fraction addition problem solving

Key to Fractions Workbooks

These workbooks by Key Curriculum Press feature a number of exercises to help your child learn about fractions. Book 1 teaches fraction concepts, Book 2 teaches multiplying and dividing, Book 3 teaches adding and subtracting, and Book 4 teaches mixed numbers. Each book has a practice test at the end.

Solving Word Problems by Adding and Subtracting Fractions and Mixed Numbers

Learn how to solve fraction word problems with examples and interactive exercises.

Example 1: Rachel rode her bike for one-fifth of a mile on Monday and two-fifths of a mile on Tuesday. How many miles did she ride altogether?

Analysis: To solve this problem, we will add two fractions with like denominators.

Solution: 

Answer: Rachel rode her bike for three-fifths of a mile altogether.

Analysis: To solve this problem, we will subtract two fractions with unlike denominators.

Answer: Stefanie swam one-third of a lap farther in the morning.

Analysis: To solve this problem, we will add three fractions with unlike denominators. Note that the first is an improper fraction.

Answer: It took Nick three and one-fourth hours to complete his homework altogether.

Analysis: To solve this problem, we will add two mixed numbers, with the fractional parts having like denominators.

Answer: Diego and his friends ate six pizzas in all.

Analysis: To solve this problem, we will subtract two mixed numbers, with the fractional parts having like denominators.

Answer: The Cocozzelli family took one-half more days to drive home.

Analysis: To solve this problem, we will add two mixed numbers, with the fractional parts having unlike denominators.

Answer: The warehouse has 21 and one-half meters of tape in all.

Analysis: To solve this problem, we will subtract two mixed numbers, with the fractional parts having unlike denominators.

Answer: The electrician needs to cut 13 sixteenths cm of wire.

Analysis: To solve this problem, we will subtract a mixed number from a whole number.

Answer: The carpenter needs to cut four and seven-twelfths feet of wood.

Summary: In this lesson we learned how to solve word problems involving addition and subtraction of fractions and mixed numbers. We used the following skills to solve these problems: 

• Add fractions with like denominators.
• Subtract fractions with like denominators.
• Find the LCD.
• Add fractions with unlike denominators.
• Subtract fractions with unlike denominators.
• Add mixed numbers with like denominators.
• Subtract mixed numbers with like denominators.
• Add mixed numbers with unlike denominators.
• Subtract mixed numbers with unlike denominators.

Directions: Subtract the mixed numbers in each exercise below.  Be sure to simplify your result, if necessary.  Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

Note: To write the fraction three-fourths, enter 3/4 into the form. To write the mixed number four and two-thirds, enter 4, a space, and then 2/3 into the form.

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Fraction Addition Word Problems Worksheets

• Pre-Algebra >
• Fractions >
• Addition >

Look to your laurels by answering our pdf fraction addition word problems worksheets, a collage of well-researched real-world word problems. Adding fractions or mixed numbers is not an alien concept. Whether it's cooking recipes; measuring lengths, weights, etc.; or sharing something among many, fraction addition and mixed-number addition are never too far away. Witness adding fractions and mixed numbers with like and unlike denominators leap into life as you solve this collection of word problems! Begin your learning journey with some of our free worksheets!

Adding Like Fractions

Pump up your practice with a pleasant potpourri of everyday situations in these adding fractions word problems worksheets for 3rd grade, 4th grade, and 5th grade. Keep at it, and summing up two like fractions will soon be a cakewalk!

Adding Unlike Fractions

An eclectic collection of word problems centering around fractions with unlike denominators, this pdf resource proves an imperative addition to your repertoire! Find equivalent like fractions and whizz through the problems!

Adding Fractions with Whole Numbers

Are you a novice wondering how to add fractions to whole numbers? Take a look at the real-life scenarios in these pdf worksheets and say goodbye to all your doubts! Put the numbers together as mixed numbers, and that's your sum.

Adding Mixed Numbers | Same Denominators

Natasha brewed a 1 1/2-ounce shot of espresso for latte and another 1 1/2-ounce shot for Americano. How much coffee did Natasha make in all? 3 shots! Keen to be explored in our printable set are a wealth of such situations!

