add and subtract fractions with the same denominator problem solving

Fractions Addition & Subtraction with Same Denominator

Add and subtract fractions having the same denominator.

When adding or subtracting fractions, consider the problem simple if the denominators are equal or the same. The rules are outlined below.

Steps on How to Add and Subtract Fractions with the Same Denominator

• To ADD  fractions with like or the same denominator, simply add the numerators then copy the common denominator. Always reduce your final answer to its lowest term.

• To SUBTRACT  fractions with like or the same denominator, just subtract the numerators then copy the common denominator. Always reduce your final answer to its lowest term.

Examples of Adding and Subtracting Fractions with Like Denominator

Example 1 : Add the fractions.

The denominators of the two fractions are both 7. Since they have the same denominator, we can easily add these fractions by adding their numerators and copying the common denominator which is 7.

We can also show the addition process using circles.

• The first fraction [latex]\Large{3 \over 7}[/latex] can be represented by a circle divided equally into seven parts with three pieces shaded in red.

Observe : The numerator tells us how many areas are shaded while the denominator tells us how many equal parts the circle is divided.

• In the same manner, the second fraction [latex]\Large{2 \over 7}[/latex]  looks like this:

• Since the two circles are both divided into seven (7) equal parts, we should be able to overlap them. The new circle after addition has five (5) shaded regions which are the accumulation of both the red and blue pieces.

Example 2 : Add the fractions.

Let’s combine these fractions using the addition rule. Again, add the numerators then copy the common denominator.

After you add fractions, always find the opportunity to simplify the result by reducing it to the lowest term. We can do so by dividing both the numerator and denominator by their greatest common divisor.

• Common divisor is a nonzero whole number that can evenly divide two or more numbers.
• Greatest Common Divisor (GCD) is the largest number among the common divisors of two or more numbers.

The numerator and denominator obviously have a common divisor of 2.  However, is there a number larger than 2 that can also evenly divide both of them?

Yes, there is! The number 4 is the greatest common divisor of 12 and 16. Therefore, we will use this number to reduce the fraction to its lowest term.

Divide the top and bottom by the GCD = 4  to get the final answer.

Example 3: Add the fractions.

Since the denominators of the two fractions are equal, add the numerators and copy the common denominator.

The top and bottom numbers of the fraction are divisible by 2 and 6. However, we always want the largest common divisor to reduce the fraction to its lowest term. Thus, the GCD = 6 .

• Divide the top and bottom numbers by 6.

Example 4: Add the fractions.

All three fractions have the same denominator. We will add as usual.

• Get the sum of the three numerators then copy the common denominator.

The greatest common divisor between the numerator and denominator is 5.

• Divide top and bottom by 5.

Example 5: Subtract the fractions.

This time around, we are going to subtract the numerators instead of adding them.

Looking at the result after subtraction, the only common divisor between the numerator and denominator is 1 . Thus, the final answer remains as [latex]\Large{{3 \over 5}}[/latex]. Think about it, dividing the top and bottom by 1 won’t change the value of the fraction.

Suppose you have a green cake. And you cut it into 5 equal portions. This can be represented as a fraction which is [latex]\Large{{5 \over 5}}[/latex].

If you ate two slices of the cake ( [latex]\Large{- {2 \over 5}}[/latex] ), you should have three leftover pieces ( [latex]\Large{{3 \over 5}}[/latex] ).

The plate should look something like this.

Example 6: Subtract the fractions.

The two fractions have the same denominator which means we should be able to easily subtract their numerators.

The answer can still be further simplified using a common divisor of 3. So, divide the numerator and denominator by 3 to reduce the fraction to its lowest terms.

Example 7: Subtract the fractions.

Since the denominators of the two fractions are equal, subtract their numerators then copy the common denominator.

The numerator and denominator are divisible by 3 and 9. However, we always want the largest common divisor to reduce the fraction to its lowest term. Thus, the GCD = 9 .

• Divide the top and bottom numbers by 9.

Example 8: Subtract the fractions.

Subtract the numerators, copy the common denominator, and reduce the resulting fraction to its lowest term using the GCD = 11 .

You might also like these tutorials:

• Add and Subtract Fractions with Different Denominators
• Multiplying Fractions
• Dividing Fractions
• Simplifying Fractions
• Equivalent Fractions
• Reciprocal of a Fraction

Addition and Subtraction of Fraction: Methods, Examples, Facts, FAQs

What is addition and subtraction of fractions, methods of addition and subtraction of fractions, addition and subtraction of mixed numbers, solved examples on addition and subtraction of fractions, practice problems on addition and subtraction of fractions, frequently asked questions on addition and subtraction of fractions.

Addition and subtraction of fractions are the fundamental operations on fractions that can be studied easily using two cases:

• Addition and subtraction of like fractions (fractions with same denominators)
• Addition and subtraction of unlike fractions (fractions with different denominators)

A fraction represents parts of a whole. For example, the fraction 37 represents 3 parts out of 7 equal parts of a whole. Here, 3 is the numerator and it represents the number of parts taken. 7 is the denominator and it represents the total number of parts of the whole.

Adding and subtracting fractions is simple and straightforward when it comes to like fractions. In the case of unlike fractions, we first need to make the denominators the same. Let’s take a closer look at both these cases.

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Before adding and subtracting fractions, we first need to make sure that the fractions have the same denominators. 

When the denominators are the same, we simply add the numerators and keep the denominator as it is. To add or subtract unlike fractions, we first need to learn how to make the denominators alike. Let’s learn how to add fractions and how to subtract fractions in both cases.

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Addition and Subtraction of Like Fractions

The rules for adding fractions with the same denominator are really simple and straightforward. 

Let’s learn with the help of examples and visual bar models.

Addition of Like Fractions

Here are the steps to add fractions with the same denominator:

Step 1: Add the numerators of the given fractions. 

Step 2: Keep the denominator the same. 

Step 3: Simplify.          

$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$  …$c \neq 0$

Example 1: Find $\frac{1}{4} + \frac{2}{4}$ .

$\frac{1}{4} + \frac{2}{4} = \frac{1 + 2}{4} = \frac{3}{4}$

We can visualize this addition using a bar model:

Example 2: $\frac{1}{8} + \frac{3}{8} = \frac{1 + 3}{8} = \frac{4}{8} = \frac{1}{2}$

Subtraction of Like Fractions

Here are the steps to subtract fractions with the same denominator:

Step 1: Subtract the numerators of the given fractions. 

Step 3: Simplify. 

$\frac{a}{c}\;-\;\frac{b}{c} = \frac{a \;-\; b}{c}$ …$c \neq 0$

Example 1: Find $\frac{4}{6} \;-\; \frac{1}{6}$.

$\frac{4}{6}\;-\;\frac{1}{6} = \frac{4-1}{6} = \frac{3}{6} = \frac{1}{2}$

Addition and Subtraction of Unlike Fractions

Addition and subtraction of fractions with unlike denominators can be a little bit tricky since the denominators are not the same. So, we need to first convert the unlike fractions into like fractions. Let’s look at a few ways to do this!

Addition of Unlike Fractions

We can make the denominators the same by finding the LCM of the two denominators. Once we calculate the LCM, we multiply both the numerator and the denominator with an appropriate number so that we get the LCM value in the denominator. 

Example: $\frac{3}{5} + \frac{3}{2}$

Step 1: Find the LCM (Least Common Multiple) of the two denominators.

The LCM of 5 and 2 is 10.

Step 2: Convert both the fractions into like fractions by making the denominators same.  

$\frac{3 \times 2}{5 \times 2} = \frac{6}{10}$  

$\frac{3 \times 5}{2 \times 5} = \frac{15}{10}$

Step 3: Add the numerators. The denominator stays the same.

$\frac{6}{10} + \frac{15}{10} = \frac{21}{10}$

Step 4: Convert the resultant fraction to its simplest form if the GCF of the numerator and denominator is not 1. 

In this case, GCF (21,10) $= 1$

The fraction $\frac{21}{10}$ is already in its simplest form. 

Thus, $\frac{3}{5} + \frac{3}{2} = \frac{21}{10}$

Subtraction of Unlike Fractions

Let’s learn how to subtract fractions when denominators are not the same. To subtract unlike fractions, we use the LCM method. The process is similar to what we discussed in the previous example.

Example: $\frac{5}{6} \;-\; \frac{2}{9}$

Step 1: Find the LCM of the two denominators.

LCM of 6 and $9 = 18$

Step 2: Convert both the fractions into like fractions by making the denominators same.

$\frac{5 \times 3}{6 \times 3} = \frac{15}{18}$   

$\frac{2 \times 2}{9 \times 2} = \frac{4}{18}$

Step 3: Subtract the numerators. The denominator stays the same.

$\frac{15}{18} \;-\; \frac{4}{18} = \frac{11}{18}$

In this case, the GCF (11,18) $= 1$

So, it is already in its simplest form. 

Thus, $\frac{5}{6}\;-\; 29 = \frac{11}{18}$

A mixed number is a type of fraction that has two parts: a whole number and a proper fraction. It is also known as a mixed fraction. Any mixed number can be written in the form of an improper fraction and vice-versa. 

Adding and subtracting mixed fractions is done by converting mixed numbers into improper fractions .

Addition and Subtraction of Mixed Fractions with Same Denominators

The steps of adding and subtracting mixed numbers with the same denominators are the same. The only difference is the operation.

Step 1: Convert the given mixed fractions to improper fractions.

Step 2: Add/Subtract the like fractions obtained in step 1.

Step 3: Reduce the fraction to its simplest form.

Step 4: Convert the resulting fraction into a mixed number.

Example 1: $2\frac{1}{5} + 1\frac{3}{5}$

$2\frac{1}{5} = \frac{(5 \times 2) + 1}{5} = \frac{11}{5}$

$1\frac{3}{5} = \frac{(5 \times 1) + 3}{5} = \frac{8}{5}$

Thus, $2\frac{1}{5} + 1\frac{3}{5} = \frac{11}{5} + \frac{8}{5} = \frac{19}{5}$

Converting $\frac{19}{5}$ into a mixed number, we get

$\frac{19}{5} = 3\frac{4}{5}$

Example 2: $2\frac{1}{5} + 1\frac{3}{5} = \frac{11}{5} \;-\; \frac{8}{5} = \frac{3}{5}$

Addition and Subtraction of Mixed Fractions with Unlike Denominators

Step 2: Convert both the fractions into like fractions by finding the least common denominator.

Step 3: Add the fractions. (or subtract the fractions.)

Step 4: Reduce the fraction if possible or convert back to a mixed number 

Let us understand the addition of mixed numbers with unlike denominators with the help of an example.

Example 1: Find the value of $1\frac{3}{5} + 2\frac{1}{2}$.

Convert the given mixed fractions to improper fractions.

$1\frac{3}{5} = \frac{8}{5}$ and $2\frac{1}{2} = \frac{5}{2}$

Step 2: Convert both the fractions into like fractions by making the denominators the same.

Here, LCM of 5 and 2 is 10.

Thus, $\frac{8 \times 2}{5 \times 2} = \frac{16}{10}$ and $\frac{5\times 5}{2 \times 5} = \frac{25}{10}$

Step 3: Add the fractions by adding the numerators.

$\frac{16}{10} + \frac{25}{10} = \frac{41}{10}$

Step 4: Convert back into a mixed number. 

Thus, $\frac{41}{10}$ will become  $4\frac{1}{10}$

Therefore, $1\frac{3}{5} + 2\frac{1}{2} =  4\frac{1}{10}$

Here’s an example for subtraction. It follows the same steps.

Example 2 : $6\frac{1}{2} \;-\; 1\frac{3}{4}$

Step 1: Convert the mixed numbers into improper fractions.

     $6\frac{1}{2} \;-\; 1\frac{3}{4} = \frac{13}{2} \;-\; \frac{7}{4}$

Step 2: Make the denominators equal.

LCM of 2 and 4 is 4. 

   $\frac{13 \times 2}{2 \times 2} = \frac{26}{4}$ 

Step 3: Subtract the fractions.

        $\frac{26}{4} \;-\;  \frac{7}{4} = \frac{19}{4}$

Step 4: Convert the fraction as a mixed number.

            $\frac{19}{4}  = 4\frac{3}{4}$  

Thus, $6\frac{1}{2} \;-\; 1\frac{3}{4}  =   4\frac{3}{4}$  

Facts about Addition and Subtraction of Fractions

• We cannot add or subtract fractions without converting them into like fractions.
• Like fractions are fractions that have the same denominator, and unlike fractions are fractions that have different denominators.
• Equivalent fractions are two different fractions that represent the same value.
• The LCD (least common denominator) of two fractions is the LCM of the denominators.

In this article, we have learned about addition and subtraction of fractions (like fractions, unlike fractions, mixed fractions), methods of addition and subtraction of these fractions along with the steps. Let’s solve some examples on adding and subtracting fractions to understand the concept better.

• Solve: $\frac{2}{4} + \frac{1}{4}$ .

Solution: 

Here, the denominators are the same.

Thus, we add the numerators by keeping the denominators as it is.

$\frac{2}{4} + \frac{1}{4} = \frac{2 + 1}{4}$ 

$\frac{2}{4} + \frac{1}{4} = \frac{3}{4}$

2. Find the sum of the fractions $\frac{3}{5}$ and $\frac{5}{2}$ by using the LCM method.

$\frac{3}{5}$ and $\frac{5}{2}$ are unlike fractions.

The LCM of 2 and 5 is 10.

Thus, we can write

$\frac{3}{5} + \frac{5}{2} = \frac{3 \times 2}{5 \times 2} + \frac{5 \times 5}{2 \times 5}$

$= \frac{6}{10} + \frac{25}{10}$

            $= \frac{6}{10} + \frac{25}{10}$

            $= \frac{31}{10}$

Thus, $\frac{3}{5} + \frac{5}{2} =  \frac{31}{10}$

3. Find $\frac{4}{16} + \frac{5}{8}$.

Solution:  

To add two fractions with different denominators, we first need to find the LCM of the denominators.

The LCM of 16 and 8 is 16.

$\frac{4}{16} + \frac{5}{8} = \frac{4 \times 1}{16\times 1} + \frac{5 \times 2}{8 \times 2}$ 

            $= \frac{10}{16} + \frac{4}{16}$ 

            $= \frac{14}{16}$

$= \frac{7}{8}$

4. From a rope $12\frac{1}{2}$ ft. long, a $7 \frac{6}{8}\;-$ ft-long piece is cut off. Find the length of the remaining rope.

Total length of the rope $= 12\frac{1}{2}$ ft.

Length of the rope that was cut off $= 7 \frac{6}{8}$ ft. 

The length of the remaining rope $= 12\frac{1}{2} \;-\; 7 \frac{6}{8}$

$12\frac{1}{2} \;-\; 7 \frac{6}{8} = \frac{25}{2} \;-\; \frac{62}{8}$

         $= \frac{25 \times 4}{2 \times 4} \;-\; \frac{62 \times 1}{8\times 1}$

         $= \frac{100}{8} \;-\; \frac{62}{8}$

         $= \frac{38}{8}$

         $= \frac{19}{4}$

Converting it into a mixed fraction, $\frac{19}{4}$ becomes $4 \frac{3}{4}$.

Thus, the length of the remaining rope is $4\frac{3}{4}$ ft.

Attend this quiz & Test your knowledge.

Find $\frac{2}{4} + \frac{2}{4}$.

$\frac{7}{24} + \frac{5}{16} =$, what is the least common denominator of $\frac{1}{2}$ and $\frac{1}{3}$, $\frac{3}{6} \;-\; \frac{1}{6} =$, what equation does the following figure represent.

How do we add and subtract negative fractions?

Negative fractions are simply fractions with a negative sign. The steps to add and subtract the negative fractions remain the same. We need to follow the rules for addition/subtraction with negative signs.

How can we convert an improper fraction into a mixed number?

To convert an improper fraction into a mixed number, we divide the numerator by the denominator. The denominator stays the same. The quotient represents the whole number part. The remainder represents the numerator of the mixed number.

Example: $\frac{14}{3} = 4\; \text{R}\; 2$

Quotient $= 4$

Remainder $= 2$

$\frac{14}{3} = 4\frac{2}{3}$

How do we divide two fractions?

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction.

$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$

For example, $\frac{1}{2} \div \frac{3}{5} = \frac{1}{2} \times \frac{5}{3} = \frac{5}{6}$

What are the rules of adding and subtracting fractions?

• Before adding or subtracting, we check if the fractions have the same denominator.
• If the denominators are equal, then we add/subtract the numerators keeping the common denominator.
• If the denominators are different, then we make the denominators equal by using the LCM method. Once the fractions have the same denominator, we can add/subtract the numerators keeping the common denominator as it is.

How do we add and subtract fractions with whole numbers?

• Convert the whole number to a fraction. To do this, give the whole number a denominator of 1.
• Convert to fractions of like denominators. 
• Add/subtract the numerators. Now that the fractions have the same denominators, you can treat the numerators as a normal addition/subtraction problem.

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Add and Subtract Fractions Online practice for grades 3-7

On this page, you can practice addition and subtraction of fractions. Each practice set will automatically include both addition and subtraction problems.

The options are:

• You can limit the fractions in the problems to like fractions (fractions with the same denominator), for example: 1/6 + 4/6.
• You can limit the script to use only proper fractions—fractions that are less than 1. With this option, the script will make problems such as 1/4 + 2/5, but will not make problems such as 8/5 − 4/5.
• When you choose problems that use simplified fractions, the script will only include fractions in the problems that are in lowest terms. For example, you could get a problem such as 5/6 + 3/5, but you would not see 2/4 + 6/8.
• The last option, when chose, allows or accepts answers to not be in lowest terms. In other words, the script will accept an answer such as 8/10.

Note: ALL answers have to be given as mixed numbers, when possible. In other words, your answer cannot be left as an improper fraction.

You may use the space below to write the intermediate step. Your work in this area will not be checked.

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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 .

Addition and Subtraction of Fractions

While adding and subtracting fractions , we need to check whether the fractions have the same denominators or different denominators and then the calculation starts. Let us learn more about the addition and subtraction of fractions in this article.

How to Add and Subtract Fractions?

Addition and subtraction of fractions is done using similar rules in which the denominators are checked before the addition or subtraction starts. After the denominators are checked, we can add or subtract the given fractions accordingly. The denominators are checked in the following way.

• If the denominators of the given fractions are the same, we add or subtract only the numerators and we retain the denominator.
• If the denominators are different, we convert the fractions to like fractions so that the denominators become the same, and then we add or subtract, whatever is required.

Let us learn about these in the following sections.

Adding and Subtracting Fractions with Like Denominators

The process for adding and subtracting fractions with like denominators is quite simple because we just need to work with the numerators.

Adding Fractions with Like Denominators

Let us add the fractions 1/5 and 2/5 using rectangular models. In this case, both the fractions have the same denominators. These fractions are called like fractions . The following figure represents both the fractions in the same model.

• 1/5 indicates that 1 out of 5 parts are shaded yellow.
• 2/5 indicates that 2 out of 5 parts are shaded blue.

Out of the 5 parts, 3 parts are shaded. In the fractional form, this can be represented as 3/5.

Now, let us add the fractions with like denominators in numerical terms. In this case, we need to add 1/5 + 2/5. Let us use the following steps to understand the addition.

• Step 1: Add the numerators of the given fractions. Here, the numerators are 1 and 2, so it will be 1 + 2 = 3
• Step 2: Retain the same denominator. Here, the denominator is 5.
• Step 3: Therefore, the sum of 1/5 + 2/5 = (1 + 2)/5 = 3/5

It should be noted that we use the same method for subtracting fractions.

Subtracting Fractions with Like Denominators

Let us subtract the fractions 2/5 and 1/5 using rectangular models. We will represent 2/5 in this model by shading 2 out of 5 parts. We will further shade out 1 part from our shaded parts of the model which would represent removing 1/5.

We are now left with 1 part in the shaded parts of the model.

