Fraction denominators: What is Denominator? Definition, Types, Examples, Facts

What is Denominator? Definition, Types, Examples, Facts

What Is the Denominator of a Fraction?

A denominator is the bottom number in a fraction. A denominator is a number below the horizontal bar of a fraction. 
A fraction represents a part of a whole. If you divide a pizza into 4 equal parts, each part represents a fraction $\frac{1}{4}$. Three parts represent the fraction $\frac{3}{4}$. Here, 4 is the denominator of the fraction and it represents the total number of parts the whole is divided into.

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Denominator: Definition

Denominator of a fraction is the part of a fraction (a number) written below the horizontal bar of the fraction. It represents the total parts of a whole.

A fraction is of the form $\frac{a}{b},\; b \neq 0$ and both a and b are whole numbers. Here, b is the denominator. Note that a denominator can never be zero!

Fraction and denominator examples:

Fractions	Denominator
$\frac{1}{2}$	2
$\frac{3}{5}$	5
$\frac{x}{y}$	y
$\frac{5}{9}$	9

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Parts of a Fraction: Numerator and Denominator

Numerator: The number written above the fraction bar is known as numerator.

It defines the number of parts of the whole we have taken.

Denominator: The number that’s written below the fraction bar is known as denominator.

It defines the total parts the whole is divided into.
The fraction bar is the small horizontal line between the numerator and denominator. It is known as “vinculum.”

Properties of a Fraction

• If two fractions have the same numerator, the fraction with the larger denominator is the smaller fraction.

Example: $\frac{2}{99} \lt \frac{2}{5}$

• If two fractions have the same denominator, the fraction with the larger numerator is the larger fraction.

Example: $\frac{2}{99} \lt \frac{7}{99}$

Denominator > Numerator

When the numerator is less than the denominator, the fraction is called a proper fraction.

Examples: $\frac{3}{4},\; \frac{4}{5},\; \frac{5}{7}$

Denominator Numerator

When the numerator is greater than (or equal to) the denominator, the fraction is called an improper fraction.

Examples: $\frac{7}{4},\; \frac{23}{5},\; \frac{15}{7},\; \frac{8}{8} = 1$

Types of Fractions Based on the Denominator

Two or more fractions are categorized as “like fractions” and “unlike fractions” based on the value of the denominators.

Fractions with Like Denominators

Observe the fractions $\frac{3}{7},\; \frac{4}{7}$ and $\frac{5}{7}$. We see that they all have the same denominator, 7. These fractions are called fractions with like denominators. The fractions with same or like denominators are called the like fractions.

In the figure above, the denominator in all the free fractions is the same, i.e., 4. So, the denominators are like denominators. 

To compare like fractions, we only compare numerators since the denominator is the same.

Example: $\frac{3}{7} \lt \frac{4}{7} \lt \frac{5}{7}$ since $3 \lt 4 \lt 5$

Fractions with Unlike Denominators

When two or more fractions have different denominators, then the fractions are called unlike fractions or fractions with unlike denominators.

To compare, add, or subtract unlike fractions, we first need to make the denominators of all the fractions the same. In order to do this, we can use the LCM method or fraction bar models.

For example, to compare the fractions $\frac{3}{4}$ and $\frac{5}{7}$, we first draw the fraction models and then compare the two fractions. You can compare the shaded regions and conclude that $\frac{3}{4} \gt \frac{5}{7}$

Least Common Denominator

The smallest or the least number among all the common multiples of the denominators of the given set of fractions is known as the Least Common Denominator or Lowest Common Denominator. 

Note that the least number of the common multiples of the denominators simply means the LCM of the denominators. Thus, to find the LCD of the given fractions, we have to calculate the LCM of the denominators. Let’s understand how to calculate LCD.

How to Find the Least Common Denominator

To find the least common denominator, we can list the multiples of both the denominators and identify the smallest common multiple. This method is convenient to use when the denominators are small numbers.

For example, $\frac{3}{4}$ and $\frac{5}{6}$. 

The multiples of $4 = 4,\; 8,\; 12,\; 16,\; 20,\; 24$, …

The multiples of $6 = 6,\; 12,\; 18,\; 24$, … 

Common multiples $= 12,\; 24,\; 36$, …

LCM of denominators$ = 12$

Thus, the least common denominator is 12. 

