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Fraction word prob.

Fraction word problems

Here you will learn about fraction word problems, including solving math word problems within a real-world context involving adding fractions, subtracting fractions, multiplying fractions, and dividing fractions.

Students will first learn about fraction word problems as part of number and operations—fractions in 4 th grade.

What are fraction word problems?

Fraction word problems are math word problems involving fractions that require students to use problem-solving skills within the context of a real-world situation.

To solve a fraction word problem, you must understand the context of the word problem, what the unknown information is, and what operation is needed to solve it. Fraction word problems may require addition, subtraction, multiplication, or division of fractions.

After determining what operation is needed to solve the problem, you can apply the rules of adding, subtracting, multiplying, or dividing fractions to find the solution.

For example,

Natalie is baking 2 different batches of cookies. One batch needs \cfrac{3}{4} cup of sugar and the other batch needs \cfrac{2}{4} cup of sugar. How much sugar is needed to bake both batches of cookies?

You can follow these steps to solve the problem:

Step-by-step guide: Adding and subtracting fractions

Step-by-step guide: Adding fractions

Step-by-step guide: Subtracting fractions

Step-by-step guide: Multiplying and dividing fractions

Step-by-step guide: Multiplying fractions

Step-by-step guide: Dividing fractions

Common Core State Standards

How does this relate to 4 th grade math to 6 th grade math?

• Grade 4: Number and Operations—Fractions (4.NF.B.3d) Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.
• Grade 4: Number and Operations—Fractions (4.NF.B.4c) Solve word problems involving multiplication of a fraction by a whole number, e.g., by using visual fraction models and equations to represent the problem. For example, if each person at a party will eat \cfrac{3}{8} of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?
• Grade 5: Number and Operations—Fractions (5.NF.A.2) Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result \cfrac{2}{5}+\cfrac{1}{2}=\cfrac{3}{7} by observing that \cfrac{3}{7}<\cfrac{1}{2} .
• Grade 5: Number and Operations—Fractions (5.NF.B.6) Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.
• Grade 5: Number and Operations—Fractions (5.NF.B.7c) Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share \cfrac{1}{2} \: lb of chocolate equally? How many \cfrac{1}{3} cup servings are in 2 cups of raisins?
• Grade 6: The Number System (6.NS.A.1) Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for \cfrac{2}{3} \div \cfrac{4}{5} and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that \cfrac{2}{3} \div \cfrac{4}{5}=\cfrac{8}{9} because \cfrac{3}{4} of \cfrac{8}{9} is \cfrac{2}{3}. (In general, \cfrac{a}{b} \div \cfrac{c}{d}=\cfrac{a d}{b c} \, ) How much chocolate will each person get if 3 people share \cfrac{1}{2} \: lb of chocolate equally? How many \cfrac{3}{4} cup servings are in \cfrac{2}{3} of a cup of yogurt? How wide is a rectangular strip of land with length \cfrac{3}{4} \: m and area \cfrac{1}{2} \: m^2?

[FREE] Fraction Operations Worksheet (Grade 4 to 6)

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

How to solve fraction word problems

In order to solve fraction word problems:

Determine what operation is needed to solve.

Write an equation.

Solve the equation.

State your answer in a sentence.

Fraction word problem examples

Example 1: adding fractions (like denominators).

Julia ate \cfrac{3}{8} of a pizza and her brother ate \cfrac{2}{8} of the same pizza. How much of the pizza did they eat altogether?

The problem states how much pizza Julia ate and how much her brother ate. You need to find how much pizza Julia and her brother ate altogether , which means you need to add.

2 Write an equation.

3 Solve the equation.

To add fractions with like denominators, add the numerators and keep the denominators the same.

4 State your answer in a sentence.

The last step is to go back to the word problem and write a sentence to clearly say what the solution represents in the context of the problem.

Julia and her brother ate \cfrac{5}{8} of the pizza altogether.

Example 2: adding fractions (unlike denominators)

Tim ran \cfrac{5}{6} of a mile in the morning and \cfrac{1}{3} of a mile in the afternoon. How far did Tim run in total?

The problem states how far Tim ran in the morning and how far he ran in the afternoon. You need to find how far Tim ran in total , which means you need to add.

To add fractions with unlike denominators, first find a common denominator and then change the fractions accordingly before adding.

\cfrac{5}{6}+\cfrac{1}{3}= \, ?

The least common multiple of 6 and 3 is 6, so 6 can be the common denominator.

That means \cfrac{1}{3} will need to be changed so that its denominator is 6. To do this, multiply the numerator and the denominator by 2.

\cfrac{1 \times 2}{3 \times 2}=\cfrac{2}{6}

Now you can add the fractions and simplify the answer.

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

Tim ran a total of 1 \cfrac{1}{6} miles.

