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Fractions: mixed operations
Fraction word problems with the 4 operations.
These word problems involve the 4 basic operations ( addition, subtraction, multiplication and division ) on fractions . Mixing word problems encourages students to read and think about the questions, rather than simply recognizing a pattern to the solutions.
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Word Problems with Fractions
Today we are going to look at some examples of word problems with fractions.
Although they may seem more difficult, in reality, word problems involving fractions are just as easy as those involving whole numbers. The only thing we have to do is:
- Read the problem carefully.
- Think about what it is asking us to do.
- Think about the information we need.
- Simplify, if necessary.
- Think about whether our solution makes sense (in order to check it).
As you can see, the only difference in fraction word problems is step 5 (simplify) .
There are some word problems which, depending on the information provided, we should express as a fraction. For example:
In my fruit basket, there are 13 pieces of fruit, 5 of which are apples.
How can we express the number of apples as a fraction?
5 – The number of apples (5) corresponds to the numerator (the number which expresses the number of parts that we wish to represent).
13 – The total number of fruits (13) corresponds to the denominator (the number which expresses the number of total possible parts).
The solution to this problem is an irreducible fraction (a fraction which cannot be simplified). Therefore, there is nothing left to do.
Word problems with fractions: involving two fractions
In these problems, we should remember how to carry out operations with fractions.
Carefully read the following problem and the steps we have taken to solve it:
What fraction of the payment has Maria spent?
We find the common denominator:
We calculate:
Word problems with fractions: involving a fraction and a whole number
Finally, we are going to look at an example of a word problem with a fraction and a whole number. Now we will have to convert all the information into a fraction with the same denominator (as we did in the example above) in order to calculate
We convert 1 into a fraction with the same denominator:
What do you think of this post? Do you see how easy it is to solve word problems with fractions?
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- Fraction Word Problems: Addition, Subtraction, and Mixed Numbers
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- Review and Practice the Two Methods of Dividing Fractions
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29 Comments
Thank you for such good explanations, it helped me a lot
It is really good it helped me improve my math a lot.
same it helps me in my math too
Wow, it really helps a lot
Good exercises
Interesting
wow it worked
Hi can you not show the answer till the bottom of the page or your giving away the answer so if you solved number one problem the number one aware to the question will be there at the bottom of the page because it is way to easy if it is right there
I like that you are doing for as Thank you
I really want to be part of this
wow, this help me a lot
A big help for my kids lesson
Thank you for all the homework you have given us. God bless you
Thank you for this problems that involved fractions
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I like it… but you can level up please 🙄
Roll two dices, the first dice is the numerator, the second is the denominator, this is the first fraction. Roll both dices again and repeat the process to generate the second fraction. Write a division story problem that incorporates these two fractions.
Seems easy of the examples but when I have fraction word promblems in front of me then its still hard for me to figure it out.The examples on this site still is helpful.I will use the site that you give on here to get further practice.Thank you for the examples on here
Interesting and very helpful. I’m going to continue using this site and tell others about it too.
I really like it
Hey I am in grade five and it is super helpful for my exams thanks and maybe if you could make more it would be appriciated thx 🙂
Good efforts
i kinda like it pls write some more problems
I think it was really good how you are helping fellow students! But I think you can improve if there were more problems for solving! Thanks
it is helpfull
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Fraction Word Problems (Difficult)
Here are some examples of more difficult fraction word problems. We will illustrate how block models (tape diagrams) can be used to help you to visualize the fraction word problems in terms of the information given and the data that needs to be found.
Related Pages Fraction Word Problems Singapore Math Lessons Fraction Problems Using Algebra Algebra Word Problems
Block modeling (also known as tape diagrams or bar models) are widely used in Singapore Math and the Common Core to help students visualize and understand math word problems.
Example: 2/9 of the people on a restaurant are adults. If there are 95 more children than adults, how many children are there in the restaurant?