Adding Mixed Numbers | Different Denominators

Evaluate 5th grade and 6th grade students' skills in adding mixed numbers with different denominators in this part of the fraction addition word problems worksheets. Convert to mixed numbers with the same denominators, and press on!

Related Printable Worksheets

▶ Adding Like Fractions

▶ Adding Unlike Fractions

▶ Fraction Word Problems

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Fractions  - Adding and Subtracting Fractions

Fractions  -, adding and subtracting fractions, fractions adding and subtracting fractions.

Fractions: Adding and Subtracting Fractions

Lesson 3: adding and subtracting fractions.

/en/fractions/comparing-and-reducing-fractions/content/

Adding and subtracting fractions

In the previous lessons, you learned that a fraction is part of a whole. Fractions show how much you have of something, like 1/2 of a tank of gas or 1/3 of a cup of water.

In real life, you might need to add or subtract fractions. For example, have you ever walked 1/2 of a mile to work and then walked another 1/2 mile back? Or drained 1/4 of a quart of gas from a gas tank that had 3/4 of a quart in it? You probably didn't think about it at the time, but these are examples of adding and subtracting fractions.

Click through the slideshow to learn how to set up addition and subtraction problems with fractions.

Let's imagine that a cake recipe tells you to add 3/5 of a cup of oil to the batter.

You also need 1/5 of a cup of oil to grease the pan. To see how much oil you'll need total, you can add these fractions together.

When you add fractions, you just add the top numbers, or numerators .

That's because the bottom numbers, or denominators , show how many parts would make a whole.

We don't want to change how many parts make a whole cup ( 5 ). We just want to find out how many parts we need total.

So we only need to add the numerators of our fractions.

We can stack the fractions so the numerators are lined up. This will make it easier to add them.

And that's all we have to do to set up an addition example with fractions. Our fractions are now ready to be added.

We'll do the same thing to set up a subtraction example. Let's say you had 3/4 of a tank of gas when you got to work.

If you use 1/4 of a tank to drive home, how much will you have left? We can subtract these fractions to find out.

Just like when we added, we'll stack our fractions to keep the numerators lined up.

This is because we want to subtract 1 part from 3 parts.

Now that our example is set up, we're ready to subtract!

Try setting up these addition and subtraction problems with fractions. Don't try solving them yet!

You run 4/10 of a mile in the morning. Later, you run for 3/10 of a mile.

You had 7/8 of a stick of butter and used 2/8 of the stick while cooking dinner.

Your gas tank is 2/5 full, and you put in another 2/5 of a tank.

Solving addition problems with fractions

Now that we know how to write addition problems with fractions, let's practice solving a few. If you can add whole numbers , you're ready to add fractions.

Click through the slideshow to learn how to add fractions.

Let's continue with our previous example and add these fractions: 3/5 of cup of oil and 1/5 of a cup of oil.

Remember, when we add fractions, we don't add the denominators.

This is because we're finding how many parts we need total. The numerators show the parts we need, so we'll add 3 and 1 .

3 plus 1 equals 4 . Make sure to line up the 4 with the numbers you just added.

The denominators will stay the same, so we'll write 5 on the bottom of our new fraction.

3/5 plus 1/5 equals 4/5 . So you'll need 4/5 of a cup of oil total to make your cake.

Let's try another example: 7/10 plus 2/10 .

Just like before, we're only going to add the numerators. In this example, the numerators are 7 and 2 .

7 plus 2 equals 9 , so we'll write that to the right of the numerators.

Just like in our earlier example, the denominator stays the same.

So 7/10 plus 2/10 equals 9/10 .

Try solving some of the addition problems below.

Solving subtraction problems with fractions

Subtracting fractions is a lot like regular subtraction. If you can subtract whole numbers , you can subtract fractions too!

Click through the slideshow to learn how to subtract fractions.

Let's use our earlier example and subtract 1/4 of a tank of gas from 3/4 of a tank.