Now, we will make the denominators the same (12).

 $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$ 

$\frac{5 \times 2}{6 \times 2} = \frac{10}{12}$ 

Thus, we get the like fractions $\frac{9}{12}$ and $\frac{10}{12}$.

Note: If the two or more denominators have $HCF = 1$, simply multiply the denominators.

For example, $\frac{1}{9}$ and $\frac{4}{7}$.

On multiplying the denominators, we get $LCD = 9 \times 7 = 63$.

The like fractions will be $\frac{7}{63}$ and $\frac{36}{63}$. 

Denominator in Division

In division problems, we divide a quantity or a number into equal parts.  

The division of a number “a” by a number “b” $(b\neq 0)$ can be represented in different ways, such as $a \div b$ or a/b or$\frac{a}{b}$. 

Here, the denominator “b” represents the divisor. It’s the number by which we divide another number.

Example: Divide 15 by 3.

$15 \div 3 = \frac{15}{3} = 5$

Here, 3 is the denominator. This is equivalent to dividing 15 objects into equal groups of 3. 

Denominator in a Ratio

Using ratios, we compare quantities of the same kind. A ratio tells us how many times one number contains another. 

In a ratio a : b, the first number “a” is called the numerator or “the antecedent,” and the second number “b” is called the denominator or the “consequent.”

Facts about Denominator

• Denominator of a fraction can never be 0.
• The term denominator in math is used in different concepts such as ratio and proportion, fractions, division problems, etc.
• If the denominator of a fraction is 1, it is called an improper fraction. Its value is the value of the numerator.
  Example: $\frac{5}{1} = 5$

Conclusion

In this article, we learned about denominators, types of fractions based on the value of the denominators. Denominator is the part of the fraction below the fractional bar. Let’s solve a few examples and practice problems.

Solved Examples on Denominator

1. An apple is cut into 8 equal pieces. Pam eats 3 pieces. Express the fraction of the apple Pam had. What is the denominator? What does it represent?

Solution: 

Total number of equal pieces of apple $= 8$

So, denominator $= 4$

Number of slices Pam had$ = 3$

So, numerator $= 1$

Fraction of apple Pam had $= 38$

Here, the denominator represents the total number of parts the whole apple is divided into.

2. What is the least common denominator of $\frac{3}{7}$ and $\frac{2}{5}$?

Solution: 

Given fractions $= \frac{3}{7}$ and $\frac{2}{5}$

Denominators $= 5$ and 7

5 and 7 are co-prime. So, we multiply them to find the LCD.

Least common denominator $= 5 \times 7 = 35$

3. Identify the fraction represented by the shaded portion. What is the denominator of the fraction?

Solution:  

The circle is divided into 12 equal parts, out of which 6 parts are shaded.

Thus, the shaded portion represents the fraction $\frac{6}{12}$.

The denominator of the fraction $\frac{6}{12}$ is 12. It represents the total number of equal parts of the circle.

Practice Problems on Denominator

1

Which of the following statements is NOT true about the denominator?

Denominator is the bottom number in a fraction.

Denominator represents the total number of equal parts of a whole.

Denominator can be 0.

Denominator of like fractions is equal.

Correct answer is: Denominator can be 0.
Denominator cannot be 0.

2

Which of the following fractions have unlike denominators?

$\frac{4}{7}\text{and}\frac{3}{7}$

$\frac{9}{11}\text{and}\frac{1}{10}$

$\frac{2}{9}\text{and}\frac{5}{9}$

$\frac{3}{13}\text{and}\frac{9}{13}$

Correct answer is: $\frac{9}{11}\text{and}\frac{1}{10}$
The denominator of $\frac{9}{11}$ is 11 and the denominator of $\frac{1}{10}$ is 10.

3

What is the denominator of the fraction represented by the shaded part?

1

4

5

$\frac{1}{4}$

Correct answer is: 4
1 part is shaded out of 4 equal parts. Thus, the shaded part represents the fraction $\frac{1}{4}$.
The denominator is 4.

Frequently Asked Questions on Denominator

Can a denominator be 0?

