Adding and Subtracting: Fractions

Intuitively one can imagine taking a whole and deconstructing it into smaller pieces. These pieces are called fractions. Like whole numbers, it is possible to add, subtract, multiply, and divide fractions.

Numbers that can be represented as fractions are called rational numbers; these are numbers in the form a/b, where a and b are integers. The top part of the fraction is called the numerator, and the bottom part is called the denominator. Irrational numbers on the other hand are numbers that cannot be written as a fraction in the form a/b. The existence of irrational numbers is thought to have been proven in ancient Greece sometime during the time of Pythagoras. Examples of irrational numbers include or π (approximately 3.14592654), or Euler’s number e (2.71821828458). All of these numbers fall into the class of real numbers, but they may not be represented as fractions.

Fractions with Like Denominators

If two numbers can be represented as fractions, it is possible to write them in terms of a common denominator. The common denominator describes that the values in the numerator represent the same amount of objects or parts. For example, 1/6 and 5/6 share a common denominator—so if a pizza has six slices and Alice eats 1 and Bob eats 5, the fractions 1/6 and 5/6 represent the amount of the entire pizza they ate.

It is important that fractions have a common denominator in order to perform addition and subtraction operations. A frequent mistake is to add (or subtract) both the numerators and denominators of fractions to obtain a result. A simple thought experiment quickly dismisses this: If Alice eats 1/2 of a cake and Bob eats 1/4, clearly the total amount of cake consumed is not 2/6.

Fractions with Unlike Denominators

Fractions that have unlike denominators are still valid numbers. When comparing them to other fractions, however, they cannot represent the same amount of objects. For example if Alice eats 1/6 of a pizza and Bob eats 1/5, it is more difficult to determine how much of the pizza they ate in total. Since the denominators are unlike, they cannot both be used to represent the same object.

To form a common denominator, multiply each fraction by a factor that will result in a denominator for all that is the same without changing the value of the fractions. To do this, multiply both the numerator and denominator by n/n. To obtain a common denominator between the fractions: 1/5 and 3/7, multiply the first fraction by the denominator of the second divided by itself and the second fraction by the denominator of the first divided by itself. This process is called cross multiplication, and it ensures the denominators will be the same without changing the value of the fractions themselves. The new fractions then become

and

.

Since fractions can be represented in an infinite number of ways, there are an infinite number of "equivalent fractions." However, it is best to simplify a fraction into a form in which it can no longer be reduced. A fraction which exists in this form is called a simplified fraction. The goal is to determine whether or not the numerator and the denominator share a common factor, and if this is the case, these values cancel and the fraction is left in a simpler form. For instance, in the fraction 4/6, the numerator and the denominator share a common factor of 2, therefore the fraction can be rewritten as

or simply 2/3.

Adding Fractions

The simplest way to add fractions is to form a common denominator between the fractions. By doing this, the fractions are being represented as partitions of the same "whole object." By adding the numerators (the amount of the whole object that each fraction represents), one can easily determine the exact value of the sum of the fractions. For instance, the fractions 2/3 and 1/5 may be added by cross multiplying, attaining a result of 10/15 and 3/15. By adding the numerators, the result yields the sum of the fractions.

If fractions are being added and already share a common denominator, there is no need to cross multiply, and one can simply add the numerators and leave the denominator untouched to obtain the sum of the two fractions. For instance, the sum of the fractions 4/7 + 2/7 is simply 6/7.

Subtracting Fractions

Subtraction in general is like addition, except that there is a factor of −1 in the expression. Therefore, the way to subtract fractions is identical to adding fractions, except values are decreasing rather than increasing. For instance, the expression 6/7 − 2/3 can be evaluated by determining the common denominator, but instead of adding the numerators, subtract them instead.

Adding and Subtracting Mixed Numbers

Mixed numbers are numbers which contain both a whole part and a fractional part. To add and subtract mixed numbers, one must transform these into a single fraction called an improper fraction. An improper fraction will have a numerator with greater value than the denominator because the expression contains a whole number, which is greater than or equal to one. Thus any fraction which is greater than one is called improper.

For example, 1 1/2 is a mixed number and simply states that it is the sum of the two numbers 1 ( or expessed as a fraction 2/2) and 1/2. It can therefore be expressed as the improper fraction 3/2 and added to other numbers as with proper fractions. The same approach is taken with subtraction, except the mixed number will be in the form .

Bibliography

"Fractions Review: Adding and Subtracting Fractions." Fractions Review: Adding and Subtracting Fractions. Elizabeth Stapel, 17 Oct. 2011. Web. 10 Jan. 2015. <http://www.purplemath.com/modules/fraction4.htm>

Petit, Marjorie M. A Focus on Fractions: Bringing Research to the Classroom. New York: Routledge, 2010. Print.

Small, Marian.Uncomplicating Fractions to Meet Common Core Standards in Math. New York: Teachers College, 2013. Print.