Example 3: subtracting fractions (like denominators)

Pia walked \cfrac{4}{7} of a mile to the park and \cfrac{3}{7} of a mile back home. How much farther did she walk to the park than back home?

The problem states how far Pia walked to the park and how far she walked home. Since you need to find the difference ( how much farther ) between the two distances, you need to subtract.

To subtract fractions with like denominators, subtract the numerators and keep the denominators the same.

\cfrac{4}{7}-\cfrac{3}{7}=\cfrac{1}{7}

Pia walked \cfrac{1}{7} of a mile farther to the park than back home.

Example 4: subtracting fractions (unlike denominators)

Henry bought \cfrac{7}{8} pound of beef from the grocery store. He used \cfrac{1}{3} of a pound of beef to make a hamburger. How much of the beef does he have left?

The problem states how much beef Henry started with and how much he used. Since you need to find how much he has left , you need to subtract.

To subtract fractions with unlike denominators, first find a common denominator and then change the fractions accordingly before subtracting.

\cfrac{7}{8}-\cfrac{1}{3}= \, ?

The least common multiple of 8 and 3 is 24, so 24 can be the common denominator.

That means both fractions will need to be changed so that their denominator is 24.

To do this, multiply the numerator and the denominator of each fraction by the same number so that it results in a denominator of 24. This will give you an equivalent fraction for each fraction in the problem.

\begin{aligned}&\cfrac{7 \times 3}{8 \times 3}=\cfrac{21}{24} \\\\ &\cfrac{1 \times 8}{3 \times 8}=\cfrac{8}{24} \end{aligned}

Now you can subtract the fractions.

\cfrac{21}{24}-\cfrac{8}{24}=\cfrac{13}{24}

Henry has \cfrac{13}{24} of a pound of beef left.

Example 5: multiplying fractions

Andre has \cfrac{3}{4} of a candy bar left. He gives \cfrac{1}{2} of the remaining bit of the candy bar to his sister. What fraction of the whole candy bar does Andre have left now?

It could be challenging to determine the operation needed for this problem; many students may automatically assume it is subtraction since you need to find how much of the candy bar is left.

However, since you know Andre started with a fraction of the candy bar and you need to find a fraction OF a fraction, you need to multiply.

The difference here is that Andre did NOT give his sister \cfrac{1}{2} of the candy bar, but he gave her \cfrac{1}{2} of \cfrac{3}{4} of a candy bar.

To solve the word problem, you can ask, “What is \cfrac{1}{2} of \cfrac{3}{4}? ” and set up the equation accordingly. Think of the multiplication sign as meaning “of.”

\cfrac{1}{2} \times \cfrac{3}{4}= \, ?

To multiply fractions, multiply the numerators and multiply the denominators.

\cfrac{1}{2} \times \cfrac{3}{4}=\cfrac{3}{8}

Andre gave \cfrac{1}{2} of \cfrac{3}{4} of a candy bar to his sister, which means he has \cfrac{1}{2} of \cfrac{3}{4} left. Therefore, Andre has \cfrac{3}{8} of the whole candy bar left.

Example 6: dividing fractions

Nia has \cfrac{7}{8} cup of trail mix. How many \cfrac{1}{4} cup servings can she make?

The problem states the total amount of trail mix Nia has and asks how many servings can be made from it.

To solve, you need to divide the total amount of trail mix (which is \cfrac{7}{8} cup) by the amount in each serving ( \cfrac{1}{4} cup) to find out how many servings she can make.

To divide fractions, multiply the dividend by the reciprocal of the divisor.

\begin{aligned}& \cfrac{7}{8} \div \cfrac{1}{4}= \, ? \\\\ & \downarrow \downarrow \downarrow \\\\ &\cfrac{7}{8} \times \cfrac{4}{1}=\cfrac{28}{8} \end{aligned}

You can simplify \cfrac{28}{8} to \cfrac{7}{2} and then 3 \cfrac{1}{2}.

Nia can make 3 \cfrac{1}{2} cup servings.

Teaching tips for fraction word problems

• Encourage students to look for key words to help determine the operation needed to solve the problem. For example, subtracting fractions word problems might ask students to find “how much is left” or “how much more” one fraction is than another.
• Provide students with an answer key to word problem worksheets to allow them to obtain immediate feedback on their solutions. Encourage students to attempt the problems independently first, then check their answers against the key to identify any mistakes and learn from them. This helps reinforce problem-solving skills and confidence.
• Be sure to incorporate real-world situations into your math lessons. Doing so allows students to better understand the relevance of fractions in everyday life.
• As students progress and build a strong foundational understanding of one-step fraction word problems, provide them with multi-step word problems that involve more than one operation to solve.
• Take note that students will not divide a fraction by a fraction as shown above until 6 th grade (middle school), but they will divide a unit fraction by a whole number and a whole number by a fraction in 5 th grade (elementary school), where the same mathematical rules apply to solving.
• There are many alternatives you can use in place of printable math worksheets to make practicing fraction word problems more engaging. Some examples are online math games and digital workbooks.