No, a denominator cannot be zero because a fraction becomes undefined if the denominator has the value 0. We can not divide a whole into 0 equal parts.

What is a unit fraction?

A unit fraction is a fraction whose numerator is 1.

What is the fraction called when the denominator is greater than the numerator?

The fraction in which the denominator is greater than the numerator is known as improper fraction.

What does it mean when the numerator and denominator are equal?

If the numerator and denominator have the same value, the fraction becomes 1. We categorize such a fraction as an improper fraction since the numerator is not less than the denominator. An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

How to Add Fractions in 3 Easy Steps — Mashup Math

Math Skills: How to add fractions with the same denominator and how to add fractions with different denominators

Knowing how to add fractions is an important and fundamental math skill.

Since fractions are a critically important math topic, understanding how to add fractions is a fundamental building block for mastering more complex math concepts that you will encounter in the future.

(Looking to learn how to subtract fractions? Click here to access our free guide)

Luckily, learning how to add fractions with like and unlike (different) denominators is a relatively simple process. The free free How to Add Fractions Step-by-Step Guide will teach you how to add fractions when the denominators are the same and how to add fractions with different denominators using a simple and easy 3-step process.

This guide will teach you the following skills (examples included):

• What is the difference between the numerator and denominator of a fraction?

• How to add fractions with the same denominator?

• How to add fractions with different denominators?

But, before you learn how to add fractions, lets do a quick review of some key characteristics and vocabulary terms related to fractions before we move onto a few step-by-step examples of how to add fractions.

Are you ready to get started?

How to Add Fractions: Definitions and Vocabulary

In order to learn how to add fractions, it is imperative that you understand the difference between a numerator and a denominator.

Definition: The numerator of a fraction is the top number in the fraction. For example, in the fraction 3/4, the numerator is 4.

Definition: The denominator of a fraction is the bottom number in the fraction. For example, in the fraction 3/4, the numerator is 4.

Pretty simple, right? These terms are visually represented in Figure 01 below. Make sure that you understand the difference between the numerator and the denominator of a fraction before moving forward in this guide. If you mix them up, you will not learn how to add fractions correctly.

Figure 01: The numerator is the top number of a fraction and the denominator is the bottom number of a fraction.

Now that you know the difference between the numerator and the denominator of a fraction, you are ready to learn how to identify whether or not a given problem involving adding fractions falls into which of the following categories:

Fractions with like denominators have bottom numbers that equal the same value. For example, in the case of 1/5 + 3/5, you would be adding fractions with like denominators since both fractions have a bottom number of 5.

Conversely, fractions with different (or unlike) denominators have bottom numbers that do not equal the same value. For example, in the case of 1/2 + 3/7, you would be adding fractions with different denominators since the fractions do not share a common denominator (one has a denominator of 2 and the other has a denominator of 7).

These examples are featured in Figure 02 below.

Figure 02: In order to learn how to add fractions, you must be able to identify when the fractions have denominators that are the same and when they have different denominators.

Again, this concept should be simple, but a quick review was required because you will need to be able to identify whether or not a fractions addition problem involves like or unlike denominators in order to solve it correctly.

Now, lets move onto a few examples.

How to Add Fractions with Like Denominators: Example #1

Example #1: 1/4 + 2/4

Our first example is rather simple, but it is perfect for learning how to use our easy 3-step process, which you can use to solve any problem that involves adding fractions:

• Step One: Identify whether the denominators are the same or different

• Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator.

• Step Three: Add the numerators and find the sum.

Okay, let’s take our first attempt at using these steps to solve the first example: 1/4 + 2/4 = ?

Step One: Identify whether the denominators are the same or different.

Clearly, the denominators are the same since they both equal 4

Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator.

Since the denominators are the same, you can move onto Step Three.

Step Three: Add the numerators and find the sum.

To complete this first example, simply add the numerators together and express the result as one single fraction with the same denominator as follows:

Since 3/4 can not be simplified further, you can conclude that…

Final Answer: 3/4

This process is summarized in Figure 03 below.

Figure 03: How to Add Fractions: The process is relatively simple when the denominators are the same.

As you can see from this first example, learning how to add fractions when the denominators are the same is very simple.