Just like in addition, we're not going to change the denominators.

We don't want to change how many parts make a whole tank of gas. We just want to know how many parts we'll have left.

We'll start by subtracting the numerators. 3 minus 1 equals 2 , so we'll write 2 to the right of the numerators.

Just like when we added, the denominator of our answer will be the same as the other denominators.

So 3/4 minus 1/4 equals 2/4 . You'll have 2/4 of a tank of gas left when you get home.

Let's try solving another problem: 5/6 minus 3/6 .

We'll start by subtracting the numerators.

5 minus 3 equals 2 . So we'll put a 2 to the right of the numerators.

As usual, the denominator stays the same.

So 5/6 minus 3/6 equals 2/6 .

Try solving some of the subtraction problems below.

After you add or subtract fractions, you may sometimes have a fraction that can be reduced to a simpler fraction. As you learned in Comparing and Reducing Fractions , it's always best to reduce a fraction to its simplest form when you can. For example, 1/4 plus 1/4 equals 2/4 . Because 2 and 4 can both be divided 2 , we can reduce 2/4 to 1/2 .

Adding fractions with different denominators

On the last page, we learned how to add fractions that have the same denominator, like 1/4 and 3/4 . But what if you needed to add fractions with different denominators? For example, our cake recipe might say to blend 1/4 cup of milk in slowly and then dump in another 1/3 of a cup.

In Comparing and Reducing Fractions , we compared fractions with a different bottom number, or denominator. We had to change the fractions so their denominators were the same. To do that, we found the lowest common denominator , or LCD .

We can only add or subtract fractions if they have the same denominators. So we'll need to find the lowest common denominator before we add or subtract these fractions. Once the fractions have the same denominator, we can add or subtract as usual.

Click through the slideshow to learn how to add fractions with different denominators.

Let's add 1/4 and 1/3 .

Before we can add these fractions, we'll need to change them so they have the same denominator .

To do that, we'll have to find the LCD , or lowest common denominator, of 4 and 3 .

It looks like 12 is the smallest number that can be divided by both 3 and 4, so 12 is our LCD .

Since 12 is the LCD, it will be the new denominator for our fractions.

Now we'll change the numerators of the fractions, just like we changed the denominators.

First, let's look at the fraction on the left: 1/4 .

To change 4 into 12 , we multiplied it by 3 .

Since the denominator was multiplied by 3 , we'll also multiply the numerator by 3 .

1 times 3 equals 3 .

1/4 is equal to 3/12 .

Now let's look at the fraction on the right: 1/3 . We changed its denominator to 12 as well.

Our old denominator was 3 . We multiplied it by 4 to get 12.

We'll also multiply the numerator by 4 . 1 times 4 equals 4 .

So 1/3 is equal to 4/12 .

Now that our fractions have the same denominator, we can add them like we normally do.

3 plus 4 equals 7 . As usual, the denominator stays the same. So 3/12 plus 4/12 equals 7/12 .

Try solving the addition problems below.

Subtracting fractions with different denominators

We just saw that fractions can only be added when they have the same denominator. The same thing is true when we're subtracting fractions. Before we can subtract, we'll have to change our fractions so they have the same denominator.

Click through the slideshow to learn how to subtract fractions with different denominators.

Let's try subtracting 1/3 from 3/5 .

First, we'll change the denominators of both fractions to be the same by finding the lowest common denominator .

It looks like 15 is the smallest number that can be divided evenly by 3 and 5 , so 15 is our LCD.

Now we'll change our first fraction. To change the denominator to 15 , we'll multiply the denominator and the numerator by 3 .

5 times 3 equals 15 . So our fraction is now 9/15 .

Now let's change the second fraction. To change the denominator to 15 , we'll multiply both numbers by 5 to get 5/15 .

Now that our fractions have the same denominator, we can subtract like we normally do.

9 minus 5 equals 4 . As always, the denominator stays the same. So 9/15 minus 5/15 equals 4/15 .

Try solving the subtraction problems below.

