Operations with fractions
Fraction is the name for part of something as distinct from the whole of it. The word itself means a small amount as, for example, when we ask someone to "move over a fraction." We mean them to move over part of the way, not all the way.
Fractional parts such as half, quarter, eighth, and so on form a part of daily language usage. When, for example, we refer to "half an hour," "a quarter pound of coffee," or "an eighth of a pie." In arithmetic, the word fraction has a more precise meaning since a fraction is a numeral. Most fractions are called common fractions to distinguish them from special kinds of fractions like decimal fractions.
A fraction is written as two stacked numerals with a line between them, e.g.,
which refers to three-fourths (also called three quarters). All fractions are read this way.
is called five-ninths and 5, the top figure, is known as the numerator, while the bottom figure, 9, is called the denominator.
A fraction expresses a relationship between the fractional parts to the whole. For example, the fraction
shows that something has been divided into four equal parts and that we are dealing with three of them. The denominator denotes into how many equal parts the whole has been divided. A numerator names how many of the parts we are taking. If we divide something into four parts and only take one of them, we show it as
This is known as a unit fraction.
Whole numbers can also be shown by fractions. The fraction
means five wholes, which is also shown by 5.
Another way of thinking about the fraction
is to see it as expressing the relationship between a number of items set apart from the larger group. For example, if there are 16 books in the classroom and 12 are collected, then the relationship between the part taken (12) and the larger group (16) is
The fraction 8
names the same number as
Two fractions that stand for the same number are known as equivalent fractions.
A third way of thinking about the fraction
is to think of it as measurement or as a division problem. In essence the symbol
says: take three units and divide them into four equal parts. The answer may be shown graphically. The size of each part may be seen to be
To think about a fraction as a measurement problem is a useful way to help understand the operation of division with fractions which will be explained later.
A fourth way of thinking about
is as expressing a ratio. A ratio is a comparison between two numbers. For example, 3 is to 4, as 6 is to 8, as 12 is to 16, and 24 is to 32. One number can be shown by many different fractions provided the relationship between the two parts of the fraction does not change. This is most important in order to add or subtract, processes which will be considered next.
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
the problem becomes
Similarly, with subtraction of fractions that do not have the same denominator, they have to be renamed.
needs to become
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
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.
Barrow, J.D. Pi in the Sky. New York: Oxford University Press, 1992.
Hamilton, Johnny E., and Margaret S. Hamilton. Math to Build On: A Book for Those Who Build. Clinton, NC: Construction Trades Press, 1993.
Savin, Steve. All the Math You'll Ever Need. New York: John Wiley & Sons, 1989.
Selma E. Hughes