Anatomy of a fraction

You probably know what fractions look like, and even what the parts are called, but do you really know what all of the parts mean? Let's start there.

In the fraction below, the a is the numerator. To enumerate is to list or to count, so the numerator enumerates or counts. Specifically, it counts how many of the denominator there are.

The denominator, b, is the size of the things that are being counted. Those things are the equal-sized parts into which the whole is divided. The denomination is the number of equal parts that would be required to make up the whole.

1. All numbers are fractions

All numbers, even integers like 1, 2, 3, ... are fractions. For example, 2 is a fraction with a denominator of 1.

Very often it's useful to rewrite an integer or a real number as a fraction with a denominator of 1, like 2/1 or 3.14/1. We'll use representations of like that later in this section.

You might argue that an irrational number, like π = 3.14159 ... is not a fraction because,

2. Reciprocals

The reciprocal of a fraction is obtained by turning it upside down – by swapping the numerator and denominator. The table gives a few examples.

When the denominator is 1, we usually omit it, but it can come in handy when multiplying or adding fractions, at least until you get good at it (you will if you practice).

Reciprocals will be especially important when we divide by fractions.

3. Fractions are division operations

Any fraction is an expression of a division operation in which the denominator is divided into the numerator. For example:

• The fraction $\frac{8}{2}$ means "two divided into eight," or $\frac{8}{2} = 4.$ Remember the terminology: 8 is the dividend, 2 is the divisor and the answer (4) is the quotient.

4. About "improper fractions"

Fractions in which the numerator is greater than the denominator are often called "improper" fractions. Here are some examples:

Converting between improper and mixed fractions

Pure fraction → mixed

To convert an improper fraction to a mixed fraction, just divide the denominator into the numerator and express the remainder as a proper fraction. For example,

Remember that what's left over is still a fourth. Fourths are what's being counted by the numerator, and that didn't change.

Mixed → pure fraction

There are two ways to turn a mixed fraction into a pure fraction. The best is just to think of the whole part in terms of the denomination of the fraction part. For example, in the mixed number 3 ¼, we first think of it as 3 + ¼, then we simply express 3 as 12/4 and add on the remaining ¼ to get 13/4:

Practice – mixed & improper fractions

Mixed   →   Improper

Convert these mixed fractions to improper fractions. Hit [Answer] to see the solution, and [Next] to generate a new problem. Practice until you're confident.

5. Some fractions can be simplified

If both the numerator and denominator of a fraction can be divided evenly by the same integer, then the fraction can be simplified, and in most cases should be simplifed in order to make later calculations easier.

Take for example our coin values. A nickel is 5/100 of a dollar, but because both 5 and 100 are divisible by 5, we can do so to obtain the equivalent fraction, 1/20. A nickel is 1/20 of a dollar.

Here are some examples of fractions that can be simplified and one that can't. Notice that 11/19 cannot be simplifies because neither 11 nor 19 have a factor in common other than 1.

The basic idea for simplifying fractions is to find common divisors for both the numerator and denominator, and do the divisions on each. Often the divisors can be small prime numbers, and you can repeat the process with the new numerator and denominator until you get to something that can no-longer be simplified.

Practice problems – Simplifying fractions

All of these fractions simplify to 3/5. See if you can figure out how:

*Hint: Just repeatedly divide the numerator by 3 and the denominator by 5, the same number of times.

6. Multiplying fractions

Multiplying two fractions couldn't be simpler, but you should try to figure out what it really means.

One key to understanding fraction multiplication is the word "of." In mathematics, "of" means multiplication. So if we say ½ of 2, that's ½ multiplied by 2, which is 1.

This picture might help. On the left is 1 circle – a whole circle with an area of 1 unit. If we want ½ of that circle, we multiply 1 by ½ to get ½ (magenta area). If we want ½ of the half-circle, we multiply ½ by ½ to get ¼.

To multiply fractions, we multiply across: numerator multiplies numerator, and denominator multiplies denominator. It's that easy. Heres an example:

Here are a few more examples:

$$\frac{2}{7} \cdot \frac{1}{3} = \frac{2 \cdot 1}{7 \cdot 3} = \frac{2}{21}$$

and

$$\frac{-3}{5} \cdot \frac{2}{5} = \frac{-3 \cdot 2}{5 \cdot 5} = \frac{-6}{25}$$

Now we could easily have asked that last question like "what is 2/5 of 3/5 ?", or "what is 3/5 of 2/5?" The answer and process would have been the same because "of" means "multiply." That's really handy to know, then we have ½ of ½ = ¼, or ½ · ½ = ¼.

7. Dividing fractions

The division operation is really no different from multiplication. It's just multiplication by the reciprocal of the divisor (which is the denominator). If you can remember that, your life studying math will be much easier.

Here's what I mean. Let's divide 8 by 2:

I know that seems more complicated than just dividing 8 by 2 in your head, but here's why it's valuable. What is ½ divided by ¼ ?

