


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 mixednumber 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.
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.
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 \(5\times 20=100\).
Using FractionsAddition and Subtraction of Fractions
Algebraicly speaking, we might write this process as:
Multiplication of Fractions
Algebraicly speaking, we might write this process as:
Division of Fractions
Algebraicly speaking, we might write this process as:




