adding fractions with different denominators problem solving

Adding Fractions

The addition of fractions is a little different from the normal addition of numbers since a fraction has a numerator and a denominator which is separated by a bar. The addition of fractions can be easily done if the denominators are equal. While like fractions have common denominators, unlike fractions are converted to like fractions to make addition easier. Let us explore more about adding fractions , and how to add two fractions in this article.

How to Add Fractions?

Fractions are part of a whole. Before moving to the addition of fractions, let us quickly revise what are fractions. Fractions are made up of two parts, the numerator and the denominator. A general representation of a fraction is a/b, where 'a' is the numerator, 'b' is the denominator, and 'b' cannot be zero. For example, 2/3, 14/5, 6/7, 28/9, and 21/43. Just like other numbers, we can perform the arithmetic operations of addition , subtraction , multiplication , and division on fractions. The addition of fractions means finding the sum of two or more fractions. Now, let us learn the basic steps of the addition of fractions with the help of the following example.

Example: Add 1/4 + 2/4

Solution: Let us add these fractions using the following steps.

• Step 1: Check if the denominators are the same. (Here, the denominators are the same, so we move to the next step)
• Step 2: Add the numerators and place the sum over the common denominator . This means (1 + 2)/4 = 3/4
• Step 3: Simplify the fraction to its lowest form, if needed. Here, it is not needed. So, the sum of the given fractions is, 1/4 + 2/4 = 3/4

There are different types of fractions in Mathematics. While adding fractions we need to check whether they are like fractions or unlike fractions. Like fractions are a group of fractions with a common denominator, while unlike fractions are a group of fractions having different denominators. While learning about the addition of fractions, we might come across the following scenarios.

• Addition of fractions with same denominators: 3/4 + 1/4
• Addition of fractions with different denominators: 3/5 + 1/2
• Addition of fractions with whole numbers: 1/2 + 2
• Adding fractions with variables: 3/5y + 1/4y

Now, let us learn more about the above cases in detail.

Adding Fractions with Like Denominators

Adding fractions with the same denominators is done by writing the sum of the numerators over the common denominator. Let us understand how to add fractions with the same denominator with the help of an example.

Example: Add the fractions 2/4 + 1/4

Solution: We can see that the denominators of the given fractions are the same. These fractions are called like fractions .

The addition of like fractions can be done by adding the numerators of the given fractions and retaining the common denominator. In this case, we keep the denominator as 4, and we add the numerators. This can be expressed as 2/4 + 1/4 = (2 +1)/4 = 3/4. This gives the sum as 3/4.

Adding Fractions with Unlike Denominators

We just learned how to add fractions with like denominators. Now let us understand how to do the addition of fractions with different or unlike denominators. When the denominators are different, the fractions are called unlike fractions. In such fractions, the first step is to convert them to like fractions so that the denominators become common. This is done by finding the Least Common Multiple (LCM) of the denominators. Let us understand this with the help of the following example.

Example: Add the fractions 1/3 and 3/5.

Solution: We will use the following steps to add these fractions.

• Step 1: Since the denominators in the given fractions are different, we find the LCM of 3 and 5 to make them the same. LCM of 3 and 5 = 15.
• Step 2: Now, multiply 1/3 with 5/5, (1/3) × (5/5) = 5/15, and 3/5 with 3/3, (3/5) × (3/3) = 9/15, which will convert them to like fractions with the same denominators.
• Step 3: Now, the denominators are the same, so we simply add the numerators and write the sum over the common denominator. The new fractions with common denominators are 5/15 and 9/15. So, 5/15 + 9/15 = (5 + 9)/15 = 14/15.

Adding Fractions with Whole Numbers

An easy way to add a whole number and a proper fraction is to combine and express them as a mixed fraction . For example, 5 + 1/2 can be combined and expressed as 5½ = 11/2. Similarly, 3 + 1/7 = \(3\frac{1}{7} \) = 22/7. However, there is another method for adding fractions with whole numbers. Let us understand that with the help of the following example.

Example: Add 3 + 4/5

Solution: Let us add these numbers using the following steps:

• Step 1: In this method, we change the whole number to its fraction form by writing 1 as its denominator. Here, 3 is the whole number and this can be written as 3/1
• Step 2: Now, 3/1 can be added to 4/5, that is, 3/1 + 4/5. We will add these by making the denominators the same because they are unlike fractions. This implies, (3/1) + (4/5) = (3/1) × (5/5) + (4/5) × (1/1) = 15/5 + 4/5 = 19/5 = \(3\frac{4}{5} \)

Adding Fractions with Variables

Now that we have seen the addition of fractions with like and unlike fractions, we can extend the same concept for adding fractions with variables. Let us understand this with the help of the following example.

Example: Add y/5 + 2y/5 where 'y' is the variable.

Solution: Let us add these fractions using the following steps:

• Step 1: The given fractions, y/5 + 2y/5 are like fractions since they have the same denominator and we can see that 'y' is common.
• Step 2: We can take the common factor out and rewrite it as: y/5 + 2y/5 = (1/5 + 2/5)y = 3y/5
• Step 3: Therefore, the sum of y/5 + 2y/5 = 3y/5

Now, let us learn how to add unlike fractions using the following example.

Example: Add y/2 + y/3

Solution: Let us add the fractions using the following steps.

• Step 1: Since the given fractions, y/2 + y/3 are unlike fractions, we will take the LCM of the denominators and convert them into like fractions.
• Step 4: Next, we need to take the common variable out and rewrite it as follows: LCM (2, 3) = 6; y/2 = (y/2) × (3/3) = 3y/6 and y/3 = (y/3 × (2/2) = 2y/6
• Step 5: We got two fractions with common denominators, (3y/6) + (2y/6) = (3y + 2y)/6 = 5y/6. Therefore, the sum of y/2 + y/3 = 5y/6

It should be noted that in some cases, when we have different variables, like 'x' and 'y', they are treated as unlike terms and cannot be simplified further, for example, x/2 + y/3

Tips and Tricks on Addition of Fractions

The following points are helpful and should be remembered while working with the addition of fractions:

• For unlike fractions, we do not add the numerators and denominators directly. 1/5 + 2/3 ≠ 3/8
• To add unlike fractions, first, convert the given fractions to like fractions by taking the LCM of the denominators.
• Add the numerators and retain the common denominator to get the sum of the fractions.