Now, let us subtract the fractions with like denominators in numerical terms. In this case, we need to subtract 2/5 - 1/5. Let us understand the procedure using the following steps.

• Step 1: We will subtract the numerators of the given fractions. Here, the numerators are 2 and 1, so it will be 2 - 1 = 1
• Step 3: Therefore, the difference of 2/5 - 1/5 = (2 - 1)/5 = 1/5

Adding and Subtracting Fractions with Unlike Denominators

For adding and subtracting fractions with unlike denominators, we need to convert the unlike fractions to like fractions by writing their equivalent fractions in such a way that their denominators become the same. Let us understand this with the help of an example.

Example: Add 1/5 + 1/3

Solution: For adding unlike fractions we need to use the following steps

• Step 1: Find the Least Common Multiple (LCM) of the denominators. Here, the LCM of 5 and 3 is 15.
• Step 2: Convert the given fractions to like fractions by writing the equivalent fractions for the respective fractions such that their denominators remain the same. Here, it will be \(\frac {1}{5}\)×\(\frac {3}{3}\)=\(\frac {3}{15}\)
• Step 3: Similarly, an equivalent fraction of 1/3 with denominator 15 is \(\frac {1}{3}\)×\(\frac {5}{5}\)=\(\frac {5}{15}\)
• Step 4: Now, that we have converted the given fractions to like fractions we can add the numerators and retain the same denominator. This will be 3/15 + 5/15 = 8/15
• Subtracting Fractions with Unlike Denominators

For subtracting unlike fractions, we follow the same steps as we did for the addition of unlike fractions. Let us understand this with the help of an example.

Example: Subtract 5/6 - 1/3

Solution: For subtracting unlike fractions we need to use the following steps.

• Step 1: Find the Least Common Multiple (LCM) of the denominators. Here, the LCM of 6 and 3 is 6.
• Step 2: Convert the given fractions to like fractions by writing the equivalent fractions for the respective fractions such that their denominators remain the same. Here, it will be \(\frac {5}{6}\)×\(\frac {1}{1}\)=\(\frac {5}{6}\)
• Step 3: Similarly, an equivalent fraction of 1/3 with denominator 6 is \(\frac {1}{3}\)×\(\frac {2}{2}\)=\(\frac {2}{6}\)
• Step 4: Now, that we have converted the given fractions to like fractions we can subtract the numerators and retain the same denominator. This will be 5/6 - 2/6 = 3/6. This can be further reduced to 1/2

Adding and Subtracting Mixed Fractions

Adding and subtracting mixed fractions is done by converting the mixed fractions to improper fractions and then the addition or subtraction is done as per the requirement. Let us understand these with the help of the following examples.

Example: Add the mixed fractions: \(2\dfrac{1}{4}\) + \(1\dfrac{3}{4}\)

Solution: First let us convert the mixed fractions to improper fractions.

• Step 1: Convert the given mixed fractions to improper fractions. So, \(2\dfrac{1}{4}\) will become 9/4; and \(1\dfrac{3}{4}\) will become 7/4
• Step 2 : Add the fractions by adding the numerators because the denominators are the same. This will be 9/4 + 7/4= 16/4.
• Step 3: Reduce the fraction, if required. This will become, 16/4 = 4. Therefore, \(2\dfrac{1}{4}\) + \(1\dfrac{3}{4}\) = 4.

Now, let us understand the subtraction of mixed fractions using the same method.

Example: Subtract the mixed fractions: \(5\dfrac{1}{3}\) - \(2\dfrac{1}{3}\)

• Step 1: Convert the given mixed fractions to improper fractions. So, \(5\dfrac{1}{3}\) will become 16/3; and \(2\dfrac{1}{3}\) will become 7/3
• Step 2 : Subtract the fractions by subtracting the numerators because the denominators are the same. This will be 16/3 - 7/3 = 9/3
• Step 3: Reduce the fraction, if required. This will become, 9/3 = 3. Therefore, \(5\dfrac{1}{3}\) - \(2\dfrac{1}{3}\) = 3

Adding and Subtracting Fractions with Whole Numbers

Adding and subtracting fractions with whole numbers can be done using the following method. Let us understand this using an example.

Example: Add 7/4 + 5

Solution: Let us add 7/4 + 5 using the following steps.

• Step 1: Write the whole number in the form of a fraction. In this case the whole number is 5 which can be written as 5/1. So, now we need to add 7/4 + 5/1
• Step 2: Now, find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 4 and 1 is 4. And after converting them to like fractions we get, (7 × 1)/(4 × 1) + (5 × 4)/(1 × 4) = 7/4 + 20/4
• Step 3: Add the numerators while the denominator remains the same. Here, 7/4 + 20/4 = 27/4 = \(6\dfrac{3}{4}\)

Now, let us understand the subtraction of a fraction from a whole number with the help of the following example.

Example: Subtract 6 - 3/5

Solution: Let us subtract 6 - 3/5 using the following steps.

• Step 1: Write the whole number in the form of a fraction. In this case the whole number is 6 which can be written as 6/1. So, now we need to subtract 6/1 - 3/5
• Step 2: Now, find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 1 and 5 is 5. And after converting them to like fractions we get, (6 × 5)/(1 × 5) - (3 × 1)/(5 × 1) = 30/5 - 3/5
• Step 3: Subtract the numerators while the denominator remains the same. Here, 30/5 - 3/5 = 27/5 = \(5\dfrac{2}{5}\)

Important Notes on Adding and Subtracting Fractions

• For adding and subtracting like fractions, we can directly work with the numerators while the denominators remain the same.
• For adding and subtracting unlike fractions, never add or subtract the numerators and denominators directly. Convert them to like fractions and then add or subtract.

☛ Related Topics

• Adding Fractions
• Subtraction of Fractions
• Multiplying Fractions
• Division of Fractions
• Adding Fractions with Unlike Denominators
• Like Fractions Calculator
• Fractions Calculator

Adding and Subtracting Fractions Examples

Example 1: Find the sum of 1/7 + 3/7 Solution: The given fractions are like fractions so we will add the numerators and retain the same denominator.

1/7 + 3/7 = (1 + 3)/7 = 4/7

Therefore, the sum is 4/7

Example 2: Subtract 2/3 - 2/5 Solution: The given fractions are unlike fractions. So, we need to find the LCM of the denominators and convert 2/5 and 2/3 to equivalent fractions of the same denominator and then subtract.