To add fractions with the same denominator, simply add the numerators and keep the same denominator.

Let’s take a look at one more example of adding fractions when the denominators are the same before you learn how to add fractions with different denominators.

How to Add Fractions with Like Denominators: Example #2

Example #2: 2/9 + 4/9

To solve this second example, lets apply the 3-step process like we did in the previous example as follows:

Step One: Identify whether the denominators are the same or different.

The denominators in this example are the same since they both equal 9.

Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator.

Again, you can skip the second step because the denominators are the same.

Step Three: Add the numerators and find the sum.

The final step is to add the numerators and keep the denominator the same:

In this case, 6/9 is the correct answer, but this fraction can actually be reduced. Since both 6 and 9 are divisible by 3, 6/9 can be reduced to 2/3.

Final Answer: 2/3

This process is summarized in Figure 04 below.

Figure 04: How to Add Fractions: 6/9 can be reduced to 2/3

Next, lets learn how to add fractions with different denominators.

How to Add Fractions with Different Denominators: Example #1

Example #1: 1/3 + 1/4

Step One: Identify whether the denominators are the same or different.

In this case, the denominators are different (one is 3 and the other is 4)

Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator.

For this example, you can not skip the second step. Before you can continue on, you will need to find a common denominator—a number that both denominators can divide into evenly.

The easier way to do this is to multiply the denominator of the first fraction by the second fraction and the denominator of the second fraction by the first fraction (i.e. multiply the denominators together).

This process is shown in Figure 05 below.

Figure 05: How to Add Fractions with Different Denominators: Get a common denominator by multiplying the denominators together.

(If you need some help with multiplying fractions, click here to access our free guide).

Now, we have transformed the original question into a scenario involving adding two fractions that do have common denominators, which means that the hard work is over and we can solve by adding the numerators and keep the same denominator:

Since 7/12 can not be simplified further, you can conclude that…

Final Answer: 7/12

Figure 06: Once you have common denominators, you can simply adding the numerators together and keep the same denominator.

Now, lets work through one final example of adding fractions with unlike denominators.

How to Add Fractions with Different Denominators: Example #2

Example #1: 3/5 + 4/11

For this last example, lets again apply the 3-step process:

Step One: Identify whether the denominators are the same or different.

The denominators are clearly different (one is 5 and the other is 11)

Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator.

Just like the last example, the second step is to find a common denominator by multiplying the denominators together as follows:

This process is shown in Figure 07 below.

Figure 07: How to Add Fractions with Different Denominators: Get a common denominator by multiplying the denominators together.

Finally, now that you have common denominators, you can solve the problem as follows:

Since there is no value that divides evenly into both 53 and 55, you can not simplify the fraction further.

Final Answer: 53/55

Figure 08: How to Add Fractions with Different Denominators: 53/55 can not be simplified further.

Conclusion: How to Add Fractions

To add fractions with the same denominator, simply add the numerators (top values) and keep the same denominator (bottom value).

To add fractions with different denominators, you need to find a common denominator. A common denominator is a number that both denominators can divide into evenly.

You can solve problems involving adding fractions for either scenario by applying the following 3-step process:

• Step Two: If they are the same, move onto Step Three. If they are different, find a common denominator.

• Step Three: Add the numerators and find the sum.

Keep Learning:

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Comparing fractions with different denominators — examples (6th grade, mathematics)

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Comparing fractions with different denominators is not the most difficult topic in 6th grade mathematics. Nevertheless, it is worth sorting out the issue once and for all in order to avoid annoying mistakes.

What is a fraction?

A fraction is an unfinished division operation. Very often in practical calculations there are situations when it is impossible to divide a number even using decimal fractions. But such an operation can be written as an ordinary fraction and the calculation can be continued without losing accuracy.

If you cannot divide one number by another, you can round the result. But then the accuracy of the calculation is lost, so it is worth rounding only the final result of the calculations.

What is the denominator of a fraction?

A common fraction is written with a horizontal bar.

The number on top is called the numerator. The numerator is the dividend of the unfinished division operation.

The number below is the denominator. The divisor of the unfinished division operation is written as the denominator.

