Common Fraction

Fractions represent numbers and, as numbers, they can be combined by addition, subtraction, multiplication, and division. Addition and subtraction of fractions present no problems when the fractions have the same denominator. For example

We are adding like fractional parts, so we ignore the denominators and add the numerators. The same holds for subtraction. When the fractions have the same denominator we can subtract the numerators and ignore the denominators. For example

To add and subtract fractions with unlike denominators, the numbers have to be renamed. For example, the problem

requires us to change the fractions so that they have the same denominator. We try to find the lowest common denominator since this makes the calculation easier. If we write

and

the problem becomes

Similarly, with subtraction of fractions that do not have the same denominator, they have to be renamed.

needs to become

which leaves

Now consider:

which is known as an improper fraction. It is said to be improper because the numerator is bigger than the denominator. Often an improper fraction is renamed as a mixed number which is the sum of a whole number and a fraction. Take six of the parts to make a whole (1) and show the part left over as

A fraction is not changed if you can do the same operation to the numerator as to the denominator. Both the numerator and denominator of

can be divided by four to reduce the fraction to

Both terms can also be multiplied by the same number and the number represented by the fraction does not change. This idea is helpful in understanding how to do division of fractions which will be considered next. When multiplying fractions the terms above the line (numerators) are multiplied, and then the terms below the line (denominators) are multiplied, e.g.,

We can also show this graphically. What we are asking is if I have half of something, (e.g., half a yard) what is

of that? The answer is

of a yard.

It was mentioned earlier that a fraction can be thought of as a division problem. Division of fractions such as

may be shown as one large division problem

The easiest problem in the division of fractions is dividing by one because in any fraction that has one as the denominator, e.g.,

we can ignore the denominator because we have 7 wholes. So in our division problem, the question becomes what can we do to get 1 in the denominator? The answer is to multiply

by its reciprocal

and it will cancel out to one. What we do to the denominator we must do to the numerator. The new equation becomes

We can also show this graphically. What we want to know is how many times will a piece of cord

fit into a piece that is

The answer is

Fractions are of immense use in mathematics and physics and the application of these to modern technology. They are also of use in daily life. If you understand fractions you know that

is bigger than

so that shutter speed in photography becomes understandable. A screw of

is smaller than one of

so tire sizes shown in fractions become meaningful rather than incomprehensible. It is more important to understand the concepts than to memorize operations of fractions.