LCM of (3, 5) = 15

\(\begin{align} &= \left(\frac {2}{3} \times \frac {5}{5} \right) - \left(\frac {2}{5} \times \frac {3}{3} \right) \\ &= \frac {10}{15} - \frac {6}{15} \\ &= \frac {4}{15} \end{align}\)

Therefore, the difference is 4/15

Example 3: State true or false with respect to adding and subtracting fractions.

a.) 4/5 + 3/5 = 7/5

b.) 7/8 - 2/8 = 9/8 Solution:

a.) True, 4/5 + 3/5 = 7/5

b.) False, 7/8 - 2/8 = 5/8

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Practice Questions on Addition and Subtraction of Fractions

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FAQs on Addition and Subtraction of Fractions

For adding and subtracting fractions , we first need to check the denominators. If the denominators are the same, we simply add or subtract the numerators and retain the same denominator. In the case of unlike fractions, when the denominators are not the same, we convert the unlike fractions to like fractions by finding the LCM of the denominators. This helps in writing their respective equivalent fractions and then they are added or subtracted, as required.

How to Add and Subtract Fractions with Different Denominators?

In order to add and subtract fractions with different denominators, we need to convert the fractions to like fractions so that the denominators become the same. Once the denominators are the same, we can add or subtract the numerators. In order to convert the given fractions to like fractions, we need to find the LCM of the denominators and then write their respective equivalent fractions. The equivalent fractions with the same denominators can then be added or subtracted, as the case may be.

How to Add and Subtract Fractions with Whole Numbers?

For adding and subtracting fractions with whole numbers we use the following method.

• Write the whole number in the form of a fraction by writing 1 as its denominator. For example, if we need to add 8/7 + 5, we will write the whole number in the form of a fraction. In this case the whole number is 5 which can be written as 5/1. So, now we need to add 8/7 + 5/1. We will find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 7 and 1 is 7. And after converting them to like fractions we get, (8 × 1)/(7 × 1) + (5 × 7)/(1 × 7) = 8/7 + 35/7 = 43/7 = \(6\dfrac{1}{7}\)
• The same method will be used for subtraction, for example, if we need to subtract 7 - 2/5, we will write the whole number 7 as 7/1 and then subtract. This will make it 7/1 - 2/5. We will find the LCM of the denominators and convert the given fractions to like fractions. Here the LCM of 5 and 1 is 5. And after converting them to like fractions we get, (7 × 5)/(1 × 5) - (2 × 1)/(5 × 1) = 35/5 - 2/5 = 33/5 = \(6\dfrac{3}{5}\)

How to Add and Subtract Fractions with Mixed Numbers?

To add and subtract fractions with mixed numbers, we convert the mixed numbers to improper fractions . Now, if they are like fractions, we can simply add or subtract the numerators and retain the same denominator. For adding or subtracting unlike fractions, we convert them to like fractions. We find the LCM of the denominators and convert the addends to their equivalent fractions and add them in the same way as we add like fractions.

What are the Rules for Adding and Subtracting Fractions?

The basic rules for adding and subtracting fractions are given below:

• We need to check if the denominators of the fractions are same or different.
• If the denominators are the same, we can simply add or subtract the numerators.
• If the denominators are not the same, we need to convert them to like fractions and then we add or subtract.

Calculator   Soup ®

Online Calculators

Adding Fractions Calculator

Calculator use.

Add and subtract fractions with this calculator and see the work to solve the problem.

Select the number of fractions in your problem and input numerators (top numbers) and denominators (bottom numbers) in the calculator fields. Click the Calculate button to solve the equation and show the work.

You can add and subtract 3 fractions, 4 fractions, 5 fractions or up to 9 fractions at once.

How to Add and Subtract Fractions

With like denominators.

When fractions have the same bottom numbers you can add or subtract the top numbers and put the result over the common denominator. If possible you can simplify the fraction to lowest terms or a mixed number.

Watch the CalculatorSoup® YouTube video How to Add Fractions with the Same Denominator .

With Unlike Denominators

When fractions have unlike denominators (bottom numbers) the first step is to find equivalent fractions so all of the denominators become the same number. Find the Least Common Denominator (LCD) and rewrite all fractions as equivalent fractions using the LCD as the denominator.

Once the denominators are the same you can add or subtract the numerators and put the result over the common denominator. If possible you can simplify the fraction to lowest terms or a mixed number.

Watch the CalculatorSoup® YouTube video How to Add Fractions with Different Denominators .

How to Work With Negative Fractions

When you are adding a negative fraction you can rewrite the equation as subtracting a positive fraction. For example 4/5 + -3/10 = 4/5 - 3/10.

When you are subtracting a negative fraction it is the same as adding a positive fraction. You can rewrite the problem this way. For example 4/5 - -3/10 = 4/5 + 3/10.

This calculator rewrites negative fractions when it shows the work to find the answer.

Rules for Math with Negative Numbers

Whether you are working with fractions, whole numbers or decimals, use these guidelines when adding and subtracting positive and negative numbers.

Why is it Important to Understand Fractions?

Fractions come up everywhere in everyday life. When following cooking recipes you'll usually see fractions in ingredient quantities. You might need a 1/2 cup of flour or 3/4 teaspoon of salt. If you're making twice the recipe you will need to know how to add fractions together so your dish turns out as expected.

Or say you wanted to make 2/3 a recipe. You would need to subtract 1/3 from each of the ingredient amounts to come up with accurate measurements for your adjusted recipe.

Fractions can also be important in budgeting your finances. Say you wanted to allocate your income into 6 categories of expenses: housing, utilities, groceries, transportation, entertainment and savings. Maybe you already know that 1/4 of your income goes toward housing, 1/8 goes toward utilites, and another 1/8 goes toward groceries. You need to be able to work with fractions to figure out how to optimize your budget to put the rest of your money toward the remaining categories where you have flexibility: transportation, entertainment and savings.

You can add 1/8 + 1/8 to get 2/8. And you can reduce this fraction to 1/4, so you see that the amount of your income that goes toward predictable expenses is 1/4 + 1/4 = 1/2. If half of your income is accounted for you can continue to use fractions to make the most of your remaining money, in ways that are important to you.

Visit the CalculatorSoup® YouTube channel for a demo of How to Add Fractions and more.

Cite this content, page or calculator as:

Furey, Edward " Adding and Subtracting Fractions Step by Step - CalculatorSoup " at https://www.calculatorsoup.com/calculators/math/adding-fractions-calculator.php from CalculatorSoup, https://www.calculatorsoup.com - Online Calculators

Last updated: April 10, 2024

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5.1: Addition and Subtraction of Fractions with Like Denominators

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

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Learning Objectives

• be able to add and subtract fractions with like denominators

Addition of Fraction With Like Denominators

Let's examine the following diagram.

2 one-fifths and 1 one fifth is shaded.

It is shown in the shaded regions of the diagram that

(2 one-fifths) + (1 one-fifth) = (3 one-fifths)

\(\dfrac{2}{5} + \dfrac{1}{5} = \dfrac{3}{5}\)

From this observation, we can suggest the following rule.

Method of Adding Fractions Having Like Denominators To add two or more fractions that have the same denominators, add the numer­ators and place the resulting sum over the common denominator. Reduce, if necessary.

