Fractions

### 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.

In the fraction $\frac{1}{4},$ four parts make up the whole, and we have one of them.

A familiar example of denomination might be the denomination of money. The denomination of a coin or bill is its value in dollars (U.S. dollar-based system). A penny has a denomination of 1/100* because it's value is 1/100 of a dollar. A nickel has a denomination of 1/20 of a dollar, a dime 1/10 and a quarter 1/4 of a dollar (thus the name "quarter"). If we have four quarters, we have 4/4 of a dollar or one whole dollar.

Two halves of a pie make a whole pie, if we have 87 potatoes, then each is 1/87th of the total number ... and so on.

* Note: From time to time I use the slanted fraction bar ( / ) in typed text, but that's just a matter of convenience when typing. I strongly advise you to use a horizontal bar ( — ) in writing fractions. It will help you immensely later.

### 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,

by the definition of irrational numbers, it cannot be represented as a fraction, but we can (and often do) still write a fraction like π/2 or π/1 for convenience.

### 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.
• The fraction 1/2 means 1 divided by 2. If you think about that, 1 divided into two parts gives two equal parts that are 1/2 of the whole.

• The fraction 1/3 means 1 divided by 3. A whole divided into three parts yields parts that are 1/3 the size of the whole ... and so on.

Here is a recap of the ways that we know to write division operations.

* An algorithm is a method or recipe that can be repeated to solve similar problems.

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

I think that the name "improper" is unfortunate. They're perfectly legitimate fractions, and almost always more useful for doing calculations than mixed fractions.

One important thing to keep in mind is that the numerator of a "proper" fraction is less than the denominator, so a proper fraction is a number less than one. Conversely, an improper fraction is always a number greater than 1.

#### Improper fractions are proper

It's almost always easier and better to work with "improper" fractions than to convert them to mixed fractions. You should at least hold off on converting to mixed fractions until the end of a calculation.

### 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:

A shortcut for converting a mixed fraction is just to multiply the whole number by the denominator of the fraction, add the numerator of the fraction, then write that over the original denominator. It looks like this:

### 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.

##### Improper   →   Mixed

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

### 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.

 $$\frac{11}{121}$$ Both numerator and denominator are divisible by 11, so this fraction simplifies to $\frac{1}{11}.$ $$\frac{11}{19}$$ Both numerator and denominator are prime numbers, so they can't have any common factors other than 1. This fraction is in its simplest form. $$\frac{100}{300}$$ Both numerator and denominator are divisible by 100, so this fraction can be simplified to $\frac{1}{3}.$ $$\frac{5}{130}$$ Both numerator and denominator are divisible by 5, so this fraction can be simplified to $\frac{1}{26}.$

### 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.

#### Pro tip

In mathematics, the word "of" almost always means "multiplied by."

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.

### Practice – multiplication

Use this tool to generate random fraction-multiplication problems. Multiply (don't forget to simplify) and check your answer. Keep going until you have a lot of confidence multiplying and simplifying fractions. (Note: the word "of" is used in some of these problems instead of the multiplication dot.)

#### Multiplying fractions

To multiply fractions, multiply numerators together and multiply denominators together.

### 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:

Thinking of division as multiplication by the reciprocal of the divisor (or denominator) will help you to solve a great many algebra problems. It will really grease the wheels for further learning in math, and I strongly encourage you to master it.

### Practice – division

Use this problem generator to practice dividing a fraction by a fraction. Do as many as you need to get confident at dividing fractions.

Note: I'm using the division sign ( ÷ ) here because I want you to practice setting up divisions as fractions (a ÷ b = a/b). It really isn't used much in real mathematics.

Below is an example with a picture. 1 divided into four parts is 1·¼ = ¼. Now ¼ divided into 4 parts gives parts that are ¼·¼ of the whole, and 1/16 divided into 4 parts is 1/16 · ¼ = 1/64.

See if you can follow the divisions through the figure below. It might help you to understand what's going on when we divide by a fraction.

#### Division by a fraction

Division is just multiplication by the reciprocal of the divisor.

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 48ths. 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 48ths:

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

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.

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

While the method above always works, there are other cases where finding a common denominator can be done a little easier, and you should probably look for them. One is where the denominator of one fraction is a multiple of the other. Take this addition for example.

#### Example 2

$$= \frac{1}{5} + \frac{4}{15} \color{#E90F89}{\leftarrow \; \text{5 divides into 15}}$$

5 is a factor of (divides evenly into) the denominator 15, so if we multiply the ⅕ by $\frac{3}{3},$ we get:

$$= \left( \frac{3}{3} \right) \frac{1}{5} + \frac{4}{5} \color{#E90F89}{\leftarrow \; multiply \; \frac{1}{5} \; by \; \frac{3}{3}}$$

When both fractions have the same denominator, it's legal to add them.

$$= \frac{3}{15} + \frac{4}{15}$$

Now both fractions have the same denominator.

3 15ths and 4 15ths add to 7 15ths.

$$= \frac{7}{15} \color{#E90F89}{\leftarrow \; \text{irreducible}}$$

The trick to adding fractions is that you have to be adding fractions of the same denomination. It doesn't make any sense to add ½ to ¼. What would you call the result? Would it be some kind of strange hybrid of halves and fourths? What we'll need to do is to convert halves to fourths (½ = 2/4), then we can add up all of the fourths to get ¾.

Two fractions can be added (or subtracted) only if their denominators are the same. A common denominator can always be found for two fractions.

### 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:

Here are a couple of examples:

$$\frac{11}{9} + \frac{4}{5} = \frac{11(5) + 4(9)}{9(5)} = \frac{91}{45}$$

$$\frac{4}{7} + \frac{3}{2} = \frac{4(2) + 3(7)}{7(2)} = \frac{29}{14}$$

Here's one with variables:

$$\frac{a}{3} + \frac{2}{b} = \frac{a(b) + 2(3)}{3(b)} = \frac{ab + 6}{3b}$$

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.

#### Fraction → percent

Now let's say we want to know how many percent ⅞ is. We can use the simple algebra of proportions to do this. In words we might say, "7 is to 8 as what is to 100 ?". The "what" will be our variable, x. Here it is algebraically:

$$\frac{7}{8} = \frac{x}{100}$$

That's just a proportion, and we solve it by cross multiplication: 7(100) = 8x:

$$700 = 88x$$

... and we finish by dividing both sides by 8 and simplifying the resulting fraction until we can't any more (I like to repeatedly by 2 if both numerator and denominator are even):

$$x = \frac{700}{8} = \frac{350}{4} = \frac{175}{2} = 87.5\%$$

So 7/8 of an inch is 87.5% of an inch. 7/8 of a Kilogram is 87.5% of a Kilogram, and so on.

### Practice problems – Percents

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.

Practice converting as many fractions to percents as you need to get confident doing so. Hit [answer] to see the solution, and [next] to generate a new problem.

#### Percents are fractions

A percent is just a fraction with a denominator of 100.

• To convert from a% to a simplified fraction just simplify the fraction a/100.
• To convert from a fraction, a/b to a percent, use a proportion:

### 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:

If you're interested in the history of mathematics and more interesting things like this, you might want to check out the books of Princeton Prof. Victor Katz.

*The U.S., Myanmar and Liberia, as of 2015, are the only countries of Earth who have not officially adopted the metric system.

### Video Examples

#### 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.

#### 2. Multiplication of fractions

When multiplying fractions, multiply numerators and multiply denominators. In this video, we'll see how to interpret multiplied fractions. What is half of half?

#### 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.