Adding and subtracting mixed numbers

Over the last few pages, you've practiced adding and subtracting different kinds of fractions. But some problems will need one extra step. For example, can you add the fractions below?

In Introduction to Fractions , you learned about mixed numbers . A mixed number has both a fraction and a whole number . An example is 2 1/2 , or two-and-a-half . Another way to write this would be 5/2 , or five-halves . These two numbers look different, but they're actually the same.

5/2 is an improper fraction . This just means the top number is larger than the bottom number. Even though improper fractions look strange, you can add and subtract them just like normal fractions. Mixed numbers aren't easy to add, so you'll have to convert them into improper fractions first.

Let's add these two mixed numbers: 2 3/5 and 1 3/5 .

We'll need to convert these mixed numbers to improper fractions. Let's start with 2 3/5 .

As you learned in Lesson 2 , we'll multiply the whole number, 2 , by the bottom number, 5 .

2 times 5 equals 10 .

Now, let's add 10 to the numerator, 3 .

10 + 3 equals 13 .

Just like when you add fractions, the denominator stays the same. Our improper fraction is 13/5 .

Now we'll need to convert our second mixed number: 1 3/5 .

First, we'll multiply the whole number by the denominator. 1 x 5 = 5 .

Next, we'll add 5 to the numerators. 5 + 3 = 8 .

Just like last time, the denominator remains the same. So we've changed 1 3/5 to 8/5 .

Now that we've changed our mixed numbers to improper fractions, we can add like we normally do.

13 plus 8 equals 21 . As usual, the denominator will stay the same. So 13/5 + 8/5 = 21/5 .

Because we started with a mixed number, let's convert this improper fraction back into a mixed number.

As you learned in the previous lesson , divide the top number by the bottom number. 21 divided by 5 equals 4, with a remainder of 1 .

The answer, 4, will become our whole number.

And the remainder , 1, will become the numerator of the fraction.

So 2 3/5 + 1 3/5 = 4 1/5 .

/en/fractions/multiplying-and-dividing-fractions/content/

Fraction Word Problems: Addition, Subtraction, and Mixed Numbers

In today’s post, we’re going to see how to solve some of the problems that we’ve introduced in Smartick: fraction word problems. They appear during the word problems section at the end of the daily session.

We’re going to look at how to solve problems involving addition and subtraction of fractions, including mixed fractions (the ones that are made up of a whole number and a fraction).

Try and solve the fraction word problems by yourself first, before you look for the solutions and their respective explanations below.

Fraction Word Problems

Problem nº 1.

Problem nº 2

Problem nº 3

Solution to Problem nº 1

This is an example of a problem involving the addition of a whole number and a fraction.

The simplest way to show the number of cookies I ate is to write it as a mixed number. And the data given in the word problem gives us the result: 9 biscuits and 5 / 6 of a biscuit = 9 5 / 6 biscuits.

Solution to Problem nº 2

In this example, we have to subtract two fractions with the same denominator.

To calculate how full the gas tank is, we have to subtract both fractions. Since we are given fractions, the best way to present the solution is in the form of a fraction. Additionally, we’re dealing with two fractions with the same denominator, so we just have to subtract the numerators of both fractions to get the result. 8 / 10 – 4 / 10 = 4 / 10

Solution to Problem nº 3

This problem requires us to subtract a mixed number and a fraction.

To solve this problem, we need to subtract the number of episodes that were downloaded this morning from the total number of episodes that are now downloaded.

To do this, we need to change the mixed number into a fraction: the 5 becomes 60 / 12 (5 x 12 = 60) and we add it to the fraction 60 / 12 + 8 / 12 = 68 / 12 .

We’ve converted the mixed number 5 8 / 12 to 68 / 12 . Now we just have to subtract the number of episodes that were downloaded yesterday ( 7 / 12 ),   68 / 12 – 7 / 12 = 61 / 12 .

Hopefully, you didn’t need the explanations and were able to solve them yourself without any help!

Fraction Video Tutorials

In the following video tutorials, you can learn a bit more about fractions. And if you would like to learn more math concepts, check out Smartick’s Youtube channel !