Sample Set A

Find the following sums.

\(\dfrac{3}{7} + \dfrac{2}{7}\). The denominators are the same. Add the numerators and place that sum over 7.

\(\dfrac{3}{7} + \dfrac{2}{7} = \dfrac{3 + 2}{7} = \dfrac{5}{7}\)

\(\dfrac{1}{8} + \dfrac{3}{8}\). The denominators are the same. Add the numerators and place the sum over 8. Reduce.

\(\dfrac{1}{8} + \dfrac{3}{8} = \dfrac{1 + 3}{8} = \dfrac{4}{8} = \dfrac{1}{2}\)

\(\dfrac{4}{9} + \dfrac{5}{9}\). The denominators are the same. Add the numerators and place the sum over 9.

\(\dfrac{4}{9} + \dfrac{5}{9} = \dfrac{4 + 5}{9} = \dfrac{9}{9} = 1\)

\(\dfrac{7}{8} + \dfrac{5}{8}\). The denominators are the same. Add the numerators and place the sum over 8.

\(\dfrac{7}{8} + \dfrac{5}{8} = \dfrac{7 + 5}{8} = \dfrac{12}{8} = \dfrac{3}{2}\)

To see what happens if we mistakenly add the denominators as well as the numerators, let's add

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

Adding the numerators and mistakenly adding the denominators produces

\(\dfrac{1}{2} + \dfrac{1}{2} = \dfrac{1 + 1}{2 + 2} = \dfrac{2}{4} = \dfrac{1}{2}\)

This means that two \(\dfrac{1}{2}\)'s is the same as one \(\dfrac{1}{2}\). Preposterous! We do not add denominators .

Practice Set A

\(\dfrac{1}{10} + \dfrac{3}{10}\)

\(\dfrac{2}{5}\)

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

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

\(\dfrac{7}{11} + \dfrac{4}{11}\)

\(\dfrac{3}{5} + \dfrac{1}{5}\)

\(\dfrac{4}{5}\)

Show why adding both the numerators and denominators is preposterous by adding \(\dfrac{3}{4}\) and \(\dfrac{3}{4}\) and examining the result.

\(\dfrac{3}{4} + \dfrac{3}{4} = \dfrac{3 + 3}{4 + 4} = \dfrac{6}{8} = \dfrac{3}{4}\), so two \(\dfrac{3}{4}\)'s = one \(\dfrac{3}{4}\), whihch is preposterous.

Subtraction of Fractions With Like Denominators

We can picture the concept of subtraction of fractions in much the same way we pictured addition.

From this observation, we can suggest the following rule for subtracting fractions having like denominators:

Subtraction of Fractions with Like Denominators To subtract two fractions that have like denominators, subtract the numerators and place the resulting difference over the common denominator. Reduce, if possible.

Sample Set B

Find the following differences.

\(\dfrac{3}{5} - \dfrac{1}{5}\). The denominators are the same. Subtract the numerators. Place the difference over 5.

\(\dfrac{3}{5} - \dfrac{1}{5} = \dfrac{3 - 1}{5} = \dfrac{2}{5}\)

\(\dfrac{8}{6} - \dfrac{2}{6}\). The denominators are the same. Subtract the numerators. Place the difference over 6.

\(\dfrac{8}{6} - \dfrac{2}{6} = \dfrac{8 - 2}{6} = \dfrac{6}{6} = 1\)

\(\dfrac{16}{9} - \dfrac{2}{9}\). The denominators are the same. Subtract numerators and place the difference over 9.

\(\dfrac{16}{9} - \dfrac{2}{9} = \dfrac{16 - 2}{9} = \dfrac{14}{9}\)

To see what happens if we mistakenly subtract the denominators, let's consider

\(\dfrac{7}{15} - \dfrac{4}{15} = \dfrac{7 - 4}{15 - 15} = \dfrac{3}{0}\)

We get division by zero, which is undefined. We do not subtract denominators.

Practice Set B

\(\dfrac{10}{13} - \dfrac{8}{13}\)

\(\dfrac{2}{13}\)

\(\dfrac{5}{12} - \dfrac{1}{12}\)

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

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

\(\dfrac{26}{10} - \dfrac{14}{10}\)

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

Show why subtracting both the numerators and the denominators is in error by performing the subtraction \(\dfrac{5}{9} - \dfrac{2}{9}\)

\(\dfrac{5}{9} - \dfrac{2}{9} = \dfrac{5 - 2}{9 - 9} = \dfrac{3}{0}\), which is undefined

For the following problems, find the sums and differences. Be sure to reduce.

Exercise \(\PageIndex{1}\)

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

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

Exercise \(\PageIndex{2}\)

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

Exercise \(\PageIndex{3}\)

\(\dfrac{9}{10} + \dfrac{1}{10}\)

Exercise \(\PageIndex{4}\)

\(\dfrac{3}{11} + \dfrac{4}{11}\)

Exercise \(\PageIndex{5}\)

\(\dfrac{9}{15} + \dfrac{4}{15}\)

\(\dfrac{13}{15}\)

Exercise \(\PageIndex{6}\)

\(\dfrac{3}{10} + \dfrac{2}{10}\)

Exercise \(\PageIndex{7}\)

\(\dfrac{5}{12} + \dfrac{7}{12}\)

Exercise \(\PageIndex{8}\)

\(\dfrac{11}{16} - \dfrac{2}{16}\)

Exercise \(\PageIndex{9}\)

\(\dfrac{3}{16} - \dfrac{3}{16}\)

Exercise \(\PageIndex{10}\)

\(\dfrac{15}{23} - \dfrac{2}{23}\)

Exercise \(\PageIndex{11}\)

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

Exercise \(\PageIndex{12}\)

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

Exercise \(\PageIndex{13}\)

\(\dfrac{3}{11} + \dfrac{1}{11} + \dfrac{5}{11}\)

\(\dfrac{9}{11}\)

Exercise \(\PageIndex{14}\)

\(\dfrac{16}{20} + \dfrac{1}{20} + \dfrac{2}{20}\)

Exercise \(\PageIndex{15}\)

\(\dfrac{12}{8} + \dfrac{2}{8} - \dfrac{1}{8}\)

Exercise \(\PageIndex{16}\)

\(\dfrac{11}{16} + \dfrac{9}{16} - \dfrac{5}{16}\)

Exercise \(\PageIndex{17}\)

\(\dfrac{4}{20} - \dfrac{1}{20} + \dfrac{9}{20}\)

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

Exercise \(\PageIndex{18}\)

\(\dfrac{7}{10} - \dfrac{3}{10} + \dfrac{11}{10}\)

Exercise \(\PageIndex{19}\)

\(\dfrac{16}{5} - \dfrac{1}{5} - \dfrac{2}{5}\)

\(\dfrac{13}{5}\)

Exercise \(\PageIndex{20}\)

\(\dfrac{21}{35} - \dfrac{17}{35} + \dfrac{31}{35}\)

Exercise \(\PageIndex{21}\)

\(\dfrac{5}{2} + \dfrac{16}{2} - \dfrac{1}{2}\)

Exercise \(\PageIndex{22}\)

\(\dfrac{1}{18} + \dfrac{3}{18} + \dfrac{1}{18} + \dfrac{4}{18} - \dfrac{5}{18}\)

Exercise \(\PageIndex{23}\)

\(\dfrac{6}{22} - \dfrac{2}{22} + \dfrac{4}{22} - \dfrac{1}{22} + \dfrac{11}{22}\)

The following rule for addition and subtraction of two fractions is preposterous. Show why by performing the operations using the rule for the following two problems.