Special cases of comparing fractions

Let’s analyze several special cases of comparing fractions:

• If the denominators of fractions are equal, then the greater is the fraction whose numerator is greater.
• If the numerators of fractions are equal, then the larger fraction is the one whose denominator is smaller.
• If the numerators and denominators of fractions are equal, then such fractions are also equal.

Please note that the calculation may result in a fraction with a numerator equal to zero. In such a case, the whole fraction is zero and the fraction should be converted to zero immediately. But at the same time, there cannot be zero in the denominator.

Comparison of fractions with different denominators

Let’s write an algorithm for comparing fractions with different denominators and numerators:

• Pay attention to the denominator. Find the least common multiple of two numbers. This is the smallest number that is divisible by both denominators. To do this, you need to perform factorization. In the case when both denominators are prime numbers, you should simply multiply them to find the desired number.
• The found number is a common denominator, to which two fractions must be reduced. You need to use the basic rule of a fraction and multiply the numerator and denominator of each of the fractions so that you end up with fractions with the same denominators.
• We got two numbers with the same denominators, which means that the larger fraction is the one whose numerator is larger.

Example

Compare two fractions:

${7\over{8}}$ and ${5\over{10}}$

• Factoring the number 8 into prime factors:

2*2*2=8

• Factoring the number 10:

5*2=10

• Find the greatest common multiple:

5*2*2*2=40

• We bring both fractions to one denominator:

$${7\over{8}} ={{7*5}\over{8*5}}={35\over{40}}$$

$${5\over{10}}={{5*4}\over{10*4}}={20\over{40}}$$

• Compare the resulting fractions:

${35\over{40}} >{20\over{40}}$, so

$${7\over{8}}>{5\over{10}}$$

What have we learned?

We talked about comparing fractions. We considered the algorithm and gave an example of comparing fractions with different denominators.

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Least Common Denominator Rules

Basic Definition of Common Denominator

Definition

The common denominator of a value is any positive given number that is a multiple of all fraction values.

In other words, we can say that the common denominator of a fraction will be characterized by a natural simple numerical value. It must be divisible without a remainder by all the values ​​​​of the denominators of these fractions.

Natural numbers have the property of infinity and therefore a number of ordinary fractional values ​​has a characteristic set of common denominator values. To determine the common denominator for a fraction, you need to apply its basic definition.

Consider two values ​​of fractional expressions: ¹ / ₆ and ³ / ₅

The common fractional denominator will be any number with a positive value. It must be a multiple of 6 and 5.

We list the appropriate values: 30,35,65,95,125,155,185,215 and so on.

An example of solving a problem of this type: Set three fractional values ​​for the solution: ¹ / ₃ ²¹ / ₆ ⁵ / ₁₂ .

They need to be analyzed and reduced to a common denominator, which is equal to 150.

To do this, you need to find out if 150 is divisible by all the numerical denominators of the fraction and is a multiple of them.

This means that 150 must be divisible by 3,6,12 without a remainder.

Compose expressions and perform calculations: 150/3=50; 150/6=25; 150/12=12.5.

When divided by 12, the remainder is obtained.

It follows from this that the number 150 will not be a common multiple denominator for the given fractions.

Least common denominator of fractions

This definition reads as follows: the minimum value of the number by which the denominator of a fraction can be divided, without a residual value.

Abbreviation for this value, looks like NOC.

In a certain list of numerical values ​​that are common denominators of given fractions, the smallest prime value will take place. It will be characterized as the lowest common denominator. Let us formulate the definition of the least common denominator of these fractions.

How to correctly determine the least common denominator of a fraction?

Since LCM, will have the value of the least positive common divisor of the given set of numbers. Then the LCM of the denominators of any fractions is represented as the minimum common denominator of the fraction.

From this it follows that the determination of the smallest denominator of a fraction will be reduced to the determination of the LCM of the denominator of a fraction.

Consider this rule on the example of a solution. The denominators are are 10 and 28, respectively.

The smallest denominator will be determined as the LCM of the numbers 10 and 28.

Let’s factorize the numbers into prime factors: 10=2*5, 28=2*2*7, therefore LCM (15 and 28)=2*2*5* 7=140.

Answer of the problem: 140.

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Reduction of fractions to the lowest common denominator

When simple ordinary fractions have the same denominators, this is characterized as fractions reduced to a common denominator.

