1.3 Working with Fractions

Introduction

In section 1.1 we reintroduced you to several different sets of numbers that we use regularly in math. Of all these sets of numbers, it is quite possible that the fractions are the ones that have caused the most frustration for students.

In fact, a 2008 report to the US Department of Education found that "the most important foundational skill not presently developed appears to be proficiency with fractions". So if you struggle with fractions, you are not alone.

It is widely believed that the ancient Greek mathematician Pythagoras was the first person to rigorously explore fractions; though he did so in a purely geometric way, often with triangles.

We've come a long way since then--no doubt you have a calculator or smart phone that can do all sorts of things with fractions without you having to think much about it. Even so, having a basic understanding of how fractions work, without having to rely on technology, will speed you on your way to success.

Reviewing Basic Fraction Concepts

Before we get into the nuts and bolts of working with fractions, let's start by reviewing some basic concepts and terminology.

Fractions are a way of representing parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). For example, in the fraction \( \frac{3}{4}\), the number \(3\) is the numerator and \(4\) is the denominator.

Fractions that represent values less than 1 are called proper fractions, and those with values greater than or equal to 1 are referred to as improper fractions.

Earlier in your education you may have come across mixed numbers, ones in which there is a whole number part and a fraction part, like \(3 \frac{1}{2}\). Almost without exception, mixed numbers are not used in algebra, calculus or statistics.

And now on to some examples of things you might be asked to do.

Simplifying Fractions

Sometimes the numerator and denominator of a fraction are both multiples of the same number. To simplify a fraction means expressing it in its simplest form by dividing both the numerator and denominator by that number, which is called the greatest common divisor (GCD). Once you've done that, the numerator and denominator will have been reduced to their smallest possible values.

Let's say we want to simplify the fraction \( \frac{20}{45}\). Here's how we do that.

1. Find the GCD: Since \(20=4\cdot5\) and \(45=9\cdot5\), the GCD of \(20\) and \(45\) is \(5\).
2. Divide the numerator by the GCD: \(20 \div 5 = 4\)
3. Divide the denominator by the GCD: \(45 \div 5 = 9\)
4. Rewrite the fraction: \(\frac{20}{45} = \frac{4}{9}\).

and now we know \( \frac{20}{45}\) simplifies to \( \frac{4}{9}\). Both fractions represent the same number, just in different formats.

It's standard practice to reduce fractions as much as possible as you work through a problem.

Multiplying Fractions

Multiplying fractions is very straightforward and has a simple rule. To multiply fractions, you simply multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator.

For example:

\[ \frac{2}{3} \cdot \frac{5}{7} = \frac{10}{21} \]

Dividing Fractions

To divide fractions, you multiply the first fraction by the reciprocal (flipped version) of the second fraction.

For example:

\begin{align} \frac{3}{4} \div \frac{2}{5} &= \frac{3}{4} \cdot \frac{5}{2} \\ &= \frac{15}{8} \end{align}

Adding and Subtracting Fractions with the Same Denominator

When adding or subtracting fractions with the same denominator, you simply add or subtract the numerators while keeping the denominator unchanged.

As long as you do not add or subtract the denominators, you should be alright. Here's an example of addition

\begin{align} \frac{1}{5} + \frac{2}{5} &= \frac{1+2}{5} \\ &= \frac{3}{5} \end{align}

and one with subtraction

\begin{align} \frac{7}{8} - \frac{3}{8} &= \frac{7-3}{8} \\ &= \frac{4}{8} \\ &= \frac{1}{2} \end{align}

Adding and Subtracting Fractions with Different Denominators

If we need to add fractions that do not have the same denominator, what do we do? The answer is that we rewrite them so that they do have the same denominator. The process is kind of like simplifying but in reverse. A quick example will illustrate the idea, then we'll go through the details. For this example, all you need to know is that \(\frac{20}{40}={1}{2}\) (which you can verify by simplifying.)

\begin{align} \frac{1}{2}+\frac{13}{40} &= \frac{20}{40}+\frac{13}{40} && \small \color{#5fa2ce} {\text{replacing } \frac{1}{2} \text{ with } \frac{20}{40}} \\ &= \frac{20+13}{40} && \small \color{#5fa2ce} {\text{add the numerators}} \\ &= \frac{33}{40} \end{align}

The trick was being able to rewrite \(\frac{1}{2}\) as \(\frac{20}{40}\) so that both fractions had a denominator of \(40\). In this case, \(40\) was the smallest multiple that both denominators share, or least common multiple (LCM).

The LCM can always be used as the common denominator, but so can any common multiple of the denominators. As long as we can rewrite one (or both) of the fractions so that they have the same denominator, we can do the addition or subtraction.

The fastest way to rewrite both fractions so they have a common denominator is to multiply each part of the fraction by the denominator of the other one. Let's see how that works.

\begin{align} \frac{1}{2}+\frac{4}{3} &= \frac{1}{2}\cdot \frac{3}{3}+\frac{4}{3} \cdot \frac{2}{2} && \small \color{#5fa2ce} {\text{multiplying each part by the denominator of the other fraction}}\\ &= \frac{3}{6} + \frac{8}{6} && \small \color{#5fa2ce} {\text{now our factions have the same denominator}} \\ &= \frac{11}{6} \end{align} We summarize this technique below.

Powers and Roots of Fractions

Lastly, we'll touch on something that will come up again in a later section: powers and roots of fractions.

The rules for these are straighfoward: we simply apply the power or root to both parts of the fraction.

For instance, \begin{align} \left(\frac{5}{3}\right)^2 &= \frac{5^2}{3^2} \\ &= \frac{25}{9} \end{align}