☛ Related Topics

• Subtraction of Fractions
• Multiplying Fractions
• Division of Fractions
• Like Fractions and Unlike Fractions
• Adding Fractions Calculator

Addition of Fractions Examples

Example 1: Add the following fractions: 1/7 and 3/7

The given fractions are like fractions. For the addition of like fractions, we add the numerators and retain the common denominator. This means, 1/7 + 3/7 = (1 + 3)/7 = 4/7

Example 2: Add the following fractions: 2/5 and 2/3

The given fractions are unlike fractions. For adding fractions with different denominators, we have to find the LCM of the denominators and convert 2/5 and 2/3 to fractions with a common denominator. LCM of 3 and 5 is 15. 2/5 + 2/3 = (2/5 × 3/3) + (2/3 × 5/5)

= 6/15 + 10/15

= (6 + 10)/15

= \(1 \dfrac{1}{15}\)

Therefore, the sum is \(1 \dfrac{1}{15}\)

Example 3: How to add a whole number and a fraction: 3 + 1/3?

This question is based on adding fractions with whole numbers. The whole number 3 can be written in the form of a fraction as 3/1. Now,

3 + 1/3 = 3/1 + 1/3

= (3/1 × 3/3) + 1/3

= 9/3 + 1/3

= (9 + 1)/3

= \(3 \frac{1}{3} \)

Therefore, the sum is \(3\frac{1}{3}\)

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Practice Questions on Adding Fractions

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FAQs on Addition of Fractions

The process of addition of fractions is a little different from normal addition of whole numbers. The first step in adding fractions is to check if the denominators of the given fractions are the same. Then we use the following procedure to add them.

• If the fractions have common denominators then we can easily add the numerators and keep the same denominator to get the sum. For example, 2/4 + 1/4 = (2 + 1)/4 = 3/4
• If the denominators are different, we make the denominators equal by converting them to equivalent fractions by finding the LCM of the denominators. Then the addition can be done. For example, 1/2 + 2/3 = (1/2 × 3/3) + (2/3 × 2/2) = 3/6 + 4/6 = (3 + 4)/6 = 7/6 = \(1 \dfrac{1}{6}\)

What is the Rule for Adding Fractions?

The basic rule for the addition of fractions is to make the denominators of the fractions the same. If the fractions have the same denominator we can simply add the numerators keeping the same denominator. However, if the denominators are different, we need to convert them to like fractions with the same denominators. This is done by writing their equivalent fractions by taking the LCM of the denominators. Once they are converted to like fractions, the fractions can be added easily because we just need to work with the numerators while we keep the same denominator.

How to Add Fractions with Whole Numbers?

To add a fraction with a whole number, we first convert the whole number into a fraction. For example, if we need to add 3 and 1/2, the whole number 3 can be easily converted into a fraction like 3/1 and added to the other fraction. Let us see how this works. (3/1) + (1/2) = (3/1) × (2/2) + (1/2) = 6/2 + 1/2 = 7/2 = 3½. Another way to add fractions and whole numbers is to simply combine and express them as mixed fractions. For example, 6 + 1/2 can be combined and written as \(6 \dfrac{1}{2}\)

How to Add Fractions with Different Denominators?

The fractions with different denominators can be added by making the denominators common. This is done by multiplying the numerator and denominator of each of the fractions with a suitable number such that all the fractions become like fractions. To add the fractions 3/5 + 4/3, we need to multiply both the fractions with a number that makes the denominators equal. For this, we need the LCM of the denominators, which is 15 in this case. The numerator and denominator of the first fraction 3/5 have to be multiplied by 3, and the numerator and denominator of the second fraction 4/3 have to be multiplied by 5. Hence, we have (3/5 × 3/3) + (4/3 × 5/5) = (9/15) + (20/15) = (9 + 20)/15 = 29/15 = \(1 \dfrac{14}{15}\)

How to Add 3 Fractions with Different Denominators?

The addition of three fractions is the same as the addition of two fractions with different denominators . First of all, we need the LCM of all three denominators. Accordingly, the denominators of all the three fractions are made common by multiplying the numerator and denominator of each of the fractions with a suitable number so that they are converted to like fractions. Now, once the denominators are common, the numerators are added to get the sum of the fraction. Let us understand this with the help of this addition problem: 2/3 + 4/5 + 1/6. The LCM of 3, 5, and 6 is 30. Now, we will multiply each fraction with the suitable number to make their denominators common: (2/3 × 10/10) + (4/5 × 6/6) + (1/6 × 5/5) = (20/30) + (24/30) + (5/30) = (20 + 24 + 5)/30 = 49/30 = \(1 \dfrac{19}{30}\)

What is the Identity Element For the Addition of Fractions?

The identity element for addition is 0, which means, for any real number 'a', a + 0 = a. Similarly, for the addition of fractions, the identity element is 0. For a fraction of the form a/b, we have a/b + 0 = 0 + a/b = a/b. The use of the identity element for addition does not change the value of the fraction .

What is Subtraction and Addition of Fractions?

In the subtraction and addition of fractions, first, the denominators of the fractions should be made equal. If the denominators are the same, we can simply add or subtract the fractions easily. However, if the fractions have different denominators, then the process starts with the LCM (Least Common Multiple) of the denominators. Then, the fractions are multiplied with a suitable number which makes all the denominators equal. Finally, the numerators are added or subtracted as per the question and the new denominator remains the same.

How to Add Fractions with the Same Denominators?

