An Introduction to Fraction arithmetic

By Robert O

Fraction, which means breaking in Latin, is a way of representing a part or several parts of a whole unit. Fractions are noted or written down with two numbers, one at the top (aka numerator) and the other at the bottom (denominator), separated by a bar.

For example,

    \[\frac{3}{7}\]

is a fraction read as “three over seven.” 3 is the numerator and 7 is the denominator.

Types of fractions

Fractions take three forms namely proper (numerator<denominator), improper (numerator>denominator), and mixed fractions (consist of two parts: a whole number and a proper fraction).

Proper fraction: When the numerator is less than the denominator. 5/7  is an example of a proper fraction. In general, A/B is a proper fraction if and only if A<B.

When you reverse a proper fraction, you get an improper fraction, where the numerator value is always greater than the denominator value. For example, A/B is an improper fraction if and only if A>B.  Take 7/5 as an example, where 7>5, and hence an improper fraction.

Since the numerator value is greater than the denominator value, you can convert an improper fraction into a whole number and a proper fraction. What you get now is a mixed fraction in the form of

    \[A \frac{b}{c}\]

where A is the whole number part and b is always less than c (b<c). Here is an illustration:

    \[\frac{7}{5}=1 \frac{2}{5}\]

Operation with Fractions

The three different fraction types are subject to basic math operations such as addition, subtraction, multiplication and division. But before we get into that, let us first discuss the simplification of fractions. Every fraction must be expressed in its simplest form unless otherwise stated.

Here are a few examples of simplified fractions:

    \[\frac{4}{8}=\frac{2 x 2}{2 x 2 \times 2}=\frac{1}{2}\]


    \[\frac{12}{27}=\frac{3 x 2 \times 2}{3 \times 3 \times 3}=\frac{2 x 2}{3 \times 3}=\frac{4}{9}\]


All improper fractions require conversion to mixed fractions. I will explain to you with the following examples:

    \[\frac{12}{7}=1 \frac{5}{7}\]


    \[\frac{11}{3}=3 \frac{2}{3}\]

You convert an improper fraction to a mixed fraction by dividing the numerator by the denominator. The resulting figure is the whole number and the quotient becomes the new numerator. NOTE: The denominator remains the same.

Addition and Subtraction of Fractions

The operation is performed the same way it is done on whole numbers. The only difference is that our fractions have to be on a common base, that is, finding the LCM for the numerator values if they are not equal for the fractions to be added or subtracted.

How to add and subtract fractions with the same denominator values?

In general, if you have two fractions A and B where A = a / b and B = c / b, then A+B is given by:

    \[\frac{a}{b}+\frac{c}{b}=\frac{a+c}{b}\]

and

    \[\frac{a}{b}-\frac{c}{b}=\frac{a-c}{b}\]

Example:

Evaluate

    \[\frac{1}{4}+\frac{1}{4} \text { and } \frac{2}{5}+\frac{1}{5}\]

Solution:

    \[\frac{1}{4}+\frac{1}{4}=\frac{1+1}{4}=\frac{2}{4}=\frac{1}{2}\]

and

    \[\frac{2}{5}+\frac{1}{5}=\frac{2+1}{5}=\frac{3}{5}\]


Follow the same procedure when subtracting fractions with a common denominator. For example:

    \[\frac{3}{5}-\frac{1}{5}=\frac{3-1}{5}=\frac{2}{5}\]

How to add and subtract fractions with different denominator values?

In this case, you will have to find a common base i.e. the least common multiple (LCM) of the denominators of all the given fractions. For example:

    \[\frac{2}{5}+\frac{1}{4}=\frac{2 x 4+1 x 5}{20}=\frac{8+5}{20}=\frac{13}{20}\]

and

    \[\frac{5}{6}-\frac{1}{3}=\frac{1 x 5-1 x 2}{6}=\frac{5-2}{6}=\frac{3}{6}=\frac{1}{2}\]


Note: Convert any mixed fractions to improper fractions before carrying out any operation. You can do that by getting the product of the whole number and the denominator and then adding the numerator to it. The result becomes the new numerator while the denominator remains the same.
For example:

    \[2 \frac{1}{2}+1 \frac{1}{3}=\frac{5}{2}+\frac{4}{3}=\frac{3 x 5+2 x 4}{6}=\frac{15+8}{6}=\frac{23}{6}=3 \frac{5}{6}\]


You can apply the same procedure for subtraction operations.