Solution: Draw a diagram with 9 equal parts: 2 parts to represent the adults and 7 parts to represent the children.
5 units = 95 1 unit = 95 ÷ 5 = 19 7 units = 7 × 19 = 133
Answer: There are 133 children in the restaurant.
Example: Gary and Henry brought an equal amount of money for shopping. Gary spent $95 and Henry spent $350. After that Henry had 4/7 of what Gary had left. How much money did Gary have left after shopping?
350 – 95 = 255 3 units = 255 1 unit = 255 ÷ 3 = 85 7 units = 85 × 7 = 595
Answer: Gary has $595 after shopping.
Example: 1/9 of the shirts sold at Peter’s shop are striped. 5/8 of the remainder are printed. The rest of the shirts are plain colored shirts. If Peter’s shop has 81 plain colored shirts, how many more printed shirts than plain colored shirts does the shop have?
Solution: Draw a diagram with 9 parts. One part represents striped shirts. Out of the remaining 8 parts: 5 parts represent the printed shirts and 3 parts represent plain colored shirts.
3 units = 81 1 unit = 81 ÷ 3 = 27 Printed shirts have 2 parts more than plain shirts. 2 units = 27 × 2 = 54
Answer: Peter’s shop has 54 more printed colored shirts than plain shirts.
Solve a problem involving fractions of fractions and fractions of remaining parts
Example: 1/4 of my trail mix recipe is raisins and the rest is nuts. 3/5 of the nuts are peanuts and the rest are almonds. What fraction of my trail mix is almonds?
How to solve fraction word problem that involves addition, subtraction and multiplication using a tape diagram or block model
Example: Jenny’s mom says she has an hour before it’s bedtime. Jenny spends 3/5 of the hour texting a friend and 3/8 of the remaining time brushing her teeth and putting on her pajamas. She spends the rest of the time reading her book. How long did Jenny read?
How to solve a four step fraction word problem using tape diagrams?
Example: In an auditorium, 1/6 of the students are fifth graders, 1/3 are fourth graders, and 1/4 of the remaining students are second graders. If there are 96 students in the auditorium, how many second graders are there?
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What to Do When Students Struggle with Fraction Operations
Are you smarter than a 5th grader? When it comes to fraction operations, for many adults, the answer is ‘no.’
If you don’t teach math, you might be surprised at how much we expect of 5th graders. Here’s a sample grade 5 test question from the Louisiana Department of Education.
I don’t know about you, but this problem had me scratching my head. I’m pretty sure the answer is D. But I certainly couldn’t have solved this when I was 10.
So if your students struggle with fraction operations, you’re not alone. Many rely on tricks like keep, change, flip, but never develop conceptual understanding.
These students have a hard time applying their understanding to real world problems. And they forget what they learn much more quickly.
Ensuring students are fluent in fractions is essential for their overall success in math. Fraction understanding provides the basis for decimals, percents, and ratios. They also allow students to understand proportions, slope, and rational numbers.
When students don’t master fractions in elementary school, they struggle through middle school, Algebra, Statistics, and even Calculus (if they get there).
To help your students, first you’ll want to know why they’re struggling with fraction operations. Then, be sure to have the tools and strategies to address their misconceptions.
Why Do Students Struggle with Fraction Operations?
An over-reliance on “tricks” isn’t unique to fraction operations. Throughout K-12 education (and beyond), students often learn to calculate without learning to reason.
So why is the problem felt most acutely with fractions? Part of the reason is that fractions are harder to connect to their lived experiences.
Fractions also require that students build upon their prior knowledge with whole number operations. So when students lack those foundations, fractions are twice as hard (or ½ as easy).
The actual answer is different for each student. But here are the common pitfalls to look out for.
Fractions Aren’t Intuitive
One reason students struggle with fraction operations is that fractions are just less intuitive than whole numbers. Students constantly add and subtract whole numbers in their everyday lives, even without realizing it. On occasion, they even multiply and divide.