In order to add fractions with the same denominators, we can simply add the numerators and keep the denominator the same. For example, let us add 3/7 + 2/7. Since the fractions have the same denominators, we just need to add the numerators. So, this will be 3/7 + 2/7 = (3 + 2)/7 = 5/7

How to Add Improper Fractions?

In order to add improper fractions, we use the same rules of adding fractions. For example, let us add 8/3 + 7/3. Since the fractions have the same denominators , we just need to add the numerators. So, this will be 8/3 + 7/3 = (8 + 7)/3 = 15/3 = 5

How to Add Mixed Fractions Step by Step?

Adding mixed numbers follows the same rules of adding fractions. The only extra step is to convert the mixed fractions to improper fractions. Let us understand this with an example. Let us add \(6 \dfrac{1}{2}\) + \(3 \dfrac{3}{4}\) using the following steps.

• Step 1: To add \(6 \dfrac{1}{2}\) + \(3 \dfrac{3}{4}\), let us convert these mixed fractions to improper fractions. This will be 13/2 + 15/4
• Step 2: Now, we will use the basic rules of addition. Here, the denominators are different, so we will convert these to their equivalent fractions so that their denominators become the same.
• Step 3: The LCM of 2 and 4 is 4. Now, we will multiply each fraction with the suitable number to make their denominators common: (13/2 × 2/2) + (15/4 × 1/1) = (26/4) + (15/4) = (26 + 15)/4 = 41/4 = \(1 0\dfrac{1}{4}\)

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5.2: Addition and Subtraction of Fractions with Unlike Denominators

• Last updated
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• Page ID 48858

• Denny Burzynski & Wade Ellis, Jr.
• College of Southern Nevada via OpenStax CNX

Learning Objectives

• be able to add and subtract fractions with unlike denominators

A Basic Rule

There is a basic rule that must be followed when adding or subtracting fractions.

A Basic Rule Fractions can only be added or subtracted conveniently if they have like denomi­nators.

To see why this rule makes sense, let's consider the problem of adding a quarter and a dime.

\(\text{1 quarter + 1 dime = 35 cents}\)

\(\left \{ \begin{array} {r} {\text{1 quarter } = \dfrac{25}{100}} \\ {\text{1 dime } = \dfrac{10}{100}} \end{array} \right \} \text{ same denominations}\)

\(35, \cancel{c} = \dfrac{35}{100}\)

\(\dfrac{25}{100} + \dfrac{10}{100} = \dfrac{25 + 10}{100} = \dfrac{35}{100}\)

In order to combine a quarter and a dime to produce 35¢, we convert them to quantities of the same denomination.

Same denomination \(\rightarrow\) same denominator

Addition and Subtraction of Fractions

Least Common Multiple (LCM) and Least Common Denominator (LCD) In [link] , we examined the least common multiple (LCM) of a collection of numbers. If these numbers are used as denominators of fractions, we call the least common multiple, the least common denominator (LCD).

Method of Adding or Subtracting Fractions with Unlike Denominators To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as a denominator the least common denomina­tor ( LCD) of the original denominators.

Sample Set A

Find the following sums and differences.

\(\dfrac{1}{6} + \dfrac{3}{4}\). The denominators are not the same. Find the LCD of 6 and 4.

\(\left\{ \begin{array} {c} {6 = 2 \cdot 3} \\ {4 = 2^2} \end{array} \right \} \text{ The LCD = } 2^2 \cdot 3 = 4 \cdot 3 = 12\)

Write each of the original fractions as a new, equivalent fraction having the common denomina­tor 12.

\(\dfrac{1}{6} + \dfrac{3}{4} = \dfrac{}{12} + \dfrac{}{12}\)

To find a new numerator, we divide the original denominator into the LCD. Since the original denominator is being multiplied by this quotient, we must multiply the original numerator by this quotient.

\(12 \div 6 = 2\)

\(12 \div 4 = 3\)

\(\begin{array} {rcll} {\dfrac{1}{6} + \dfrac{3}{4}} & = & {\dfrac{1 \cdot 2}{12} + \dfrac{3 \cdot 3}{12}} & {} \\ {} & = & {\dfrac{2}{12} + \dfrac{9}{12}} & {\text{ Now the denominators are the same.}} \\ {} & = & {\dfrac{2 + 9}{12}} & {\text{ Add the numerators and place the sum over the common denominator.}} \\ {} & = & {\dfrac{11}{12}} & {} \end{array}\)

\(\dfrac{1}{2} + \dfrac{2}{3}\). The denominators are not the same. Find the LCD of 2 and 3.

\(\text{LCD} = 2 \cdot 3 = 6\)

Write each of the original fractions as a new, equivalent fraction having the common denominator 6.

\(\dfrac{1}{2} + \dfrac{2}{3} = \dfrac{}{6} + \dfrac{}{6}\)

\(6 \div 2 = 3\) Multiply the numerator 1 by 3. \(6 \div 2 = 3\) Multiply the numerator 2 by 2.

\(\begin{array} {rcl} {\dfrac{1}{2} + \dfrac{2}{3}} & = & {\dfrac{1 \cdot 3}{6} + \dfrac{2 \cdot 3}{6}} \\ {} & = & {\dfrac{3}{6} + \dfrac{4}{6}} \\ {} & = & {\dfrac{3 + 4}{6}} \\ {} & = & {\dfrac{7}{6} \text{ or } 1 \dfrac{1}{6}} \end{array}\)

\(\dfrac{5}{9} - \dfrac{5}{12}\). The denominators are not the same. Find the LCD of 9 and 12.

\(\left \{ \begin{array} {l} {9 = 3 \cdot 3 = 3^2} \\ {12 = 2 \cdot 6 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3} \end{array} \right \} \text{ LCD} = 2^2 \cdot 3^2 = 4 \cdot 9 = 36\)

\(\dfrac{5}{9} - \dfrac{5}{12} = \dfrac{}{36} - \dfrac{}{36}\)

\(36 \div 9 = 4\) Multiply the numerator 5 by 4. \(36 \div 12 = 3\) Multiply the numerator 5 by 3.