Simplifying Fractions

Simplification Using the GCD

Equivalent Fractions

If you would like to practice more fraction word problems like these and others, log in to Smartick and enjoy learning math.

Learn More:

• Word Problems with Fractions
• What Is a Fraction? Learn Everything There Is to Know!
• Using Mixed Numbers to Represent Improper Fractions
• Learning How to Subtract Fractions
• Learn How to Subtract Fractions
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Fraction Word Problem Worksheets

Featured here is a vast collection of fraction word problems, which require learners to simplify fractions, add like and unlike fractions; subtract like and unlike fractions; multiply and divide fractions. The fraction word problems include proper fraction, improper fraction, and mixed numbers. Solve each word problem and scroll down each printable worksheet to verify your solutions using the answer key provided. Thumb through some of these word problem worksheets for free!

Represent and Simplify the Fractions: Type 1

Presented here are the fraction pdf worksheets based on real-life scenarios. Read the basic fraction word problems, write the correct fraction and reduce your answer to the simplest form.

• Download the set

Represent and Simplify the Fractions: Type 2

Before representing in fraction, children should perform addition or subtraction to solve these fraction word problems. Write your answer in the simplest form.

Adding Fractions Word Problems Worksheets

Conjure up a picture of how adding fractions plays a significant role in our day-to-day lives with the help of the real-life scenarios and circumstances presented as word problems here.

(15 Worksheets)

Subtracting Fractions Word Problems Worksheets

Crank up your skills with this set of printable worksheets on subtracting fractions word problems presenting real-world situations that involve fraction subtraction!

Multiplying Fractions Word Problems Worksheets

This set of printables is for the ardently active children! Explore the application of fraction multiplication and mixed-number multiplication in the real world with this exhilarating practice set.

Fraction Division Word Problems Worksheets

Gift children a broad view of the real-life application of dividing fractions! Let them divide fractions by whole numbers, divide 2 fractions, divide mixed numbers, and solve the word problems here.

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» Decimal Word Problems

» Ratio Word Problems

» Division Word Problems

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Adding & subtracting fractions word problems

Word problem worksheets: addition & subtraction of fractions.

Below are three versions of our grade 4 math worksheet on adding and subtracting fractions and mixed numbers.  All fractions have like denominators.  Some problems will include irrelevant data so that students have to read and understand the questions, rather than simply recognizing a pattern to the solutions.  These worksheets are pdf files .

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Fraction Worksheets

Conversion :: Addition :: Subtraction :: Multiplication :: Division

Conversions

Fractions - addition, fractions - subtraction, fractions - multiplication, fractions - division.

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.

Recommended Worksheets

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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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Adding and Subtracting Fraction Word Problems

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Addition Lesson: Strategies, Techniques, and Practical Applications

Lesson overview, introduction to addition lesson, what is addition, how is addition classified, what are the basic principles of addition, what are the important addition rules, how do you add fractions, how do you add decimals, how do you add powers, how do you add reciprocal numbers, what are the properties of addition, how is addition used in real-world problems.

Addition is one of the four fundamental arithmetic operations, alongside subtraction, multiplication, and division. It forms the basis of many mathematical concepts and is essential for understanding how numbers interact. This addition lesson will guide you through the fundamental principles of addition, starting with the basics and progressing to more advanced techniques. We will cover addition with whole numbers, fractions, decimals, powers, and reciprocal numbers, ensuring you develop a comprehensive understanding of each concept.

Mastering addition is crucial for both academic success and practical applications in everyday life. From calculating totals while shopping to solving complex mathematical problems, addition is a skill that is frequently used. In this lesson, we will explore various addition strategies, techniques, and tips to help you solve addition problems efficiently and accurately.

Addition is a basic arithmetic operation that involves combining two or more numbers to find their total. This operation is essential for understanding relationships between quantities and is widely used in various mathematical and real-life applications. In addition, the numbers being combined are called addends, and the result is known as the sum.For example, in the equation, 4 + 3 = 7, 4 and 3 are the addends, and 7 is the sum. Understanding the basics of addition is crucial for tackling more complex problems and applying addition techniques in everyday situations. A solid grasp of addition forms the foundation for learning other mathematical operations and enhances overall mathematical skills.