Just in case you could have figured that one out in your head (because ¼ is half of ½), here's a tougher one that's still very easy to do if you remember that division by a denominator is the same thing as multiplication by the reciprocal of the denominator:

8. Adding fractions

Adding fractions is probably one of the scariest things that many math students can think of doing, and that's a shame, because it's not really so bad. With just a little practice, you'll be an expert and it won't trouble you any more. Be sure to put in that practice. Get good at adding fractions now and you'll be good at it forever.

Example 1

Here's an example. Let's add ⅙ and ⅞. Notice that these fractions have different denominators, so we'll need to convert them both to the same one. The easiest way is just to multiply the two denominators: 6 · 8 = 48. That way we know that both 6 and 8 have to be factors of 48.

$$\frac{1}{6} + \frac{7}{8} \; \color{#E90F89}{\leftarrow \; \text{Different denominators}}$$

We'll convert both fractions to 48^ths. Now we don't want to change our sum, so to convert, we'll multiply ⅙ by 8/8 (which is just one because a number divided by itself is 1) and ⅞ by 6/6 for the same reason. It looks like this:

$$= \left(\frac{8}{8} \right) \frac{1}{6} + \frac{7}{8} \left( \frac{6}{6} \right) \; \; \color{#E90F89}{\frac{6}{6} = 1 \; \& \; \frac{8}{8} = 1}$$

Multiplication gives us a sum of 48^ths:

$$= \frac{8}{48} + \frac{42}{48}$$

Now that our fractions have a common denominator (which means that the numerators are counting the same thing, like apples alone instead of apples and oranges), we just add the numerators and simplify the fraction if we can.

$$= \frac{50}{48} = \frac{25}{24} \color{#E90F89}{\leftarrow \; \text{simplify}}$$

A handy shortcut for adding two fractions

Let's review a general method for adding fractions. It's the brute-force method that always works. We don't look for a common denominator; we just multiply the denominators to get one.

We begin by adding two fractions with different denominators, b and d:

$$\frac{a}{b} + \frac{c}{d}$$

Multiply by d/d = 1 on the left and b/b = 1 on the right:

$$\left( \frac{d}{d} \right) \frac{a}{b} + \frac{c}{d} \left( \frac{b}{b} \right)$$

... and we get our sum:

$$\frac{ad}{bd} + \frac{bc}{bd} = \frac{ab + bc}{bd}$$

Now these are just letters, so this has to work for any two fractions. If you look closely, you'll see a pattern in there. On the top is the sum of two "diagonal products," ad and bc, and on the bottom is just the product of the denominators. This picture might help:

Practice – addition

This box will generate as many random fraction addition problems as you can do. Try a bunch and check your answers. You should be able to get to the point where you get most of them right. You might even be able to do some in your head. That's when you know you've got it.

Adding fractions is really an important skill, so make sure you practice and gain confidence.

Note: because this widget generates random fractions, once in a while you'll get a problem like 2 - 1.

Convenient denominators – Percent

Some denominations (denominators) are very convenient and commonly used, so we ought to be familiar with them.

Percents are no big mystery, they're just fractions. The word percent is from the Latin per centum, which roughly means "by the hundred." The denominator of the percent system is always 100. So 20% means 20 out of every 100, or

$$20\% = \frac{20}{100} = \frac{2}{10} = \frac{1}{5}$$

Percent → fraction

It's always pretty easy to convert from percent to a simplified fraction: Just put the percentage over 100 and simplify if you can. Here are some examples:

$$8\% = \frac{8}{100} = \frac{2}{25}$$

$$73\% = \frac{73}{100} \; \color{#E90F89}{\leftarrow \; \text{not reducible}}$$

If you're at a store and something is 10% off, that's 10 cents out of every dollar (100 cents), or \$10 out of every \$100, or \$100 out of every \$1000, ... you get the point.

Generate as many percents as you need to get confident at converting them to fractions. Make sure to simplify your fractions as much as possible.

Other denominators

Why do we use base 60 and base 12?

Did you ever wonder why our clocks have 60 seconds per minute and 60 minutes per hour, why we use base 12 in our length measurements here in the U.S.?*

These fractions come to us from a time when cultures, such as the Mesopotamians, were developing mathematics, and when much of that math revolved around supporting commerce – the exchange of money and goods.

Now in our base 10 system, consider the even fractions of 10. They are 1, 1/2, and multiples of 1/5 and 1/10. But consider all of the integer fractions we can get from a base of 60 divisions:

Video Examples

These videos will help you to better understand and develop your skills in adding, subtracting, multiplying and dividing fractions.

1. Where does the negative sign go?

It doesn't matter where you put the negative sign in a fraction – numerator, denominator or in an ambiguous position – but putting it in the numerator is the best practice. This video will explain why they're all the same.

3. Division of fractions

Division is really just multiplication by the reciprocal, so dividing by ½ is the same is multiplication by 2. Here are a couple of example problems.