And while they know what it means to eat half a cookie, or measure ½ cup of flour, they don’t usually operate with fractions. Many think of a ‘half’ as just a half , not as a number with one in the numerator and two in the denominator.
Many Students Lack Foundational Concepts
Another reason students struggle is that they haven’t mastered foundational concepts. Namely, the meaning of fractions, and whole number operations.
If you ask your students to multiply 3 by 13, do they envision an array ? An area model? Skip count by 10 three times, then skip count by 3 three times? Or do they just stack and calculate?
Students who can visualize whole number operations can build on their understanding of division to imagine a whole cut into four parts. They can extend that idea to add one fourth to another fourth. Or to multiply a fifth by three .
But since many students lack these conceptual foundations, they just see fractions as one number stacked atop another. The more tricks and procedures they learn, the more likely they are to forget them or use the wrong trick for the wrong problem.
But there’s an even bigger problem with algorithms sans concepts . Students don’t develop the ability to see the math that is all around them. They don’t recognize math as a language , or a study of space and quantity. They can’t see the beauty and the art within mathematics .
Instead, it becomes a collection of “ math facts ,” random rules, and secret symbols.
Fraction Operations Contradict Whole Number Rules
Even students who understand whole number concepts may struggle with fractions.
In early elementary, students focus on mastering the Base-10 system . They count by tens, combine tens and ones, and break up numbers by place value to operate. It’s all about making tens, breaking tens, and using zero as a placeholder.
But fractions aren’t built on tens. Fractions can split a whole into 3rds, 7ths, or 45ths. And as the denominator changes, the size of the “unit” changes. As does the units needed to make a group (one whole).
Fractions also contradict the ‘rules’ students learn for whole numbers. When you multiply 3 x 5, the result is larger than both. But when multiplying fractions, you never know. The product of ⅓ and ¼ is smaller than both. But multiplying ⅕ by 4 creates a product in-between the two factors.
When students have strong conceptual foundations, they can generalize what they know about whole numbers to understand fractions. But without this understanding, they might feel like they’re starting over, with a whole new set of rules!
Three Tips for Teaching Fraction Operations
The first step to helping your students with fraction operations is to learn why they struggle .
The next is to have specific, actionable steps to address those challenges.
First, make sure your students really understand the meaning of a fraction. Then, connect that to what they already know about whole number operations.
Both of these can be accomplished with the The Three Vehicles — inquiry-based lesson models that can support almost any math concept.
1. Review and Assess Fraction Foundations
Simply put, a denominator tells us how many pieces a whole is divided into. A numerator tells us how many of those pieces we have. Though the numerator is on top , it doesn’t mean much unless you know the denominator. That’s why I teach fractions from the bottom up.
At some point (usually around grade 5), students learn that a fraction can also be thought of as a quotient . The result of a numerator divided by denominator. So 3 ÷ 4 is the same as ¾. This can be shown visually as 3 wholes, each split into 4 equal parts, with the parts regrouped into a single fraction.
To assess their understanding, students must be able to do this without direction . If you have to tell them to cut or to shade, they don’t get the concept. They’re just drawing.
2. Connect Fraction Operations to Whole Number Operations
Once students understand fractions, they can operate with fractions the same way they operate with whole numbers. Assuming, of course, they understand the meaning of the operations.
Addition and Subtraction
When students understand the meaning of ⅓, they can begin counting up and counting down to add and subtract fractions with like denominators.
The next step is to make a whole, such as by counting up by 4ths or adding ⅓ to ⅔. Next, they can subtract to break a whole (1 – ⅙).
By counting up beyond a whole , they can begin to operate with mixed numbers (⅔ + ⅔ = 1 ⅓).
Adding and subtracting fractions with unlike denominators is more complicated. Students need to convert to equivalent fractions, which relies on skills developed by multiplying fractions. So students first learn those concepts before returning to addition and subtraction with unlike denominators.