Addition can be classified into different categories based on the types of numbers being added and the methods used. Here's how addition is classified

• Addition of Whole Numbers Addition of whole numbers is the most basic form of addition, involving positive integers. This is often the first type of addition learned in early mathematics education. Example: 5 + 7 = 12
• Addition of Fractions When adding fractions, it is important to have a common denominator. This classification deals with combining parts of whole numbers, focusing on ensuring that the fractions have a shared base before performing the addition.

                                             Example: 1/2+​1/3 = 3/6+2/6  = 5/6

• Addition of Decimals Addition of decimals requires aligning the decimal points and then adding the numbers as with whole numbers. This classification handles numbers with fractional parts expressed in decimal form.                                 Example: 3.75 + 2.5 = 6.25
• Addition of Powers Adding powers typically involves combining terms with the same base and exponent. If the exponents or bases differ, the addition is handled differently, usually resulting in an expression that cannot be simplified further.                                         Example : 4x 2 +3x 2 =7x 2
• Addition of Reciprocal Numbers This classification deals with adding numbers that are inverses of each other. The addition of reciprocal numbers often involves finding a common denominator and then performing the addition.

                                        Example: 1/4 + 1/6 =  3/12 + 2/12 =  5/12

The basic principles of addition provide the foundation for understanding how this operation works in various mathematical contexts. Here are the detailed explanations of these principles

• Commutative Property The commutative property of addition states that changing the order of the addends does not change the sum. For example, 3 + 4 = 4 + 3 = 7. This property shows that addition can be performed in any order, making calculations flexible and straightforward.
• Associative Property The associative property of addition indicates that when adding three or more numbers, the way the numbers are grouped does not affect the sum. For example, (2 + 3) + 4 = 2 + (3 + 4) = 9. This property is useful in simplifying complex addition problems by grouping numbers in a way that makes calculation easier.
• Identity Property The identity property of addition states that adding zero to any number leaves the number unchanged. For example, 5 + 0 = 5. This property highlights the role of zero as the identity element in addition, maintaining the original value of the addend.
• Additive Inverse The additive inverse principle states that for every number, there exists another number that when added together, equals zero. For example, 7 + (-7) = 0. This principle is essential for understanding subtraction as the inverse of addition.
• Distributive Property The distributive property of addition over multiplication states that a(b + c) = ab + ac. This property is particularly useful in algebra and helps in simplifying expressions and solving equations.

Addition has several important rules that help in understanding and performing the operation correctly. Here are the detailed explanations of these rules

• Order of Operations (PEMDAS/BODMAS) When addition is combined with other operations such as multiplication or division, the order of operations must be followed. Addition is typically performed after multiplication or division unless parentheses indicate otherwise.
• Carrying Over In addition involving multi-digit numbers, when the sum of digits in a column exceeds 9, the extra value is carried over to the next column. For example, in adding 27 + 35, the sum of the units column (7 + 5 = 12) requires carrying over 1 to the tens column, resulting in 62.
• Handling Negative Numbers When adding negative numbers, the process is similar to adding positive numbers, but with special attention to the signs. For example, adding -3 + 4 is equivalent to subtracting 3 from 4, resulting in 1.
• Decimal Alignment When adding decimals, it is crucial to align the decimal points to ensure that digits are added in the correct place value columns. This alignment prevents errors and ensures accurate results.
• Fraction Addition Adding fractions requires finding a common denominator before adding the numerators. This ensures that the fractions represent comparable quantities, making the addition accurate.

Take These Quizzes

Adding fractions involves several steps to ensure accuracy. Here is a detailed explanation along with three examples to illustrate the process

• Find a Common Denominator Ensure both fractions have the same denominator. If they do not, find the least common denominator (LCD) and adjust the fractions accordingly.
• Add the Numerators Once the fractions have the same denominator, add the numerators while keeping the denominator the same.
• Simplify Reduce the fraction to its simplest form, if possible, by dividing both the numerator and the denominator by their greatest common divisor (GCD).