\(\begin{array} {rcl} {\dfrac{5}{9} - \dfrac{5}{12}} & = & {\dfrac{5 \cdot 4}{36} - \dfrac{5 \cdot 3}{36}} \\ {} & = & {\dfrac{20}{36} - \dfrac{15}{36}} \\ {} & = & {\dfrac{20 - 15}{6}} \\ {} & = & {\dfrac{5}{36}} \end{array}\)

\(\dfrac{5}{6} - \dfrac{1}{8} + \dfrac{7}{16}\). The denominators are not the same. Find the LCD of 6, 8, and 16

\(\left \{ \begin{array} {l} {6 = 2 \cdot 3} \\ {8 = 2 \cdot 4 = 2 \cdot 2 \cdot 2 = 2^3} \\ {16 = 2 \cdot 8 = 2 \cdot 2 \cdot 4 = 2 \cdot 2 \cdot 2 \cdot 2 = 2^4} \end{array} \right \} \text{ The LCD is} = 2^4 \cdot 3 = 48\)

\(\dfrac{5}{6} - \dfrac{1}{8} + \dfrac{7}{16} = \dfrac{}{48} - \dfrac{}{48} + \dfrac{}{48}\)

\(48 \div 6 = 8\) Multiply the numerator 5 by 8 \(48 \div 8 = 6\) Multiply the numerator 1 by 6 \(48 \div 16 = 3\) Multiply the numerator 7 by 3

\(\begin{array} {rcl} {\dfrac{5}{6} - \dfrac{1}{8} + \dfrac{7}{16}} & = & {\dfrac{5 \cdot 8}{48} - \dfrac{1 \cdot 6}{48} + \dfrac{7 \cdot 3}{48}} \\ {} & = & {\dfrac{40}{48} - \dfrac{6}{48} + \dfrac{21}{48}} \\ {} & = & {\dfrac{40 - 6 + 21}{48}} \\ {} & = & {\dfrac{55}{48} \text{ or } 1 \dfrac{7}{48}} \end{array}\)

Practice Set A

\(\dfrac{3}{4} + \dfrac{1}{12}\)

\(\dfrac{5}{6}\)

\(\dfrac{1}{2} - \dfrac{3}{7}\)

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

\(\dfrac{7}{10} - \dfrac{5}{8}\)

\(\dfrac{3}{40}\)

\(\dfrac{15}{16} + \dfrac{1}{2} - \dfrac{3}{4}\)

\(\dfrac{11}{16}\)

\(\dfrac{1}{32} - \dfrac{1}{48}\)

\(\dfrac{1}{96}\)

Exercise \(\PageIndex{1}\)

A most basic rule of arithmetic states that two fractions may be added or subtracted conveniently only if they have .

The same denominator

For the following problems, find the sums and differences.

Exercise \(\PageIndex{2}\)

\(\dfrac{1}{2} + \dfrac{1}{6}\)

Exercise \(\PageIndex{3}\)

\(\dfrac{1}{8} + \dfrac{1}{2}\)

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

Exercise \(\PageIndex{4}\)

\(\dfrac{3}{4} + \dfrac{1}{3}\)

Exercise \(\PageIndex{5}\)

\(\dfrac{5}{8} + \dfrac{2}{3}\)

\(\dfrac{31}{24}\)

Exercise \(\PageIndex{6}\)

\(\dfrac{1}{12} + \dfrac{1}{3}\)

Exercise \(\PageIndex{7}\)

\(\dfrac{6}{7} - \dfrac{1}{4}\)

\(\dfrac{17}{28}\)

Exercise \(\PageIndex{8}\)

\(\dfrac{9}{10} - \dfrac{2}{5}\)

Exercise \(\PageIndex{9}\)

\(\dfrac{7}{9} - \dfrac{1}{4}\)

\(\dfrac{19}{36}\)

Exercise \(\PageIndex{10}\)

\(\dfrac{8}{15} - \dfrac{3}{10}\)

Exercise \(\PageIndex{11}\)

\(\dfrac{8}{13} - \dfrac{5}{39}\)

\(\dfrac{19}{39}\)

Exercise \(\PageIndex{12}\)

\(\dfrac{11}{12} - \dfrac{2}{5}\)

Exercise \(\PageIndex{13}\)

\(\dfrac{1}{15} + \dfrac{5}{12}\)

\(\dfrac{29}{60}\)

Exercise \(\PageIndex{14}\)

\(\dfrac{13}{88} - \dfrac{1}{4}\)

Exercise \(\PageIndex{15}\)

\(\dfrac{1}{9} - \dfrac{1}{81}\)

\(\dfrac{8}{81}\)

Exercise \(\PageIndex{16}\)

\(\dfrac{19}{40} + \dfrac{5}{12}\)

Exercise \(\PageIndex{17}\)

\(\dfrac{25}{26} - \dfrac{7}{10}\)

\(\dfrac{17}{65}\)

Exercise \(\PageIndex{18}\)

\(\dfrac{9}{28} - \dfrac{4}{45}\)

Exercise \(\PageIndex{19}\)

\(\dfrac{22}{45} - \dfrac{16}{35}\)

\(\dfrac{2}{63}\)

Exercise \(\PageIndex{20}\)

\(\dfrac{56}{63} + \dfrac{22}{33}\)

Exercise \(\PageIndex{21}\)

\(\dfrac{1}{16} + \dfrac{3}{4} - \dfrac{3}{8}\)

\(\dfrac{7}{16}\)

Exercise \(\PageIndex{22}\)

\(\dfrac{5}{12} - \dfrac{1}{120} + \dfrac{19}{20}\)

Exercise \(\PageIndex{23}\)

\(\dfrac{8}{3} - \dfrac{1}{4} + \dfrac{7}{36}\)

