2.1: Properties of Fractions and Reducing to Lowest Terms

In this section we deal with fractions, numbers or expressions of the form a/b.

Definition: Fractions

A number of the form

\[ \dfrac{a}{b}\nonumber \]

where \(a\) and \(b\) are numbers is called a fraction. The number \(a\) is called the numerator of the fraction, while the number \(b\) is called the denominator of the fraction.

Near the end of this section, we’ll see that the numerator and denominator of a fraction can also be algebraic expressions, but for the moment we restrict our attention to fractions whose numerators and denominators are integers. We start our study of fractions with the definition of equivalent fractions.

Equivalent Fractions

Two fractions are equivalent if they represent the same numerical value.

But how can we tell if two fractions represent the same number? Well, one technique involves some simple visualizations. Consider the image shown in Figure 4.1, where the shaded region represents 1/3 of the total area of the figure (one of three equal regions is shaded).

Figure 4.1: The shaded region is 1/3 of the whole region.

In Figure 4.2, we’ve shaded 2/6 of the entire region (two of six equal regions are shaded).

Figure 4.2: The shaded region is 2/6 of the whole region.

In Figure 4.3, we’ve shaded 4/12 of the entire region (four of twelve equal regions are shaded).

Figure 4.3: The shaded region is 4/12 of the whole region.

Let’s take the diagrams from Figure 4.1, Figure 4.2, and Figure 4.3 and stack them one atop the other, as shown in Figure 4.4.

Figure 4.4: One of three equals two of six equals four of twelve.

Figure 4.4 provides solid visual evidence that the following fractions are equivalent.

\[ \dfrac{1}{3} = \dfrac{2}{6} = \dfrac{4}{12}\nonumber \]

Key Observations

1. If we start with the fraction 1/3, then multiply both numerator and denominator by 2, we get the following result.

\[ \begin{aligned} \dfrac{1}{3} = \dfrac{1 \cdot 2}{3 \cdot 2} ~ & \textcolor{red}{ \text{ Multiply the numerator and denominator by 2.}} \\ = \dfrac{2}{6} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]

This is precisely the same thing that happens going from Figure 4.1 to 4.2, where we double the number of available boxes (going from 3 available to 6 available) and double the number of shaded boxes (going from 1 shaded to 2 shaded).

2. If we start with the fraction 1/3, then multiply both numerator and denominator by 4, we get the following result.

\[ \begin{aligned} = \dfrac{1}{3} = \dfrac{1 \cdot 4}{3 \cdot 4} ~ & \textcolor{red}{ \text{ Multiply numerator and denominator by 4.}} \\ = \dfrac{4}{12} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]

This is precisely the same thing that happens going from Figure 4.1 to 4.3, where we multiply the number of available boxes by 4 (going from 3 available to 12 available) and multiply the number of shaded boxes by 4 (going from 1 shaded to 4 shaded).

The above discussion motivates the following fundamental result.

Creating Equivalent Fractions

If you start with a fraction, then multiply both its numerator and denominator by the same number, the resulting fraction is equivalent (has the same numerical value) to the original fraction. In symbols,

\[ \dfrac{a}{b} = \dfrac{a \cdot x}{b \cdot x}\nonumber \]

Arguing in Reverse

Reversing the above argument also holds true.

1. If we start with the fraction 2/6, then divide both numerator and denominator by 2, we get the following result.

\[ \begin{aligned} \dfrac{2}{6} = \dfrac{2 \div 2}{6 \div 2} ~ & \textcolor{red}{ \text{ Divide numerator and denominator by 2.}} \\ = \dfrac{1}{3} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]

This is precisely the same thing that happens going backwards from Figure 4.2 to 4.1, where we divide the number of available boxes by 2 (going from 6 available to 3 available) and dividing the number of shaded boxes by 2 (going from 2 shaded to 1 shaded).

2. If we start with the fraction 4/12, then divide both numerator and denominator by 4, we get the following result.

\[ \begin{aligned} \dfrac{4}{12} = \dfrac{4 \div 4}{12 \div 4} ~ & \textcolor{red}{ \text{ Multiply numerator and denominator by 4.}} \\ = \dfrac{1}{3} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]

This is precisely the same thing that happens going backwards from Figure 4.3 to 4.1, where we divide the number of available boxes by 4 (going from 12 available to 3 available) and divide the number alignof shaded boxes by 4 (going from 4 shaded to 1 shaded).

The above discussion motivates the following fundamental result.

Creating Equivalent Fractions

If you start with a fraction, then divide both its numerator and denominator by the same number, the resulting fraction is equivalent (has the same numerical value) to the original fraction. In symbols,

\[ \dfrac{a}{b} = \dfrac{a \div x}{b \div x}.\nonumber \]

The Greatest Common Divisor

We need a little more terminology.

Divisor

If d and a are natural numbers, we say that “d divides a” if and only if when a is divided by d, the remainder is zero. In this case, we say that “d is a divisor of a.”

