Fractions

Fractions are a useful way to represent numererical quantities that are not whole numbers.

What is a Fraction?

A fraction is a way of showing a numerical quantity that is not a whole number (for example, a value less than 1 but greater than 0). In reality, a fraction is really just a divide. When we write a fraction like \(\frac{2}{3}\), we are really writing "2 divided by 3". Fractions are useful in mathematics, because the value represented is a precise value without the precision loss that might exist if the divide it represents were actually performed. One example of this is the fraction \(\frac{5}{7}\), which in decimal form would be approximately 0.714285714; it is only an approximation, because there are more digits than are shown, and the last digit may have been rounded. The fractional value \(\frac{5}{7}\) is also much easier to write (and to remember) than 0.714285714.

Fractions are generally represented as one of two forms: a pure fraction such as \(\frac{1}{2}\) and a mixed fraction (also called a mixed number or mixed-number fraction) such as \(5\frac{3}{8}\). Whenever possible, a fraction should be written in its simplest form, so we generally steer away from mixed fractions except for the final result. This is partially because pure fractions are much easier to work with than mixed fractions; however, the main reason is because the format used to write a mixed fraction can be a source of confusion once you get into higher mathematics such as algebra.

It is my opinion that one of the difficulties that people face with learning fractions is that they should learn a little bit of algebra first. This is because it is very useful to be able to substitute numbers with letters (variables) in order to specify how to perform the processes involved in working with fractions. You will see examples of this in the text below.

Numerator and Denominator

A fraction is composed of either two or three parts, depending on if it's a pure or mixed fraction: there is a top number (the numerator), a bottom number (the denominator), and an optional whole number. The easy way to remember which is on top and which is on bottom, is that the word "denominator" starts with the letter "d", which is also the first letter of "down"; so it goes on bottom.

\(Whole\,Number\frac{Numerator}{Denominator}\)

The numerator enumerates (that is, tells us a count of) "how many" we have, while the denominator tells us the type of unit we have (the denomination). For example, let's say we have 17 pennies. How many dollars do we have? Now, we don't typically thing of pocket change as having denominations as we do paper money, but it is still there. A penny has a denominaton of \(\frac{1}{100}\) of a dollar. And we have 17 of them. So we have a total of \(\frac{17}{100}\) of a dollar. If a whole number exists, then we have that number of whole units in addition to the fractional amount.

Proper and Improper Fractions

When we refer to fractions as being proper or improper, we are really refering to whether or not the fraction is in its simplest form. Any fraction whose numerator is greater than or equal to its denomintor is said to be improper; otherwise, the fraction is said to be proper.

In order to turn an improper fraction into a proper fraction, we need to perform a divide with a remainder. The whole number portion of the result is then added to any existing whole number, and the remainder is the new numerator. For example, given the fraction \(\frac{16}{3}\), we decide that 3 goes into 16 a total of 5 times with a remainder of 1. The proper fraction is then \(5\frac{1}{3}\). It's that easy!

To turn a proper mixed fraction into an improper fraction is even easier. All we need to do is multiply the whole number by the denominator and add it to the numerator. For example, given \(3\frac{7}{16}\), we multiply 3 by 16 to get 48 and add that to the 7 to get 55. The result is then \(\frac{55}{16}\). Pretty painless, if you ask me.

Algebraicly speaking, we might write this process as:

\(I\frac{N}{D}=\frac{N+ID}{D}\)

The example given would look like this:

\(3\frac{7}{16}=\frac{(7)+(3)(16)}{16}=\frac{7+48}{16}=\frac{55}{16}\)

Finding a Common Denominator

When we are adding or subtracting fractions, the denominators of the fractions must be the same. We call this denominator a common denominator, because it is common to all fractions involved. Let's start with a simple example where we select the denominator we want: I have 3 pennies, 2 nickles, 1 dime, and 3 quarters. How many dollars do I have? Because all of our change are fractions of a dollar, and we want our result to be in dollars, it makes sense to use a denominator of 100. The important thing to remember when converting fractions, is that whatever we do to the denominator, we must also do to the numerator. For example, a nickel is \(\frac{1}{20}\) of a dollar, so to convert the fraction, we must multiply both the numerator and the denominator by 5, since \(5\times 20=100\).

dollars \(=\frac{3}{100}+\frac{2}{20}+\frac{1}{10}+\frac{3}{4}\)
\(=\frac{3}{100}+\frac{10}{100}+\frac{10}{100}+\frac{75}{100}\)
\(=\frac{98}{100}\)

The easiest way to find a common denominator is to simply multiply the denominators together. This can, however, produce very large denominators, especially if multiple fractions are involved. For example, if we want to find a common denominator for the fractions \(\frac{2}{3}\), \(\frac{5}{9}\), and \(\frac{7}{16}\), we would multiply \(3\times 9\times 16\) to get a common denominator of \(432\). Then we need to convert the fractions by deciding how many times each denominator will go into the common denominator and multiplying that number by the numerator. We then get:

\(\frac{2}{3}=\frac{(2)(144)}{(3)(144)}=\frac{288}{432}\)

\(\frac{5}{9}=\frac{(5)(48)}{(9)(48)}=\frac{240}{432}\)

\(\frac{7}{16}=\frac{(7)(27)}{(16)(27)}=\frac{198}{432}\)

The other way to find a common denominator is by finding the lowest common multiple of the denominators. The easiest way to do this is to simply write out the multiples of each number. We want the first (lowest) value common to all the fractions. Using the previous example, we would get:

3,6,9,12,15,18,...,144

9,18,27,36,45,54,...,144

16,32,48,64,80,96,...,144

So the lowest common multiple of 3, 9, and 16, is 144.

Using Fractions

Addition and Subtraction of Fractions

Algebraicly speaking, we might write this process as:

\(r=\frac{a}{b}\pm\frac{c}{d}=\frac{ad\pm bc}{bd}\)

Multiplication of Fractions

Algebraicly speaking, we might write this process as:

\(r=\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}\)

Division of Fractions

Algebraicly speaking, we might write this process as:

\(r=\frac{a}{b}\div\frac{c}{d}=\frac{ad}{bc}\)