Preposterous Rule To add or subtract two fractions, simply add or subtract the numerators and place this result over the sum or difference of the denominators.

Exercise \(\PageIndex{24}\)

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

Exercise \(\PageIndex{25}\)

\(\dfrac{8}{15} + \dfrac{8}{15}\)

\(\dfrac{16}{30} = \dfrac{8}{15}\) (using the preposterous rule)

Exercise \(\PageIndex{26}\)

Find the total length of the screw.

Two months ago, a woman paid off \(\dfrac{3}{24}\) of a loan. One month ago, she paid off \(\dfrac{5}{24}\) of the total loan. This month she will again pay off \(\dfrac{5}{24}\) of the total loan. At the end of the month, how much of her total loan will she have paid off?

\(\dfrac{13}{24}\)

Exercise \(\PageIndex{27}\)

Find the inside diameter of the pipe.

Exercises for Review

Exercise \(\PageIndex{28}\)

Round 2,650 to the nearest hundred.

Exercise \(\PageIndex{29}\)

Use the numbers 2, 4, and 8 to illustrate the associative property of addition.

Exercise \(\PageIndex{30}\)

Find the prime factors of 495.

\(3^2 \cdot 5 \cdot 11\)

Exercise \(\PageIndex{31}\)

Find the value of \(\dfrac{3}{4} \cdot \dfrac{16}{25} \cdot \dfrac{5}{9}\).

Exercise \(\PageIndex{32}\)

\(\dfrac{8}{3}\) of what number is \(1 \dfrac{7}{9}\)?

\(\dfrac{2}{3}\)

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Adding Fractions with Like Denominators

Welcome to our Adding Fractions with Like Denominators page.

Here you will find our selection of worksheets designed to help your child master the skill of adding fractions with the same denominator.

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• Adding and Subtracting Fractions Support

Adding Subtracting Fractions Calculator

Adding fractions with like denominators worksheets.

• Adding Subtracting Fractions with Like Denominators Worksheets
• More related resources
• Adding Fractions with Like Denominators Online Quiz

How to Add and Subtract Fractions Support

Here you will find support pages about how to add and subtract fractions (with both like and unlike denominators).

• How do you add Fractions?
• Steps to subtract Fractions?

How to Add and Subtract Fractions with Like Denominators Video

Find out how to add and subtract fractions with like denominators using the video below.

If you want to use our Free Fraction Calculator to do the work for you then use the link below.

Our Fraction calculator will allow you to add or subtract fractions and show you the steps to work it out.

Otherwise, for more detailed support and worksheets, keep reading!

• Free Fraction Calculator

Here you will find a selection of Free Fraction worksheets designed to help your child understand how to add fractions with the same denominator. The sheets are graded so that the easier ones are at the top.

Using these sheets will help your child to:

• add (and subtract) fractions with the same denominator.
• solve a problem where the answer is given and one of the addends is missing.

We have split the worksheets into 3 sections:

• The first section contains adding fractions using circles - a visual way to understand this concept.
• The second section contains just adding fractions with like denominators with no extra support.
• The third section contains both adding and subtracting fractions with like denominators.

Adding Fractions with Like Denominators Using Circles

A great way to introduce and understand this concept is to use visual representations of fractions.

The questions on these sheets involve shading different fractions and then working out the total fraction shaded.

This is then converted into a simple fraction addition sentence!

• Adding Fractions With Like Denominators Using Circles 1
• PDF version
• Adding Fractions With Like Denominators Using Circles 2
• Adding Fractions With Like Denominators Using Circles 3

Adding Fractions with Like Denominators - problems only

• Adding Fractions With Like Denominators 1
• Adding Fractions With Like Denominators 2
• Adding Fractions With Like Denominators 3

Adding & Subtracting Fractions with Like Denominators

The first sheet involves using fraction diagrams to look at the pattern between adding and subtracting fractions.

The other 3 sheets are more traditional fraction problems involving adding and subtracting fractions.

• Adding Subtracting Fractions Like Denominators Using Circles 1
• Adding Subtracting Fractions like denominators 1
• Adding Subtracting Fractions like denominators 2
• Adding Subtracting Fractions like denominators 3

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

• Subtracting Fractions with like denominators

We also have a selection of worksheets involving subtracting fractions with like denominators.

These sheets are similar to the ones on this page, but involve subtraction instead of addition.

Adding & Subtracting Fractions with unlike denominators

These sheets are all about adding (and subtracting) fractions with different denominators.

• add and subtracting fractions with different denominators;
• apply your equivalent fractions knowledge.
• Adding Fractions Worksheets with unlike denominators
• Adding Subtracting Fractions with unlike denominators
• Fractions Adding and Subtracting Worksheets (randomly generated)

More Third Grade Fraction Worksheets

• understand what fractions are;
• relate fractions to everyday objects and quantities;
• place different fractions on a number line;
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• Finding Fractions - Fraction Spotting
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Learning Fractions Math Help Page

Here you will find the Math Salamanders free online Math help pages about Fractions.

There is a wide range of help pages including help with:

• fraction definitions;
• equivalent fractions;
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• how to add and subtract fractions;
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• how to simplify fractions.
• Learning Fractions Support

Adding Fractions with Like Denominators Quiz

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Adding and Subtracting Two Mixed Fractions with Similar Denominators, Mixed Fractions Results and Some Simplifying (Fillable) (A)

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How to Add and Subtract Fractions

Last Updated: April 6, 2024 Fact Checked

This article was co-authored by David Jia . David Jia is an Academic Tutor and the Founder of LA Math Tutoring, a private tutoring company based in Los Angeles, California. With over 10 years of teaching experience, David works with students of all ages and grades in various subjects, as well as college admissions counseling and test preparation for the SAT, ACT, ISEE, and more. After attaining a perfect 800 math score and a 690 English score on the SAT, David was awarded the Dickinson Scholarship from the University of Miami, where he graduated with a Bachelor’s degree in Business Administration. Additionally, David has worked as an instructor for online videos for textbook companies such as Larson Texts, Big Ideas Learning, and Big Ideas Math. There are 10 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 165,287 times.