Fractions Activities for Your Classroom
Converting Fractions to Percents| Interactive Digital Visual Models
Dividing Fractions Word Problem Activities – Complete Digital and Print Lesson
Fraction Essentials Bundle: e-Book and Digital Activities
Fraction Foundations Visual Models and Digital Manipulatives | Interactive Google Slides
Multiplying fractions.
Once your students can add fractions, it’s a simple step to multiply a fraction by a whole number. Just think of ¼ x 3 as ¼ + ¼ + ¼.
Next, replace the multiplication sign with the word ‘of.’ So ¼ x 3 becomes “one fourth of three.” This connects fraction multiplication to whole number multiplication. 3 x 2 means three (groups) of two. So it makes sense for ⅓ x 4 to be one third of (a group of) 4.
It also extends the concept from multiplication of a fraction to multiplication by a fraction, which allows students to solve when both factors are fractions.
This Google Slides Activity uses visual representations to demonstrate the meaning of fraction multiplication: students slide fraction models to see what happens when we multiply.
In addition to supporting conceptual understanding, fraction multiplication models show students why we multiply numerators and denominators in the algorithm.
For more on connecting fraction multiplication to whole numbers, review the five meanings of multiplication .
Dividing Fractions
Dividing fractions can also be anchored in what students know about whole numbers.
Start by dividing a fraction by its numerator, such as ⅗ divided by 3. This involves partitive division, in which the divisor determines the number of groups .
Next, divide a whole number by a fraction. Here we use quotative division, in which the divisor determines the size of each group . Dividing 2 by ⅓, involves splitting both wholes into thirds (divisor = group size), and counting the total groups (quotient = number of groups). This illustrates why we multiply by the denominator when dividing by a fraction.
Quotative division is also useful for dividing a fraction by a fraction, but only in some cases. To divide ⅔ by ⅓, simply create 2 groups of ⅓ each.
But what about ⅓ divided by ¼? It’s possible to imagine splitting a third into groups one fourth in size…but it’s not very intuitive. In this case, I return to partitive division and use what I call the “ghost copies” strategy.
If I were to divide 8 by 2 partitively, I am changing one group of 8 into two new groups, with four in each group. If, instead, I divide 8 by ½, I’m turning one group of 8 into one half of a group . To create one whole group, I have to make a ghost copy of my starting value (dividend).
The ghost copies strategy can be extended for quotients like ½ ÷ ⅓. Treat ½ as ⅓ of a complete group . Thus, we add two ghost copies of ½ to make one whole group, resulting in 1½.
The five meanings of multiplication is also useful for helping students with fraction division. Each meaning of multiplication, in reverse, applies to division.
Equivalent Fractions
Equivalent fractions may be the trickiest aspect of fraction operations. It’s hard to connect it to whole numbers, as there’s no other whole number equivalent: Eight is just eight, there’s no whole number equivalent to 8.
But a fraction like ⅓ can also be written as 2/6 or 10/30, and still have the same value. In fact, you could argue that equivalence is why we use fractions in the first place.
When students ask “why do we need both fractions and decimals,” a great answer is that fractions allow us to divide whole numbers into any size parts we want. With decimals, we are limited to factors of ten.
But I’m including equivalence here for two reasons. First, it’s critical to many later applications of fractions. Simplifying fractions and adding unlike denominators require converting equivalent fractions. As do decimal and percent conversions , working with proportions, and finding slope. The list goes on.
The second reason is that fraction multiplication can be used to teach equivalent fractions.
I teach fraction conversion as multiplying by one . To find an equivalent for ½, I can multiply by 3/3 (aka one), creating 3/6.
Students who can use area models to multiply fractions should connect the idea of multiplying by ⅓ to the idea of multiplying by 3/3. Both visually, and with the “multiply across” algorithm.