For example, when 36 is divided by 4, the remainder is zero. In this case, we say that “4 is a divisor of 36.” On the other hand, when 25 is divided by 4, the remainder is not zero. In this case, we say that “4 is not a divisor of 25.”

Greatest Common Divisor

Let a and b be natural numbers. The common divisors of a and b are those natural numbers that divide both a and b. The greatest common divisor is the largest of these common divisors.

Example 1

Find the greatest common divisor of 18 and 24.

Solution

First list the divisors of each number, the numbers that divide each number with zero remainder.

Divisors of 18 : 1, 2, 3, 6, 9, and 18

Divisors of 24 : 1, 2, 3, 4, 6, 8, 12, and 24

The common divisors are:

Common Divisors : 1, 2, 3, and 6

The greatest common divisor is the largest of the common divisors. That is,

Greatest Common Divisor = 6.

That is, the largest number that divides both 18 and 24 is the number 6.

Exercise

Find the greatest common divisor of 27 and 36.

Answer

9

Reducing a Fraction to Lowest Terms

First, a definition.

Lowest Terms

A fraction is said to be reduced to lowest terms if the greatest common divisor of both numerator and denominator is 1.

Thus, for example, 2/3 is reduced to lowest terms because the greatest common divisior of 2 and 3 is 1. On the other hand, 4/6 is not reduced to lowest terms because the greatest common divisor of 4 and 6 is 2.

Example 2

Reduce the fraction 18/24 to lowest terms.

Solution

One technique that works well is dividing both numerator and denominator by the greatest common divisor of the numerator and denominator. In Example 1, we saw that the greatest common divisor of 18 and 24 is 6. We divide both numerator and denominator by 6 to get

\[ \begin{aligned} \dfrac{18}{24} = \dfrac{18 \div 6}{24 \div 6} ~ & \textcolor{red}{ \text{ Divide numerator and denominator by 6.}} \\ = \dfrac{3}{4} ~ & \textcolor{red}{ \text{ Simplify numerator and dice.}} \end{aligned}\nonumber \]

Note that the greatest common divisor of 3 and 4 is now 1. Thus, 3/4 is reduced to lowest terms.

There is a second way we can show division of numerator and denominator by 6. First, factor both numerator and denominator as follows:

\[ \begin{aligned} \dfrac{18}{24} = \dfrac{3 \cdot 6}{4 \cdot 6} ~ & \textcolor{red}{ \text{ Factor out a 6.}} \end{aligned}\nonumber \]

You can then show “division” of both numerator and denominator by 6 by “crossing out” or “canceling” a 6 in the numerator for a 6 in the denominator, like this:

\[ \begin{aligned} = \dfrac{3 \cdot \cancel{6}}{4 \cdot \cancel{6}} ~ & \textcolor{red}{ \text{ Cancel common factor.}} \\ = \dfrac{3}{4} \end{aligned}\nonumber \]

Note that we get the same equivalent fraction, reduced to lowest terms, namely 3/4.

Exercise

Reduce the fraction 12/18 to lowest terms.

Answer

2/3

Important Point

In Example 2 we saw that 6 was both a divisor and a factor of 18. The words divisor and factor are equivalent.

We used the following technique in our second solution in Example 2.

Cancellation Rule

If you express numerator and denominator as a product, then you may cancel common factors from the numerator and denominator. The result will be an equivalent fraction.

Because of the “Cancellation Rule,” one of the most effective ways to reduce a fraction to lowest terms is to first find prime factorizations for both numerator and denominator, then cancel all common factors.

Example 3

Reduce the fraction 18/24 to lowest terms.

Solution

Use factor trees to prime factor numerator and denominator.

Once we’ve factored the numerator and denominator, we cancel common factors.

\[ \begin{aligned} \dfrac{18}{24} = \dfrac{2 \cdot 3 \cdot 3}{2 \cdot 2 \cdot 2 \cdot 3} ~ & \textcolor{red}{ \text{ Prime factor numerator and denominator.}} \\ = \dfrac{ \cancel{2} \cdot \cancel{3} \cdot 3}{ \cancel{2} \cdot 2 \cdot 2 \cdot \cancel{3}} ~ & \textcolor{red}{ \text{ Cancel common factors.}} \\ = \dfrac{3}{2 \cdot 2} ~ & \textcolor{red}{ \text{ Remaining factors.}} \\ = \dfrac{3}{4} ~ & \textcolor{red}{ \text{ Simplify denominator.}} \end{aligned}\nonumber \]

Thus, 18/24 = 3/4.

Exercise \(\PageIndex{1}\)

Reduce the fraction 28/35 to lowest terms.

Answer

4/5

Example 4

Reduce the fraction 28/42 to lowest terms.

Solution

Use factor trees to prime factor numerator and denominator.

Now we can cancel common factors.

\[ \begin{aligned} \dfrac{28}{42} = \dfrac{2 \cdot 2 \cdot 7}{2 \cdot 3 \cdot 7} ~ & \textcolor{red}{ \text{ Prime factor numerator and denominator.}} \\ = \dfrac{ \cancel{2} \cdot 2 \cdot \cancel{7}}{ \cancel{2} \cdot 3 \cdot \cancel{7}} ~ & \textcolor{red}{ \text{ Cancel common factors.}} \\ = \dfrac{2}{3} \end{aligned}\nonumber \]

Thus, 28/42 = 2/3.