\(\dfrac{47}{18}\)

Exercise \(\PageIndex{24}\)

\(\dfrac{11}{9} - \dfrac{1}{7} + \dfrac{16}{63}\)

Exercise \(\PageIndex{25}\)

\(\dfrac{12}{5} - \dfrac{2}{3} + \dfrac{17}{10}\)

\(\dfrac{103}{30}\)

Exercise \(\PageIndex{26}\)

\(\dfrac{4}{9} + \dfrac{13}{21} - \dfrac{9}{14}\)

Exercise \(\PageIndex{27}\)

\(\dfrac{3}{4} - \dfrac{3}{22} + \dfrac{5}{24}\)

\(\dfrac{217}{264}\)

Exercise \(\PageIndex{28}\)

\(\dfrac{25}{48} - \dfrac{7}{88} + \dfrac{5}{24}\)

Exercise \(\PageIndex{29}\)

\(\dfrac{27}{40} + \dfrac{47}{48} - \dfrac{119}{126}\)

\(\dfrac{511}{720}\)

Exercise \(\PageIndex{30}\)

\(\dfrac{41}{44} - \dfrac{5}{99} - \dfrac{11}{175}\)

Exercise \(\PageIndex{31}\)

\(\dfrac{5}{12} + \dfrac{1}{18} + \dfrac{1}{24}\)

\(\dfrac{37}{72}\)

Exercise \(\PageIndex{32}\)

\(\dfrac{5}{9} + \dfrac{1}{6} + \dfrac{7}{15}\)

Exercise \(\PageIndex{33}\)

\(\dfrac{21}{25} - \dfrac{1}{6} + \dfrac{7}{15}\)

\(\dfrac{221}{150}\)

Exercise \(\PageIndex{34}\)

\(\dfrac{5}{18} - \dfrac{1}{36} + \dfrac{7}{9}\)

Exercise \(\PageIndex{35}\)

\(\dfrac{11}{14} - \dfrac{1}{36} - \dfrac{1}{32}\)

\(\dfrac{1.465}{2,016}\)

Exercise \(\PageIndex{36}\)

\(\dfrac{21}{33} + \dfrac{12}{22} + \dfrac{15}{55}\)

Exercise \(\PageIndex{37}\)

\(\dfrac{5}{51} + \dfrac{2}{34} + \dfrac{11}{68}\)

\(\dfrac{65}{204}\)

Exercise \(\PageIndex{38}\)

\(\dfrac{8}{7} - \dfrac{16}{14} + \dfrac{19}{21}\)

Exercise \(\PageIndex{39}\)

\(\dfrac{7}{15} + \dfrac{3}{10} - \dfrac{34}{60}\)

\(\dfrac{1}{5}\)

Exercise \(\PageIndex{40}\)

\(\dfrac{14}{15} - \dfrac{3}{10} - \dfrac{6}{25} + \dfrac{7}{20}\)

Exercise \(\PageIndex{41}\)

\(\dfrac{11}{6} - \dfrac{5}{12} + \dfrac{17}{30} + \dfrac{25}{18}\)

\(\dfrac{607}{180}\)

Exercise \(\PageIndex{42}\)

\(\dfrac{1}{9} + \dfrac{22}{21} - \dfrac{5}{18} - \dfrac{1}{45}\)

Exercise \(\PageIndex{43}\)

\(\dfrac{7}{26} + \dfrac{28}{65} - \dfrac{51}{104} + 0\)

\(\dfrac{109}{520}\)

Exercise \(\PageIndex{44}\)

A morning trip from San Francisco to Los Angeles took \(\dfrac{13}{12}\) hours. The return trip took \(\dfrac{57}{60}\) hours. How much longer did the morning trip take?

Exercise \(\PageIndex{45}\)

At the beginning of the week, Starlight Publishing Company's stock was selling for \(\dfrac{115}{8}\) dollars per share. At the end of the week, analysts had noted that the stock had gone up \(\dfrac{11}{4}\) dollars per share. What was the price of the stock, per share, at the end of the week?

$ \(\dfrac{137}{8}\) or $ \(17 \dfrac{1}{8}\)

Exercise \(\PageIndex{46}\)

A recipe for fruit punch calls for \(\dfrac{23}{3}\) cups of pineapple juice, \(\dfrac{1}{4}\) cup of lemon juice, \(\dfrac{15}{2}\) cups of orange juice, 2 cups of sugar, 6 cups of water, and 8 cups of carbonated non-cola soft drink. How many cups of ingredients will be in the final mixture?

Exercise \(\PageIndex{47}\)

The side of a particular type of box measures \(8 \dfrac{3}{4}\) inches in length. Is it possible to place three such boxes next to each other on a shelf that is \(26 \dfrac{1}{5}\) inches in length? Why or why not?

No; 3 boxes add up to \(26 \dfrac{1''}{4}\), which is larger than \(25 \dfrac{1''}{5}\).

Exercise \(\PageIndex{48}\)

Four resistors, \(\dfrac{3}{8}\) ohm, \(\dfrac{1}{4}\) ohm, \(\dfrac{3}{5}\) ohm, and \(\dfrac{7}{8}\) ohm, are connected in series in an electrical circuit. What is the total resistance in the circuit due to these resistors? ("In series" implies addition.)

Exercise \(\PageIndex{49}\)

A copper pipe has an inside diameter of \(2 \dfrac{3}{16}\) inches and an outside diameter of \(2 \dfrac{5}{34}\) inches. How thick is the pipe?

No pipe at all; inside diameter is greater than outside diameter

Exercise \(\PageIndex{50}\)

The probability of an event was originally thought to be \(\dfrac{15}{32}\). Additional information decreased the probability by \(\dfrac{3}{14}\). What is the updated probability?

Exercises for Review

Exercise \(\PageIndex{51}\)

Find the difference between 867 and 418.

