Fraction notation

Fractions (or common fractions) are used to describe a part of a whole object. There are several notations for fractions:

a is called the numerator and b is called the denominator. The notation means that we break an object into b equal pieces and we have a of those pieces. The portion or fraction of the object that we have is a/b. For example if we break a pie into 4 equal pieces and take 1 piece then we have 1/4 of the pie:

Equivalent fractions

Notice that we get the same amount of pie as in the previous example if we divide the pie into 8 equal pieces and get 2 of them:
Fractions like 1/4 and 2/8 that have the same value are said to be equivalent fractions. This example suggests the following method for testing if two fractions are equivalent.

 Two fractions are equivalent if multiplying the numerator and denominator of one fraction by the same whole number yields the other fraction.

For example 4/5 and 24/30 are equivalent because we can start with 4/5 and multiply the numerator and denominator each by 6 to get 24/30:
Going in the opposite direction (from 24/30 to 4/5) suggests the following method for reducing a fraction to lowest terms or to its simplest equivalent fraction:

 To reduce a fraction to lowest terms or to its simplest equivalent fraction, factor both the numerator and denominator completely (i.e. into prime numbers). Then cancel every factor that occurs in both the numerator and denominator. What remains is the simplest equivalent fraction.

For example, here is how to reduce the fraction 24/42 is its simplest equivalent fraction, namely 4/7:

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Improper fractions, mixed fractions and long division

A fraction where the numerator is smaller than the denominator is called a proper fraction and a fraction where the numerator is bigger than the denominator is called an improper fraction. An example of an improper fraction is 7/4. Using the pie example this means that you have broken many pies each into 4 equal pieces and you have 7 of those pieces:
Improper fractions are sometimes expressed in mixed fraction notation, which is the sum of a whole number and a proper fraction, but with the + sign omitted. For example 7/4 in mixed fraction notation looks like this:
Mixed fraction notation is not used in this Algebra Help e-book or in the Algebra Coach program because it is too easy to confuse it with the product of a whole number and a fraction. Instead of writing we will keep the + sign and write .

Long division is the method used to convert an improper fraction to a mixed fraction. We will illustrate the method on the fraction . Carry out the following steps:
• Set up the long division format, namely .

• Since 5 into 9 goes 1 time, write a “1” above the 9, write 1 × 5 or “5” below the 9, and subtract 5 from 9 to get a difference of 4, like this:
• Then bring down the 2 like this:
Here is what we have actually done: The “1” and “5” are in the tens place so they actually represent the numbers 10 and 50 as shown here:
Therefore we have actually shown that .

• Now repeat the entire process with the remainder, 42. Since 5 into 42 goes 8 times, write an “8” above the 2, write 8 × 5 or “40” below the 42, and subtract 40 from 42 to get a remainder of 2, like this:
• This shows that that . Since the remainder, 2, is smaller than the divisor, 5, this is our final mixed fraction result.

Some special fractions

There are several special fractions that will come in very useful later:
• n = n/1.   Any number n can be turned into a fraction by writing it over a denominator of 1.

• n/n = 1.   Anything divided by itself equals 1. Some people call this a UFOO (a useful form of one).

• If the numerator of a fraction is a multiple of the denominator then the fraction is equivalent to a whole number. An example is   15/3 = 5.

• n/0   is undefined for any numerator n. Division by zero is not allowed in mathematics.

• 0/n = 0   for any denominator n except 0. (A zero numerator causes no problem.)

This picture shows that 2/8 of a pie plus 3/8 of a pie equals 5/8 of a pie:
Fractions that have the same denominator are called like fractions. If you think about this example, then the following procedure for adding or subtracting like fractions is obvious:

 To add two or subtract like fractions (fractions that have a common denominator), just add or subtract the numerators and put the result over the common denominator, like this:

But what if the fractions don’t have a common denominator? The answer is that they must then be converted to equivalent fractions that do have a common denominator. The procedure is illustrated in this example:
The steps are:
1. Find the least common multiple of the two denominators 24 and 30. When applied to fractions this number is called the least common denominator (LCD). In this example the LCD is 120.

2. Convert each fraction to an equivalent fraction that has the LCD of 120 as its denominator. To do this in this example multiply the numerator and denominator of the first fraction by 5 and the numerator and denominator of the second fraction by 4 (shown in red).

3. Add the numerators and place over the common denominator.
Sometimes there is one more step. The result should always be expressed as the simplest equivalent fraction, like this:

Here are some more examples:

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Multiplying fractions

 Multiplying fractions produces a new fraction. Multiply the numerators to get the new numerator and multiply denominators to get the new denominator, like this: Then simplify by reducing the new fraction to lowest terms. To multiply a fraction by a whole number, just multiply the fraction’s numerator by the whole number to get the new numerator, like this: Then simplify by reducing the new fraction to lowest terms.

Here is an example of why the first procedure works. Suppose that there is half a pie (the fraction 1/2) as shown on the left. Now suppose that you take 2/3 of that half pie. (The word “of” translates into the mathematical operation “multiply”.) This means that you cut the half pie into 3 equal pieces and take 2 of them. The result is 2/6 of the pie.

Here is an example of why the second procedure works. Suppose that you ate 1/4 of a pie and that your friend ate 3 times as much pie as you did. This means that your friend ate 3/4 of the pie.

Here are some more multiplication examples:

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Reciprocals and dividing fractions

Reciprocals play an essential role when dividing fractions. Two numbers or fractions are said to be the reciprocals of each other if their product is 1. For example:
4/5 and 5/4 are reciprocals because

8 and 1/8 are reciprocals because

 Dividing fractions: The procedure is to replace a division by a fraction by the multiplication by the reciprocal of that fraction, like this:

Notice that you take the reciprocal of the fraction on the bottom!

 Here is why this procedure works: The key is that instead of seeing a fraction divided by a fraction, look for a single fraction whose numerator and denominator just happen to be fractions. In the first step we multiplied this fraction by a UFOO whose numerator and denominator just happen to be fractions. The UFOO was chosen so the fractions in the denominator would cancel and give 1. After another simplification that left only the final multiplication of fractions.

Example 1: A fraction divided by a fraction:

Example 2: A fraction divided by a number. Notice that we have drawn one divide line longer than the other so you can tell which is the fraction and which is the number. The first step is to convert the whole number 4 into the fraction 4/1. After several steps you get the expression shown in blue. If you compare this expression to the original one you will notice a nice shortcut. The number 4 that you are dividing the fraction by simply becomes a new factor in the fraction’s denominator.

Example 3: A number divided by a fraction. Check the steps. This one is quite different from the previous example!

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