Multiplication and Division of Fractions

How to Multiply Fractions?

Multiplication of fractions is even more direct than the other operations that we have seen thus far. Let us use the example below to illustrate it.

Example:

Evaluate

    \[\frac{2}{5} \times \frac{1}{4} \text { and } \frac{3}{5} \times 1 \frac{1}{4}\]


Solution:

    \[\frac{2}{5} \times \frac{1}{4}=\frac{2 \times 1}{5 \times 4}=\frac{2}{20}=\frac{1}{10}\]

and

    \[\frac{3}{5} \times 1 \frac{1}{4}=\frac{3}{5} \times \frac{5}{4}=\frac{3 \times 5}{5 \times 4}=\frac{15}{20}=\frac{3}{4}\]


The following are the steps for multiplying any fractions:

Change any mixed fraction into an improper fraction

    \[\frac{3}{5} \times 1 \frac{1}{4}=\frac{3}{5} \times \frac{5}{4}\]


Compute the product of all the numerator and do the same for the denominators. You may arrive at an improper or proper fraction.

    \[\frac{3}{5} \times \frac{5}{4}=\frac{3 \times 5}{5 \times 4}\]


Simplify proper fractions by dividing both numerator and denominator by a common factor or convert improper fractions to mixed fractions

    \[\frac{15}{20}=\frac{3}{4}\]

a common factor for 15 and 20 is 5.

Note: You can simplify the process by cross-cancellation if there is a common factor. For example:

    \[\frac{2}{5} \times \frac{1}{4}=\frac{1}{5} \times \frac{1}{2}=\frac{1}{10}\]

How to Divide Fractions?

If you know how to multiply a fraction, division of fractions becomes easy. You multiply the numerator of the first fraction by the denominator of the second fraction by a process called cross multiplication. The steps are as follows:

Perform mixed fractions conversion to improper fractions

    \[\frac{3}{5} \div 1 \frac{1}{4}=\frac{3}{5} \div \frac{5}{4}\]

Find the reciprocal of the fraction to the right and then convert the division sign into multiplication sign.

    \[\frac{3}{5} \div \frac{5}{4}=\frac{3}{5} \times \frac{4}{5}\]

Compute the product of all the numerator and do the same for the denominators. You may arrive at an improper or proper fraction.

    \[\frac{3}{5} \times \frac{4}{5}=\frac{3 \times 4}{5 \times 5}=\frac{12}{25}\]

Simplify proper fractions or convert improper fractions to mixed fractions if possible

Example:

Evaluate

    \[\frac{2}{5} \div \frac{1}{4} \text { and } \frac{3}{5} \div 1 \frac{1}{4}\]

Solution:

    \[\frac{2}{5} \div \frac{1}{4}=\frac{2}{5} \times \frac{4}{1}=\frac{2 \times 4}{5 \times 1}=\frac{8}{5}=1 \frac{3}{5}\]

and

    \[\frac{3}{5} \div 1 \frac{1}{4}=\frac{3}{5} \div \frac{5}{4}=\frac{3}{5} \times \frac{4}{5}=\frac{3 \times 4}{5 \times 5}=\frac{12}{25}\]

Remarks:

That concludes the operation with fractions. What is left is the squares and square roots of fractions, but that is a topic for another day.

About the Author

This lesson was prepared by Robert O. He holds a Bachelor of Engineering (B.Eng.) degree in Electrical engineering and electronic engineering. He is a career teacher and headed the department of languages and assumed various leadership roles. He writes for Full Potential Learning Academy.