Exercise \(\PageIndex{52}\)

Is 81,147 divisible by 3?

Exercise \(\PageIndex{53}\)

Find the LCM of 11, 15, and 20.

Exercise \(\PageIndex{54}\)

Find \(\dfrac{3}{4}\) of \(4 \dfrac{2}{9}\).

Exercise \(\PageIndex{55}\)

Find the value of \(\dfrac{8}{15} - \dfrac{3}{15} + \dfrac{2}{15}\).

\(\dfrac{7}{15}\)

Adding Fractions

A fraction like 3 4 says we have 3 out of the 4 parts the whole is divided into.

To add fractions there are Three Simple Steps:

• Step 1: Make sure the bottom numbers (the denominators ) are the same
• Step 2: Add the top numbers (the numerators ), put that answer over the denominator
• Step 3: Simplify the fraction (if possible)

Step 1 . The bottom numbers (the denominators) are already the same. Go straight to step 2.

Step 2 . Add the top numbers and put the answer over the same denominator:

1 4 + 1 4 = 1 + 1 4 = 2 4

Step 3 . Simplify the fraction:

In picture form it looks like this:

... and do you see how 2 4 is simpler as 1 2 ? (see Equivalent Fractions .)

Step 1 : The bottom numbers are different. See how the slices are different sizes?

We need to make them the same before we can continue, because we can't add them like that.

The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by 2 , like this:

Important: you multiply both top and bottom by the same amount, to keep the value of the fraction the same

Now the fractions have the same bottom number ("6"), and our question looks like this:

The bottom numbers are now the same, so we can go to step 2.

Step 2 : Add the top numbers and put them over the same denominator:

2 6 + 1 6 = 2 + 1 6 = 3 6

Step 3 : Simplify the fraction:

In picture form the whole answer looks like this:

With Pen and Paper

And here is how to do it with a pen and paper (press the play button):

A Rhyme To Help You Remember

♫ "If adding or subtracting is your aim, The bottom numbers must be the same! ♫ "Change the bottom using multiply or divide, But the same to the top must be applied, ♫ "And don't forget to simplify, Before its time to say good bye"

Again, the bottom numbers are different (the slices are different sizes)!

But let us try dividing them into smaller sizes that will each be the same :

The first fraction: by multiplying the top and bottom by 5 we ended up with 5 15 :

The second fraction: by multiplying the top and bottom by 3 we ended up with 3 15 :

The bottom numbers are now the same, so we can go ahead and add the top numbers:

The result is already as simple as it can be, so that is the answer: 

1 3 + 1 5 = 8 15

Making the Denominators the Same

In the previous example how did we know to cut them into 1 / 15 ths to make the denominators the same? We simply multiplied the two denominators together (3 × 5 = 15).

Read about the two main ways to make the denominators the same here:

• Common Denominator Method , or the
• Least Common Denominator Method

They both work, use which one you prefer!

Example: Cupcakes

You want to make and sell cupcakes:

• A friend can supply the ingredients, if you give them 1 / 3 of sales
• And a market stall costs 1 / 4 of sales

How much is that altogether?

We need to add 1 / 3 and 1 / 4

First make the bottom numbers (the denominators) the same.

Multiply top and bottom of 1 / 3 by 4 :

And multiply top and bottom of 1 / 4 by 3 :

Now do the calculations:

Answer: 7 12 of sales go in ingredients and market costs.

Adding Mixed Fractions

We have a special (more advanced) page on Adding Mixed Fractions .

Adding Fractions

Learn about adding fractions., adding fractions lesson, how to add fractions.

To add fractions, we follow three simple steps. They are as follows:

• Make the denominators the same if they aren't already.
• Add the numerators, keeping the denominator the same.
• Simplify the resulting fraction.

The same three steps apply for adding mixed fractions (such as 4 1 / 2 + 1 2 / 3 ) except that we will simply add the whole number and fraction components separately.

In this lesson we will go through how to add fractions and show examples of adding fractions with like and unlike denominators.

Adding Fractions with Like Denominators

Let's go through how to add fractions with like denominators first, since it is most simple type of fraction addition. Here's an example of adding fractions with like denominators, using the three steps from earlier.

Find the sum of 3 / 5 + 1 / 5 .

• The denominators are already the same, so we can skip step 1.
• Let's add the numerators. 3 + 1 = 4, so the sum of our numerators is 4. The denominator is still 5, so our result is 4 / 5 .
• 4 / 5 is already in its simplest form, so there is no simplifying needed here.

The solution is 3 / 5 + 1 / 5 = 4 / 5 .

Adding Fractions with Unlike Denominators

Now let's go through another example but this time with unlike denominators. We will use the same exact three steps.

Find the sum of 1 / 4 + 2 / 3 .

• Let's find the lowest common denominator and convert these fractions to like denominators to make them addable. Multiplying the top and bottom of each fraction by the other fraction's denominator gives us 1 / 4  ·  3 / 3 = 3 / 12 and 2 / 3  ·  4 / 4 = 8 / 12 .
• Now let's add the numerators. 3 + 8 = 11, so the sum of our numerators is 11. The denominator is still 12, so our result is 11 / 12 .
• 11 / 12 is already in its simplest form, so there is no simplifying needed here.

The solution is 1 / 4 + 2 / 3 = 11 / 12 .

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Adding fractions

Here you’ll learn about how to add fractions with like denominators and with unlike denominators. You will also learn how to add mixed numbers.

Students will first learn about adding fractions as part of number and operations in fractions in elementary school.

What is adding fractions?

Adding fractions is when you combine two or more fractions together to find the total.

To do this, fractions need a common denominator (bottom number). Then you can add the fractions by adding the numerators (top numbers).

For example,

Since the denominators are the same, you need to add the numerators 3+7=10.

There are 10 parts. But what size are the parts? They are still eighths, so the denominator stays the same and the answer is 10 eighths, or 1 and 2 eighths

You can also write this answer as the equivalent fraction 1\cfrac{1}{4}.