Example 1 Adding Fractions with the Same Denominator

3/8+4/8 = 7/8

The fractions already have the same denominator (8).

Add the numerators 3 + 4 = 7

The result is 7/8

Example 2 Adding Fractions with Different Denominators

            3/4+1/6

• Find a common denominator. The least common denominator of 4 and 6 is 12.
• Adjust the fractions 3/4= 9/12 (by multiplying the numerator and denominator by 3).
• Adjust the fractions 1/6= 2/12 by multiplying the numerator and denominator by 2).
• Add the numerators 9 + 2 = 11.

The result is 11/12,  which is already in its simplest form.

Example 3 Adding Mixed Numbers

2 1/3+ 1 1/2

• Convert the mixed numbers to improper fractions.

            2 1/3 = 2 X 3 + 1/3 = 7/3

            1 1/2 = 1 X 2 + 1/2 = 3/2

• Find a common denominator. The least common denominator of 3 and 2 is 6.
• Adjust the fractions 7/3 = 14/6 (by multiplying the numerator and denominator by 2).
• Adjust the fractions 3/2 = 9/6 (by multiplying the numerator and denominator by 3).
• Subtract the numerators 14+9 = 23.
• The result is 23/6 

Adding decimals follows a similar process to adding whole numbers, with careful attention to the placement of the decimal point. Here is a detailed explanation along with three examples

• Align the Decimal Points Ensure the decimal points of both numbers are aligned vertically. This helps to correctly place the digits in the right columns.
• Add Normally Perform the addition as with whole numbers, starting from the rightmost digit and moving leftward.
• Place the Decimal Point Ensure the decimal point in the result is aligned with the decimal points of the numbers being added.

Example 1 Adding Simple Decimals To add 4.25 and 3.6 Align the decimal points: 4.25 +3.60 Add normally 7.85 The decimal point in the result is aligned with the numbers being added.

Example 2 Adding Decimals with Different Numbers of Decimal Places To add 5.7 and 2.345 Align the decimal points and add zeros as necessary   5.700 +2.345 Add normally 8.045 The decimal point in the result is aligned with the numbers being added.

Example 3 Adding Larger Decimals To add 123.456 and 78.123 Align the decimal points: 123.456 +78.123 Add normally 201.579 The decimal point in the result is aligned with the numbers being added.

Adding powers involves specific rules when the powers share the same base and exponent. Here's a detailed explanation along with three examples

• Ensure Same Exponents To add powers, the terms must have the same base and exponent. This allows for the coefficients to be added while keeping the base and exponent unchanged.
• Add the Coefficients When the exponents are the same, add the coefficients of the terms.
• Maintain the Base and Exponent The base and exponent of the terms remain unchanged after adding the coefficients.

Example 1 Adding Powers with the Same Base and Exponent To add 3x 2 and 5x 2 Ensure the exponents are the same. Here, both terms have the base x and exponent 2. Add the coefficients 3 + 5 = 8. Maintain the base and exponent 8x 2 .

Example 2 Adding Larger Powers with the Same Base and Exponent To add 6y 3 and 2y 3 Ensure the exponents are the same. Both terms have the base y and exponent 3. Add the coefficients 6 + 2 = 8. Maintain the base and exponent: 8y 3 .

Example 3 Adding Negative Powers with the Same Base and Exponent To add −4z 4 and −3z 4 Ensure the exponents are the same. Both terms have the base z and exponent 4. Add the coefficients -4 + (-3) = -7. Maintain the base and exponent −7z 4 .

Adding reciprocal numbers involves a few key steps to ensure the process is done correctly. Here is a detailed explanation along with three examples to illustrate the process

• Find the Common Denominator Make the denominators of the fractions the same. This allows for straightforward addition of the numerators.
• Add the Numerators Once the fractions have the same denominator, add the numerators.