Adding and subtracting fractions is an essential skill to have. Fractions show up in daily life all the time, especially in math classes, from elementary school through college. Just follow these steps to learn how to add and subtract them, whether they're like fractions, unlike fractions, mixed, or improper fractions. Once you know one way, the rest is pretty easy!

Adding and Subtracting Fractions with the Same Denominator

• In other words, 1/5 and 2/5 does not need to be written as 1/5 + 2/5 = ? It can be written as 1+2/5 = ? . The denominator is the same, so it can be written only once. Both numerators then go on top.

• Whether you have it written 1/5 + 2/5 or 1+2/5, you answer should be the same: 3! After all, 1 + 2 = 3.

• So, using the same example, our denominator is 5. That's it! That's the bottom number of our fraction. That's half the answer already!

• What was your numerator? 3. The denominator? 5. Therefore, 1/5 + 2/5, or 1+2/5, equals 3/5 .

Adding and Subtracting Fractions with Different Denominators

• Write out the multiples . The multiples of 3 are 3, 6, 9, 12, 15, 18...and so on. The multiples of 4? 4, 8, 12, 16, 20, etc. What's the lowest number seen in both of the sets? 12! That's your lowest common denominator, or LCD.
• Multiply the numbers together for small numbers. In some cases, like this one, you could just multiply the numbers together – 3 x 4 = 12. However, if your denominators are big, don't do this! You don't want to multiply 56 x 44 and have to work with 2,464 as your answer!

• You'll notice that the denominators, in this instance, are multiplied by each other. This works in this situation, but not all situations. Sometimes, instead of multiplying the two denominators together, you can multiply both denominators by different numbers to get one small number.
• And then in other cases, sometimes you only have to multiply one denominator to make it equal to the denominator of the other fraction in the equation.

• We had 2/3x4 and 3/4x3 as our first step – to add the second step, it's really 2 x 4/3 x 4 and 3 x 3/4 x 3. That means our new numbers are 8/12 and 9/12. Perfect!

• For this example, (8+9)/12 = 17/12. To turn this into a mixed fraction, simply subtract the denominator from the numerator and see what's left over. In this case, 17/12 = 1 5/12

Adding and Subtracting Mixed and Improper Fractions

• For the example for this section, let's work with 13/12 and 17/8.

• Let's figure out the multiples of our example, 12 and 8. What's the smallest number these two go into? 24. 8, 16, 24 and 12, 24 – bingo!

• So 13 x 2/12 x 2 = 26/24. And 17 x 3/8 x 3 = 51/24. We're well on our way to solving the problem!

• 26/24 + 51/24 = 77/24. There's your one fraction! That top number is mighty big, though....

• For this example, 24 goes into 77 three times. That is, 24 x 3 = 72. But there's 5 leftover! So what's your final answer? 3 5/24. That's it!

Adding and Subtracting Fractions without looking for the LCD

• e.g. ½ + ¾ + ⅝

• Multiply ¹ to the denominator/s of the other fractions.
• Multiply 1 to 4 and 8. [32]

• Multiply 3 with 2 and 8. [48]
• Lastly, multiply 5 with 4 and 2. [40]

• 32+48+40=120

• 120/64 = 1 56/64 = 1 ⅞

Joseph Meyer

To simplify fractions, you can divide both the numerator and denominator by a common factor. This creates a new, easier-to-use fraction with smaller components, but it represents the same value. For instance, if you divide both the numerator and denominator of 6/12 by 2, you get 3/6, which is equal to 1/2.

Calculator, Practice Problems, and Answers

Community Q&A

Tips from our Readers

• Simplify if the fraction can be simplified and when the fraction is an improper fraction, make sure you convert it into a mixed or whole number.
• This method might lead you to multiplying large numbers.
• This might require you a calculator.

You Might Also Like

• ↑ https://www.chilimath.com/lessons/introductory-algebra/adding-and-subtracting-fractions-with-same-or-like-denominator/
• ↑ https://www.mathsisfun.com/fractions_subtraction.html
• ↑ https://edu.gcfglobal.org/en/fractions/adding-and-subtracting-fractions/1/
• ↑ https://www.mathsisfun.com/least-common-denominator.html
• ↑ https://www.coolmath4kids.com/math-help/fractions/adding-and-subtracting-fractions-different-denominators
• ↑ https://www.bbc.co.uk/bitesize/topics/zhdwxnb/articles/z9n4k7h
• ↑ https://www.georgebrown.ca/sites/default/files/uploadedfiles/tlc/_documents/Adding_and_Subtracting_Mixed_Numbers_and_Improper_Fractions.pdf
• ↑ https://www.mathsisfun.com/numbers/fractions-mixed-addition.html
• ↑ https://www.chilimath.com/lessons/introductory-algebra/adding-and-subtracting-fractions-with-different-denominators/

About This Article

To add and subtract fractions with the same denominator, or bottom number, place the 2 fractions side by side. Add or subtract the numerators, or the top numbers, and write the result in a new fraction on the top. The bottom number of the answer will be the same as the denominator of the original fractions. To learn how to add and subtract fractions with different denominators, keep reading! Did this summary help you? Yes No

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Speedway Fractions NUMBER OF PLAYERS: 12 Add and subtract fractions. Simplify your answer. Power up your racecar and win first place!

How to Add Fractions in 3 Easy Steps

March 9, 2023 by Anthony Persico

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 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, let’s 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 3.

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

Like Denominators (the denominators are the same)

Unlike Denominators (the denominators are different)

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, let’s move onto a few examples.

How to Add Fractions with Like Denominators

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 = ?

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

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

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

1/4 + 2/4 = (1+2)/4 = 3/4

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, let’s apply the 3-step process like we did in the previous example as follows:

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

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

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

2/9 + 4/9 = (2+4)/9 = 6/9

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, let’s learn how to add fractions with different denominators.

How to Add Fractions with Different Denominators

How to add fractions with different denominators: example #1.

Example #1: 1/3 + 1/4

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

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).

1/3 + 1/4 (4x1)/(4x3) + (3x1)/(3x4) = 4/12 + 3/12

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:

4/12 + 3/12 = (4+3)/12 = 7/12

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 add the numerators together and keep the same denominator.

Now, let’s 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, let’s again apply the 3-step process:

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

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

3/5 + 4/11 (11x3)/(11x5) + (5x4)/(5x11) = 33/55 + 20/55 = 53/55

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:

33/55 + 20/55 = (33+20)/55 = 53/55

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:

Keep Learning:

How to Subtract Fractions in 3 Easy Steps

Search Tags: how to add fractions, how to add fractions with different denominators, how to add fraction , how to add fractions with unlike denominators, how to. add fractions, how to add fractions with, how to add a fraction

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