In simplest terms a fraction is a ratio between two integers. For example, \(2 \over 3\), \(3 \over 5\), and \(13 \over 37\) are all fractions. Fractions give people fits. I'm not sure why.

One fraction of particular interest is the fraction formed by the ratio of an integer to itself. For example, \(3 \over 3\) is the ratio of the integer 3 to itself. All such fractions have the same value, namely 1. \({3 \over 3} = 1\), \({2 \over 2} = 1\), \({37 \over 37} = 1\).

A curious property of \(1\) is that it is the identity element for multiplication. Any integer—in fact, any real number—multiplied by \(1\) yields the same number again. Thus, \(2 \times 1 = 2\), \(37 \times 1 = 37\), \(\pi \times 1 = \pi\). Naturally, this applies to fractions, too. \({2 \over 3} \times 1 = {2 \over 3}\), \({3 \over 5} \times 1 = {3 \over 5}\), and \({13 \over 37} \times 1 = {13 \over 37}\). But \(1\) can be expressed as a ratio between an integer and itself. So we can also have: \({2 \over 3} \times {4 \over 4} = {2 \over 3}\), \({3 \over 5} \times {12 \over 12} = {3 \over 5}\), and \({13 \over 37} \times {3 \over 3} = {13 \over 37}\), which then becomes: \({8 \over 12} = {2 \over 3}\), \({36 \over 60} = {3 \over 5}\), and \({39 \over 111} = {13 \over 37}\). In other words, some ratios between integers have the same value as other ratios between integers.

Think of doubling a recipe. If your original recipe calls for \(\frac 1 4\) cup of sugar, you know you can use the \(\frac 1 4\) cup measure twice or use the \(\frac 1 2\) cup measure once. This is because \({\frac 2 4} = {\frac 1 2}\). They have the same value. Or consider using a ruler. If you count the longer quarter-inch tick marks and then count the shorter sixteenth-inch tick marks, you can easily see that \({\frac 3 4}'' = {\frac {12} {16}}''\).

Fractions are often used to represent a part of a whole or a portion of a group. For example, you buy a pizza and find it cut into eight pieces. You eat three. You ate \(\frac 3 8\) of the pizza. Or perhaps you are out with five of your friends. Two of them order beer while three of them and you order water. A third the group drinks beer. Regardless how they are used, though, a fraction can always be expressed as a ratio between integers.

Fractions can always be expressed as a ratio between integers, but there are so many different fractions with
the same value. How can we be sure we are talking about the same value? This is where simplifying or reducing
fractions comes in. A fraction is reduced (or simplified) when the two integers that make it up have no common
factors. But how can you tell if two integers have common factors? This is a deep question indeed, and its answer
can take you into the very heart of cryptography and Internet security protocols because it turns out that finding
factors of integers is not necessarily easy. It can be nearly impossible. Luckily, however, when *you* need to find
factors of an integer, you are not usually dealing with integers having 80 digits. Usually, they have no more than 4.
What follows are some useful tips for finding and eliminating common factors to reduce fractions. When the two integers
that make up a fraction have no common factors, then the fraction is *reduced to its simplest form* or just
*simplified*.

- Look for
*common*factors first. For example, if one of the integers is a prime number, you only need to see if that number is also a factor of the other number. For example, for the fraction \(7 \over 84\), since \(7\) is prime, you only need to check if \(7\) is a factor of \(84\). It is; \(84 = 7 \times 12\). Therefore, \({7 \over 84} = {1 \over 12}\). - Pick the easiest number first. The easiest number is the one that is easiest for you to find factors for. Perhaps you instantly recognize powers of \(2\), for example. Then for \({120 \over 128}\), you might start with \(128\). Doing so tells you immediately that the only common factors you need to look for are \(2\)s. You don't care that \(120\) has factors of \(3\) and \(5\) because only powers of \(2\) matter. The reduced fraction is \(15 \over 16\).
- Use divisibility tests. Look for two even numbers. Look for two numbers that end in \(5\) or \(0\). Look for two numbers whose digits sum to a multiple of \(3\). Examples: \({12 \over 20} = {3 \over 5}\), \({15 \over 20} = {3 \over 4}\), \({144 \over 225} = {16 \over 25}\).