Example 1 Adding Simple Reciprocal Numbers

Add 1/2 and 1/3

• 3/6 + 2/6 = 5/6
• Final Answer 1/2 + 1/3 = 5/6

Example 2 Adding Reciprocal Numbers with Different Denominators

Add 1/4 and 1/5

• 5/20 + 4/20 = 9/20
• Final Answer 1/4 + 1/5 = 9/20

Example 3 Adding Larger Reciprocal Numbers

Add 1/6 and 1/8

• 4/24 + 3/24 = 7/24
• Final Answer 1/6 + 1/8 = 7/24

Addition has several important properties that help us understand and perform this operation correctly. Here are the detailed explanations of these properties

• Both equations result in 8.

This property shows that the order in which you add numbers doesn't affect the result, making calculations flexible.

• (2+3)+4=2+(3+4)
• Both groupings result in 9.

This property is useful when simplifying addition problems by grouping numbers in a way that makes the calculation easier.

• The result is 7.

This property highlights the role of zero as the identity element in addition, maintaining the original value of the addend.

• The result is 0.

This property is essential for understanding subtraction as the inverse operation of addition.

• 2×(3+4)=(2×3)+(2×4)
• Both sides result in 14.

This property is particularly useful in algebra for simplifying expressions and solving equations.

Addition is a fundamental operation that plays a critical role in many real-world scenarios. Here are detailed explanations and examples of how addition is applied in different contexts

• Suppose you spent $45 on groceries, $30 on fuel, and $25 on dining out. To find the total amount spent, you add these expenses together:
• $45 + $30 + $25 = $100
• The total amount spent is $100.

Example Balancing a Checkbook

• If you start with a balance of $1,200 and deposit $500, you add the deposit to your balance to find the new total
• $1,200 + $500 = $1,700
• Your new balance is $1,700.
• If a recipe requires 2 cups of flour and you add another 1/2 cup, you calculate the total amount of flour used by adding the quantities together
• 2 cups + 1/2 cup = 2 1/2 cups
• You now have a total of 2 1/2 cups of flour.

Example Calculating Total Distance Traveled

• If you drive 60 miles to visit a friend and then another 30 miles to reach your final destination, the total distance traveled is:
• 60 miles + 30 miles = 90 miles
• You traveled a total of 90 miles.
• A store starts with 150 units of a product. After receiving a shipment of 50 more units, the new inventory level is calculated by adding the shipment to the existing stock:
• 150 units + 50 units = 200 units
• The store now has 200 units in stock.

Example Tracking Sales and Stock Levels

• If a store sells 30 units of a product on Monday and 40 units on Tuesday, the total number of units sold is
• 30 units + 40 units = 70 units
• The store sold 70 units over two days.
• If you worked 8 hours on Monday and 7 hours on Tuesday, the total number of hours worked is
• 8 hours + 7 hours = 15 hours
• You worked a total of 15 hours over those two days.

Example Scheduling Appointments

• If an appointment starts at 2:00 PM and lasts for 45 minutes, you add the duration to the start time to find the end time:
• 2:00 PM + 45 minutes = 2:45 PM
• The appointment ends at 2:45 PM.
• If a student scores 85 on a math test, 90 on a science test, and 88 on an English test, the total score across these subjects is:
• 85 + 90 + 88 = 263
• The student's total score is 263.

Example Analyzing Survey Results

• If a survey receives responses from 120 participants in one week and 150 participants in the next week, the total number of responses is
• 120 + 150 = 270
• The survey received a total of 270 responses over the two weeks.

Great job on completing this addition lesson! You've now gained a comprehensive understanding of addition, one of the most essential mathematical operations. This lesson has taken you through the key concepts of addition, from the basics of combining numbers to more advanced techniques like adding fractions, decimals, and powers.

We explored the core properties of addition, such as commutativity, associativity, and the identity property, which are foundational to solving problems efficiently. You've also seen how addition plays a vital role in real-life situations, whether it's managing finances, calculating time, or analyzing data. With the knowledge and strategies you've acquired, you're now equipped to apply addition confidently in various contexts, making it a valuable skill both in and out of the classroom.

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