In simplest terms a fraction is a ratio between two integers. For example, ^{2}/_{3},
^{3}/_{5}, and ^{13}/_{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}/_{3} is the ratio of the integer 3 to itself. All such fractions have the same value, namely 1.
^{3}/_{3} = 1, ^{2}/_{2} = 1, ^{37}/_{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 × 1 = 2, 37 × 1 = 37, π × 1 = π.
Naturally, this applies to fractions, too. ^{2}/_{3} × 1 = ^{2}/_{3},
^{3}/_{5} × 1 = ^{3}/_{5}, and ^{13}/_{37} × 1 =
^{13}/_{37}. But 1 can be expressed as a ratio between an integer and itself. So we can also have:
^{2}/_{3} × ^{4}/_{4} = ^{2}/_{3},
^{3}/_{5} × ^{12}/_{12} = ^{3}/_{5}, and
^{13}/_{37} × ^{3}/_{3} =
^{13}/_{37}, which then becomes: ^{8}/_{12} = ^{2}/_{3},
^{36}/_{60} = ^{3}/_{5}, and
^{39}/_{111} =
^{13}/_{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 ^{1}/_{4} cup of sugar, you
know you can use the ^{1}/_{4} cup measure twice or use the ^{1}/_{2} cup measure
once. This is because ^{2}/_{4} = ^{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 ^{3}/_{4}" = ^{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 8 pieces. You eat 3. You ate ^{3}/_{8} of the pizza. Or perhaps
you are out with 5 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.

- 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}/_{84}, since 7 is prime, you only need to check if 7 is a factor of 84. It is; 84 = 7 × 12. Therefore,^{7}/_{84}=^{1}/_{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}/_{128}, you might start with 128. Doing so tells you immediately that the only common factors you need to look for are 2s. You don't care that 120 has factors of 3 and 5 because only powers of 2 matter. The reduced fraction is^{15}/_{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}/_{20}=^{3}/_{5},^{15}/_{20}=^{3}/_{4},^{144}/_{225}=^{16}/_{25}.