that you’ve conquered adding, subtracting, multiplying and dividing “normal” and “decimal” numbers, let’s advance to things a little more complicated.

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, or when less than 1, are part of something.  We saw the number line before:

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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.

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 Samantha’s birthday party and her parents order two large pizzas, which we’ll eat before the cake: Now 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.

Let’s say each pizza has 6 pieces total, and Sam’s nasty little brother Jake 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”, which we’ll talk about later.

Here are what the pizza pies look like now:

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

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

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.  Memorize this!

So 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. 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 $$\displaystyle \frac{1}{3}$$, we notice that $$\displaystyle \require{cancel} \frac{2}{6}=\frac{{1\times \cancel{2}}}{{3\,\times \cancel{2}}}$$, and we can cross out the 2 on the top and 2 on the bottom (since they divide to equal 1) to get $$\displaystyle \frac{1}{3}$$. To get from $$\displaystyle \frac{4}{6}$$ to $$\displaystyle \frac{2}{3}$$, we notice that $$\displaystyle \require{cancel} \frac{4}{6}=\frac{{2\,\times \cancel{2}}}{{3\,\times \cancel{2}}}$$, and we can cross out the 2 on the top and 2 on the bottom (since they divide to equal 1) 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.

Actually, 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), which we learned about in 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”.

Now if we had 6 girls 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 pieces eaten by the evil little brother Jake)?

Let’s say the girls starting eating the pizza without Jake’s cooties on it; each had one piece, so they ate one whole pizza ($$\displaystyle \frac{6}{6}$$ gone). And we still have the pizza Jake started ($$\displaystyle \frac{2}{6}$$ or $$\displaystyle \frac{1}{3}$$ gone). So 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). Now $$\displaystyle \frac{4}{3}$$ is called an improper fraction, since the top is bigger than the bottom (sometimes called a “Dolly Parton” fraction, for obvious reasons).

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 1 time, and we have 1 left over (which is a fractional part of 3), so the fraction is $$\displaystyle 1\frac{1}{3}$$:

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.

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

So 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 the following. 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:

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

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

$$\displaystyle 1\frac{5}{6}\,\,=\frac{{(1 \times 6)+5}}{6}\,\,=\,\,\frac{{11}}{6}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\frac{3}{6}\,\,=\,\,\frac{{(2\,\times \,6)+3}}{6}\,\,=\,\,\frac{{15}}{6}\,$$

$$\displaystyle \frac{{11}}{6}+\frac{{15}}{6}\,\,=\,\,\frac{{26}}{6}\,\,=\,\,\frac{{24+2}}{6}\,\,=\,\,\frac{{24}}{6}+\frac{2}{6}\,\,=\,\,4+\frac{2}{6}\,\,=\,\,4\frac{2}{6}\,\,=\,\,4\frac{1}{3}$$

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, you have to find what we call the Lowest Common Denominator, which is the Least Common Multiple (LCM), that we talked about in 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.  So we add:

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

We have to find to find the least common multiple of the denominators, 3 and 4.  Like we did before, we find the smallest number that they both go into:

MULTIPLES of 3:      3, 6, 9, 12, 15, 18, 21, 24, 27 . . .

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MULTIPLES of 4:      4, 8, 12, 16, 20, 24, 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 will “blow up”!)

So 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.

So 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:

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

So 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:

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

A bit more complicated example:

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

Let’s 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 12:  12, 24, 36, 48 . . .

MULTIPLES of 18:  18, 36, 54 . . .

So 36 is the least common denominator, and let’s 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:

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

# Multiplying Fractions

Now let’s work on multiplying fractions – this is actually a little 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!! So let’s say we want to get one half of your age, and let’s say you’re 8 years old.  To get one half of your age, we multiply the following fractions:

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

To multiply fractions, you simply put both fractions into fraction form (so 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 $$\displaystyle \frac{8}{1}$$. So we have:

$$\displaystyle \frac{1}{2} \,\times \,\,8=\,\,\frac{1}{2} \times \frac{8}{1}\,\,=\,\,\frac{{1\times 8}}{{2\times 1}}\,\,=\,\,\frac{8}{2}\,\,=\,\,\frac{{4\times \cancel{2}}}{{1\times \cancel{2}}}\,\,=\,\,\frac{4}{1}\,\,=4$$

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 then sometimes your teacher will make you turn them back into mixed fractions.  For example, let’s multiply two mixed fractions and make them it a little more complicated:

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

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

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

# 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:

\displaystyle \begin{align}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 \color{#800000}{{\frac{7}{3}}}=\frac{{17}}{5}\times \color{#800000}{{\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 (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 up”: 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 up” 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 a lot 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!   You have probably used a measuring cup, where on the side, it subdivides (divides up) the amounts into half cups, quarter cups, and ounces.  You have probably also used a measuring spoon, where you can 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:

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.

For Practice: Use the Mathway widget below to try a Describe a Transformation problem. Click on Submit (the blue arrow to the right of the problem) and click on Describe the Transformation to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on Tap to view steps, or Click Here, you can register at Mathway for a free trial, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to Metric System – you are ready!!

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