If the denominators are not the same, you create equivalent fractions before adding.

Since the denominators are NOT the same, the parts are NOT the same size. Use equivalent fractions to create a common denominator of 9. Multiply the numerator and denominator of \cfrac{2}{3} by 3.

You add to see how many parts there are in total: 6+5=11.

There are 11 parts. But what size are the parts? They are still ninths, so the denominator stays the same and the answer is 11 ninths, or 1 and 2 ninths

Common Core State Standards

How does this relate to 4th grade math and 5th grade math?

• Grade 4 – Number and Operations – Fractions (4.NF.B.3c) Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
• Grade 5 – Number and Operations – Fractions (5.NF.A.1) Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, \cfrac{2}{3}+\cfrac{5}{4}=\cfrac{8}{12}+\cfrac{15}{12}=\cfrac{23}{12}. (In general, \cfrac{a}{b}+\cfrac{c}{d}=\cfrac{(ad+bc)}{bd}. )

How to add fractions

In order to add fractions with like denominators:

Add the numerators (top numbers).

Write your answer as a fraction.

In order to add mixed numbers with like denominators:

Add the whole numbers.

Add the fractions.

Write your answer as a mixed number.

In order to add fractions with unlike denominators:

Create common denominators (bottom numbers).

In order to add mixed numbers with unlike denominators:

[FREE] Fraction Operations Check for Understanding Quiz (Grade 4 to 6)

Use this quiz to check your grade 4 to 6 students’ understanding of fraction operations. 10+ questions with answers covering a range of 4th to 6th grade fraction operations topics to identify areas of strength and support!

Adding fractions examples

Example 1: adding fractions with like denominators.

Solve \cfrac{3}{5}+\cfrac{4}{5}.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 3+4=7.

2 Write your answer as a fraction.

There are 7 parts. But what size are the parts? They are still fifths, so the denominator stays the same.

\cfrac{3}{5}+\cfrac{4}{5}=\cfrac{7}{5} \ or \ 1 \cfrac{2}{5}

Example 2: adding mixed numbers with like denominators

Solve 4\cfrac{5}{6}+1\cfrac{2}{6}.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5+2=7.

There are 7 parts. But what size are the parts? They are still sixths, so the denominator stays the same.

\cfrac{5}{6}+\cfrac{2}{6}=\cfrac{7}{6} \ or \ 1 \cfrac{1}{6}

Add the whole numbers and fraction together.

Example 3: adding mixed numbers with like denominators

Solve 3 \cfrac{7}{10}+3 \cfrac{5}{10}.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5+7=12.

There are 12 parts. But what size are the parts? They are still tenths, so the denominator stays the same.

\cfrac{7}{10}+\cfrac{5}{10}=\cfrac{12}{10} \ or \ 1 \cfrac{2}{10}

You can also write this answer as the equivalent fraction 7 \cfrac{1}{5}.

Example 4: adding fractions with unlike denominators

Solve \cfrac{4}{10}+\cfrac{1}{3}.

Since \cfrac{4}{10} and \cfrac{1}{3} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator. Multiply each fraction by the opposite denominator.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 12+10=22.

There are 22 parts. But what size are the parts? They are still thirtieths, so the denominator stays the same.

You can also write this answer as the equivalent fraction \cfrac{11}{15}.

Example 5: adding mixed numbers with unlike denominators

Solve 2 \cfrac{3}{8}+1 \cfrac{7}{12}.

Since \cfrac{3}{8} and \cfrac{7}{12} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator.

A common denominator of 24 can be used. Multiply the numerator and denominator of each fraction by the factor that will create a denominator of 24.

\cfrac{3}{8}=\cfrac{3 \times 3}{8 \times 3}=\cfrac{9}{24} \ and \ \cfrac{7}{12}=\cfrac{7 \times 2}{12 \times 2}=\cfrac{14}{24}

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 9+14=23.

There are 23 parts. But what size are the parts? They are still twenty-fourths, so the denominator stays the same.

\cfrac{9}{24}+\cfrac{14}{24}=\cfrac{23}{24}

Add the whole number and fraction together.

Example 6: adding mixed numbers with unlike denominators

Solve 4 \cfrac{2}{3}+5 \cfrac{5}{6}.

Since \cfrac{2}{3} and \cfrac{5}{6} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator.

A common denominator of 6 can be used. Multiply the numerator and denominator of \cfrac{2}{3} by the factor that will create a denominator of 6.

\cfrac{2}{3}=\cfrac{2 \times 2}{3 \times 2}=\cfrac{4}{6} \ and \ \cfrac{5}{6}

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 4+5=9.

There are 9 parts. But what size are the parts? They are still sixths, so the denominator stays the same.

\cfrac{4}{6}+\cfrac{5}{6}=\cfrac{9}{6} or 1 \cfrac{3}{6}

You can also write this answer as the equivalent mixed number 10 \cfrac{1}{2}.

Teaching tips for adding fractions

• Fraction work in 3rd grade centers around understanding through models, particularly area models and number lines. To build on this in 4th grade and 5th grade, always have physical or digital models available for students to use when necessary.
• To introduce this topic, give students a fraction addition problem and give them time to solve in a way that makes sense to them. Then go over the different solving strategies as a whole group.
• In the beginning, you may want to stick to smaller fractions or problems that do not involve regrouping. However, at some point it is important to transition to showing students all kinds of fraction addition problems mixed together. This gives them an opportunity to identify which solving strategy to use.
• Fraction worksheets may have their place when students are still developing understanding around addition, but once students have a successful strategy and can flexibly operate, incorporate math games or real world projects that involve addition of fractions.

Our favorite mistakes

• Adding without like denominators When fractions are added together, they must have like denominators. Before adding fractions with unlike denominators, use equivalent fractions to create a common denominator.