Example: values ​​ ⁴⁵ / ₇₆ and ¹⁴³ / ₇₆ reduced to a common denominator, the number 76. Consider a few more fractions. Consider some more fractions ¹ / ₃ , ³ / ₃ , ¹⁷ / ₃ and ¹⁰⁰⁰ / 9015 8 3 .

All these values ​​are reduced to a common denominator of 3.

If the denominators of fractional numbers are different in value and not equal to each other. You can bring them to a common numerical denominator. To do this, we multiply the value of the numerator and denominator of these values ​​with an additional factor.

For example: ² / ₅ and ⁷ / ₄ — these fractions have different denominators, so we will use reduction to a common denominator using additional factors, namely 4 and 5.

Using these values, we present and calculations and we get a common factor: a value equal to 20. When multiplying the numerator and denominator of the fraction ² / ₅ by a value equal to 4, we get a fraction of the form ⁸ / ₂₀ . We carry out similar actions, but only with a fraction When multiplying the numerator and denominator of a fraction ⁷ / ₄ by 5 and reduce it to a fraction of the form ³⁵ / ₂₀ .

Now we can formulate a definition, bringing fractions to a common denominator.

Reducing fractions to the same denominator is a computational process that includes: multiplying the numerators and denominators of any fraction values ​​by certain values ​​of additional factors so that the results of the calculations are fractions with the same denominators.

In mathematics, there is a rule that helps to reduce fractions to a common smallest denominator.

This rule includes three main points.

Least common denominator principle

Then an additional multiplier is determined for each fraction. It must comply with the rule: dividing the least common denominator by the denominator of each fraction considered in the solution.

Multiply the numerator and denominator by the accepted additional factor.

Examples of problem solving using reduction to the smallest denominator.

Example No. 1:

It is necessary to reduce the following fractions to the smallest denominator: ⁵ / ₁₄ and ⁷ / ₁₈

For the solution, we apply the solution algorithm discussed in the above paragraph.

First, let’s determine the smallest value of the common denominator, which is equal to the minimum and a multiple of 14 and 18.

Let’s decompose the values ​​of the denominators into factors: 14=2*7, 18=2*3*3, therefore, the LCM value will be equal to 2*3*3*7=126. The next step is to calculate additional multipliers. With their help, we give the fractional values ​​ ⁵ / ₁₄ and ⁷ / ₁₈ will be reduced to the number 126. The fractional value ⁵ / ₁₄ will be equal to the additional factor 126/14=9. For the value of the second fraction equal to ⁷ / ₁₈ , the same factor will be equal to 129/18=7.

Numerators and denominators of fractions are multiplied by an additional factor of 9 and 7, respectively.

Write the following expressions:

\[
\frac{5}{14}=\frac{5 \cdot 9}{14 \cdot 9}=\frac{45}{126}
\]

\[
\frac{7}{18}=\frac{7 \cdot 7}{18 \cdot 7}=\frac{49}{126}
\]

The results of the calculations: the given fractions ⁵ / ₁₄ and ⁷ / ₁₈ are reduced to a common denominator. Final value of the expression: ⁴⁵ / ₂₆ and ⁴⁹ / ₁₂₆ .

Example No. 2:

It is necessary to reduce the following fractions to the smallest denominator: ³ / ₁₂ and ⁵ / ₁₈ . In this example, we also apply a solution algorithm consisting of three main steps.

Using the solution algorithm, determine the smallest value of the common denominator, which is equal to the smallest value and a multiple of 12 and 18.

The next step is to split these values ​​into factors: 12=2*6, 18=2*3*3, therefore, the LCM value will be equal to 2*3*3*7=216. Let’s calculate additional factors. With their help, we will give the fractional values ​​ ³ / ₁₂ and ⁵ / ₁₈ will be reduced to the number 216. The fractional value ³ / ₁₂ will be equal to the additional multiplier 216/12=18. For the value of the second fraction equal to ⁷ / ₁₈ , the same multiplier will be equal to 216/18=12.

These values ​​of fractions must be multiplied by an additional numerical factor equal to the numbers 9 and 7, respectively. Let’s substitute this data and calculate the composed expressions.