3. Use The Three Vehicles to Teach Fraction Operations
The Three Vehicles are inquiry-based lesson models that promote conceptual understanding. They can be used to teach almost any math concept, at any grade level.
The vehicles are built on multiple representations theory , the idea that any mathematical concept can be represented five different ways: physically, visually, symbolically, conceptually, and verbally.
We can define fluency as the ability to translate a mathematical idea among all five representations . This includes translating an expression into a visual model. Or explaining, in words, how a manipulative works.
The five representations define how mathematical ideas can be represented. And the vehicles teach students how to translate among them.
Scale Models
The first vehicle is the scale model . It combines physical and visual representations (also known as concrete and pictorial), as both serve a similar purpose. They represent the size of numbers and the meaning of operations.
Scale models are useful for building and assessing conceptual understanding. To use scale models as a teaching tool, first teach the principles of reading and creating scale models. Then, use the models as tools for helping students to learn through inquiry.
For example, if students understand fractions, and whole number multiplication, they can use their modeling skills to “discover” fraction multiplication, even without direct instruction.
It’s important for students to create their own models , rather than just interpreting models in the textbook or made by teachers. Reading models is helpful, but isn’t enough to achieve fluency.
Three Bridges Online Workshops for Educators
Elementary Word Problems: Online Workshop for Math Teachers
Online Workshop: How to Teach Word Problems in Middle School
Teaching With Manipulatives And Visual Models (Grades 1-5)
Visual Models and Math Manipulatives for Middle School
Number sentences.
The next vehicle is the number sentence . Number sentences (equations and inequalities) build on the concepts developed with visual models, to support development of abstract understanding.
Number sentences allow students to manipulate expressions and work in multiple steps, making them much more useful than calculating with algorithms.
I use number sentence proofs to help students learn number sentences and connect them to the other representations.
Story Problems
The third vehicle, story problems, help with application skills and mathematical language.
I teach students to use the Polya Process for solving word problems . This approach emphasizes the importance of using multiple representations to solve word problems.
This vehicle comes last, because students can use scale models and number sentences as tools for solving word problems.
Teaching Fraction Operations in Your Classroom
I hope this article offered a useful overview of the main concepts needed for success with fraction operations.
You can find resources for teaching fractions with all three vehicles in our online store .
Or level up your inquiry-based math skills by enrolling in an online workshop . These are real-time sessions, conducted by a live facilitator. We offer separate sessions on each vehicle for elementary and for middle school teachers, so you can focus on the techniques and standards that are most important to your students.
Finally, if you’d like to incorporate this type of learning right away, download our Fractions Essentials Bundle . It has everything you need to get started, from interactive Google Slides activities, to lesson plans, answer keys, and more!
Get Your Copy of FRACTION ESSENTIALS
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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.
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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COMMENTS
An explanation of how to choose a strategy to solve word problems. ... Grade 6 Math #4.9, Problem Solving - multi-step fraction operations.
Sometimes a word problem will involve more than one fraction operation, more than 1 step. We can use parentheses to group different
This (FUN) lesson solves fraction problems using a bar model.
The strategy work backward can help us solve a problem with fractions that involve addition and subtraction. We can write an equation to
These word problems involve the 4 basic operations (addition, subtraction, multiplication and division) on fractions. Mixing word problems encourages
Read the problem carefully. · Think about what it is asking us to do. · Think about the information we need. · Solve it. · Simplify, if necessary.
Fraction Word Problems - using block models (tape diagrams), Solve a problem involving fractions of fractions and fractions of remaining parts, how to solve
This vehicle comes last, because students can use scale models and number sentences as tools for solving word problems. Teaching Fraction Operations in Your
Analysis: To solve this problem, we will subtract two fractions with unlike denominators. Solution: word-example2.gif. Answer: Stefanie swam one-third of a lap
Step 1: Use keywords to write the equation representing the situation in the word problem. Step 2: Employ the order of operations to solve the word problem.