Exercise

Reduce the fraction 36/60 to lowest terms.

Answer

3/5

Equivalent Fractions in Higher Terms

Sometimes the need arises to find an equivalent fraction with a different, larger denominator.

Example 6

Express 3/5 as an equivalent fraction having denominator 20.

Solution

The key here is to remember that multiplying numerator and denominator by the same number produces an equivalent fraction. To get an equivalent fraction with a denominator of 20, we’ll have to multiply numerator and denominator of 3/5 by 4.

\[ \begin{aligned} \dfrac{3}{5} ~ & \textcolor{red}{ \text{ Multiply numerator and denominator by 4.}} \\ = \dfrac{12}{20} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]

Therefore, 3/5 equals 12/20.

Exercise

Express 2/3 as an equivalent fraction having denominator 21.

Answer

14/21

Example 7

Express 8 as an equivalent fraction having denominator 5.

Solution

The key here is to note that

\[ \begin{aligned} 8 = \dfrac{8}{1} ~ & \textcolor{red}{ \text{ Understood denominator is 1.}} \end{aligned}\nonumber \]

To get an equivalent fraction with a denominator of 5, we’ll have to multiply numerator and denominator of 8/1 by 5.

\[ \begin{aligned} = \dfrac{8 \cdot 5}{1 \cdot 5} ~ & \textcolor{red}{ \text{ Multiply numerator and denominator by 5.}} \\ = \dfrac{40}{5} ~ & \textcolor{red}{ \text{ Simplify numerator and denominator.}} \end{aligned}\nonumber \]

Therefore, 8 equals 40/5.

Exercise

Express 5 as an equivalent fraction having denominator 7.

Answer

35/7

Negative Fractions

We have to also deal with fractions that are negative. First, let’s discuss placement of the negative sign.

• Positive divided by negative is negative, so

\[ \dfrac{3}{-5} = - \dfrac{3}{5}.\nonumber \]

• But it is also true that negative divided by positive is negative. Thus,

\[ \dfrac{−3}{5} = \dfrac{−3}{5}.\nonumber \]

These two observations imply that all three of the following fractions are equivalent (the same number):

\[ \dfrac{3}{-5} = - \dfrac{3}{5} = \dfrac{-3}{5}.\nonumber \]

Note that there are three possible placements for the negative sign: (1) the denominator, (2) the fraction bar, or (3) the numerator. Any one of these placements produces an equivalent fraction.

Fractions and Negative Signs

Let a and b be any integers. All three of the following fractions are equivalent (same number):

\[ \dfrac{a}{-b} = - \dfrac{a}{b} = \dfrac{-a}{b}.\nonumber \]

Mathematicians prefer to place the negative sign either in the numerator or on the fraction bar. The use of a negative sign in the denominator is discouraged.

Fractions and Negative Signs

Let \(a\) and \(b\) be any integers. Then,

\[ \dfrac{-a}{-b} = \dfrac{a}{b}.\nonumber \]

 

 

Exercises

Find the GCD of the given numbers.

2. 36, 60

3. 52, 20

8. 10, 40

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Reduce the given fraction to lowest terms.

14. \(\dfrac{28}{56}\)

15. \(\dfrac{93}{15}\)

20. \(\dfrac{10}{45}\)

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30. Express 3 as an equivalent fraction having denominator 8.

36. Express \(\dfrac{17}{22}\) as an equivalent fraction having denominator 44.

37. Express \(\dfrac{1}{3}\) as an equivalent fraction having denominator 24. 

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Reduce the given fraction to lowest terms.

42. \(\dfrac{−48}{14}\)

44. \(\dfrac{27}{−75}\)

56. \(\dfrac{−60}{−100}\)

 

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101. Hurricanes. According to the National Atmospheric and Oceanic Administration, in 2008 there were 16 named storms, of which 8 grew into hurricanes and 5 were major.

i) What fraction of named storms grew into hurricanes? Reduce your answer to lowest terms.

ii) What fraction of named storms were major hurricanes? Reduce your answer to lowest terms.

iii) What fraction of hurricanes were major? Reduce your answer to lowest terms.

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Answers

2. 12

3. 4

8. 10

14. \(\dfrac{1}{2}\)

15. \(\dfrac{31}{5}\)

20. \(\dfrac{2}{9}\)

30. \(\dfrac{24}{8}\)

36. \(\dfrac{34}{44}\)

37. \(\dfrac{8}{24}\)

42. \(\dfrac{−24}{7}\)

44. \(\dfrac{− 9}{25}\)

56. \(\dfrac{3}{5}\)

101.

i) \(\dfrac{1}{2}\)

ii) \(\dfrac{5}{16}\)

iii) \(\dfrac{5}{8}\)