• When adding fractions, only the numerators are added The denominator tells us how big the parts are. You add the number of parts (numerators) and keep the denominator (the size of the parts) the same.

Related fractions operations lessons

• Fractions operations
• Multiplying and dividing fractions
• Subtracting fractions
• Adding and subtracting fractions
• Multiplying fractions
• Dividing fractions
• Reciprocal math
• Fraction word problems
• Multiplicative inverse
• Interpret a fraction as division

Adding fractions practice questions

1. Solve \cfrac{8}{12}+\cfrac{9}{12}.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 8+9=17.

There are 17 parts. But what size are the parts? They are still twelfths, so the denominator stays the same.

2. Solve 2 \cfrac{4}{8}+2 \cfrac{7}{8}.

First, add the whole numbers

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 4+7=11.

There are 11 parts. But what size are the parts? They are still eighths, so the denominator stays the same.

\cfrac{4}{8}+\cfrac{7}{8}=\cfrac{11}{8} \ or \ 1 \cfrac{3}{8}

3. Solve 1 \cfrac{2}{5}+7 \cfrac{4}{5}.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 2+4=6.

There are 6 parts. But what size are the parts? They are still fifths, so the denominator stays the same.

\cfrac{2}{5}+\cfrac{4}{5}=\cfrac{6}{5} \ or \ 1 \cfrac{1}{5}

4. Solve \cfrac{3}{4}+\cfrac{10}{12}.

Since \cfrac{3}{4} and \cfrac{10}{12} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator.

A common denominator of 12 can be used. Multiply the numerator and denominator of \cfrac{3}{4} by the factor that will create a denominator of 12.

\cfrac{3}{4}=\cfrac{3 \times 3}{4 \times 3}=\cfrac{9}{12} \ and \ \cfrac{10}{12}

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 9+10=19.

There are 19 parts. But what size are the parts? They are still twelfths, so the denominator stays the same.

\cfrac{9}{12}+\cfrac{10}{12}=\cfrac{19}{12} \ or \ 1 \cfrac{7}{12}

5. Solve 6 \cfrac{1}{2}+1 \cfrac{3}{10}.

First, add the whole numbers.

Since \cfrac{1}{2} and \cfrac{3}{10} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator.

A common denominator of 10 can be used. Multiply the numerator and denominator of \cfrac{1}{2} by the factor that will create a denominator of 10.

\cfrac{1}{2}=\cfrac{1 \times 5}{2 \times 5}=\cfrac{5}{10} \ and \ \cfrac{3}{10}

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 5+3=8.

There are 8 parts. But what size are the parts? They are still tenths, so the denominator stays the same.

6. Solve 3 \cfrac{5}{6}+4 \cfrac{3}{5}.

Since \cfrac{5}{6} and \cfrac{3}{5} do not have like denominators, the parts are NOT the same size. Use equivalent fractions to create a common denominator.

Multiplying the numerator and denominator by the opposite denominator will create common denominators.

\cfrac{5}{6}=\cfrac{5 \times 5}{6 \times 5}=\cfrac{25}{30} \ and \ \cfrac{3}{5}=\cfrac{3 \times 6}{5 \times 6}=\cfrac{18}{30}

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 25+18=43.

You have 43 parts. But what size are the parts? They are still thirtieths, so the denominator stays the same.

\cfrac{25}{30}+\cfrac{18}{30}=\cfrac{43}{30} \ or \ 1 \cfrac{13}{30}

Adding fractions FAQs

No, although a new numerator and new denominator are created the value of the fraction remains the same. Since the new fraction has a larger denominator, the parts will be smaller. The numerator also needs to be larger so that the total area of the parts is the same as the original fraction.

No, students do not have to find the least common denominator in order to correctly answer a fraction addition question. However, as students progress in their understanding of fractions, it is a good habit to develop. It is also important to be mindful of standard expectations, as they may vary from state to state.

Yes, you follow many of the same steps to subtract fractions, which also requires common denominators. The only difference is that you subtract the numerators instead of adding.

The next lessons are

• Algebraic expression
• Converting fractions, decimals, and percentages

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Word Problems - Fraction Addition (different denominators)

Description : This packet helps students practice solving word problems that require addition with fractions with different denominators. Each page contains 6 problems. Each page also has a speed and accuracy guide to help students see how fast and how accurately they should be doing these problems. After doing all 12 problems, students should be more comfortable doing these problems and have a clear understanding of how to solve them. 

Claudia ordered pizza. So far, she has eaten $\dfrac{3}{9}$ of the large pizza. Her dad has eaten $\dfrac{1}{6}$ of the large pizza. How much of the pizza has been eaten so far? 

Practice problems require knowledge of how to add, subtract, and multiply whole numbers

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Free Equivalent Fractions Worksheets

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Equivalent fractions are fractions with different numerators and denominators that represent the same value. They can be especially tricky for students to learn because although they look different, their values are the same. So if you’re introducing the concept to your 3rd or 4th graders (or have kids who are struggling with the concept), these equivalent fractions worksheets are for you. Don’t miss out on our newest bundle! Just fill out the form on this page to get all the worksheets described below.

Equivalent Fractions Using Models

Our first worksheet has students practicing equivalent fractions using models. We used different shapes and also have students dividing the shapes themselves before shading.

Equivalent Fractions Using Number Lines

Practicing equivalent fractions on a number line provides a visual representation of how fractions occupy different points along the line while representing the same portion of the whole.

Equivalent Fractions Using Multiplication

Equivalent fractions can be found by multiplying or dividing the numerator and denominator by the same number. This worksheet provides practice for both. Get more multiplication practice here .

Mystery Picture

The equivalent fractions mystery picture worksheet is a fun and colorful way for students to test their knowledge of equivalent fractions. It’s also a great way to motivate creative kids!

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Ready to assess your students? Grab the equivalent fractions review page with some of each of the types of practice featured in the bundle plus a word problem.

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Add & subtract fractions word problems

Like & unlike denominators.

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