Easy mistakes to make

• Misinterpreting the problem Misreading or misunderstanding the word problem can lead to solving for the wrong quantity or using the wrong operation.
• Not finding common denominators When adding or subtracting fractions with unlike denominators, students may forget to find a common denominator, leading to an incorrect answer.
• Forgetting to simplify Unless a problem specifically says not to simplify, fractional answers should always be written in simplest form.

Related fractions operations lessons

• Fractions operations
• Multiplicative inverse
• Reciprocal math
• Fractions as divisions

Practice fraction word problem questions

1. Malia spent \cfrac{5}{6} of an hour studying for a math test. Then she spent \cfrac{1}{3} of an hour reading. How much longer did she spend studying for her math test than reading?

Malia spent \cfrac{1}{2} of an hour longer studying for her math test than reading.

Malia spent \cfrac{5}{18} of an hour longer studying for her math test than reading.

Malia spent \cfrac{1}{2} of an hour longer reading than studying for her math test.

Malia spent 1 \cfrac{1}{6} of an hour longer studying for her math test than reading.

To find the difference between the amount of time Malia spent studying for her math test than reading, you need to subtract. Since the fractions have unlike denominators, you need to find a common denominator first.

You can use 6 as the common denominator, so \cfrac{1}{3} becomes \cfrac{3}{6}. Then you can subtract.

\cfrac{3}{6} can then be simplified to \cfrac{1}{2}.

Finally, you need to choose the answer that correctly answers the question within the context of the situation. Therefore, the correct answer is “Malia spent \cfrac{1}{2} of an hour longer studying for her math test than reading.”

2. A square garden is \cfrac{3}{4} of a meter wide and \cfrac{8}{9} of a meter long. What is its area?

The area of the garden is 1\cfrac{23}{36} square meters.

The area of the garden is \cfrac{27}{32} square meters.

The area of the garden is \cfrac{2}{3} square meters.

The perimeter of the garden is \cfrac{2}{3} meters.

To find the area of a square, you multiply the length and width. So to solve, you multiply the fractional lengths by mulitplying the numerators and multiplying the denominators.

\cfrac{24}{36} can be simplified to \cfrac{2}{3}. 

Therefore, the correct answer is “The area of the garden is \cfrac{2}{3} square meters.”

3. Zoe ate \cfrac{3}{8} of a small cake. Liam ate \cfrac{1}{8} of the same cake. How much more of the cake did Zoe eat than Liam?

Zoe ate \cfrac{3}{64} more of the cake than Liam.

Zoe ate \cfrac{1}{4} more of the cake than Liam.

Zoe ate \cfrac{1}{8} more of the cake than Liam.

Liam ate \cfrac{1}{4} more of the cake than Zoe.

To find how much more cake Zoe ate than Liam, you subtract. Since the fractions have the same denominator, you subtract the numerators and keep the denominator the same.

\cfrac{2}{8} can be simplified to \cfrac{1}{4}. 

Therefore, the correct answer is “Zoe ate \cfrac{1}{4} more of the cake than Liam.”

4. Lila poured \cfrac{11}{12} cup of pineapple and \cfrac{2}{3} cup of mango juice in a bottle. How many cups of juice did she pour into the bottle altogether?

Lila poured 1 \cfrac{7}{12} cups of juice in the bottle altogether.

Lila poured \cfrac{1}{4} cups of juice in the bottle altogether.

Lila poured \cfrac{11}{18} cups of juice in the bottle altogether.

Lila poured 1 \cfrac{3}{8} cups of juice in the bottle altogether.

To find the total amount of juice that Lila poured into the bottle, you need to add. Since the fractions have unlike denominators, you need to find a common denominator first.

You can use 12 as the common denominator, so \cfrac{2}{3} becomes \cfrac{8}{12}.  Then you can add.

\cfrac{19}{12} can be simplified to 1 \cfrac{7}{12}. 

Therefore, the correct answer is “Lila poured 1 \cfrac{7}{12} cups of juice in the bottle altogether.”

5. Killian used \cfrac{9}{10} of a gallon of paint to paint his living room and \cfrac{7}{10} of a gallon to paint his bedroom. How much paint did Killian use in all?

Killian used \cfrac{2}{10} gallons of paint in all.

Killian used \cfrac{1}{5} gallons of paint in all.

Killian used \cfrac{63}{100} gallons of paint in all.

Killian used 1 \cfrac{3}{5} gallons of paint in all.

To find the total amount of paint Killian used, you add the amount he used for the living room and the amount he used for the kitchen. Since the fractions have the same denominator, you add the numerators and keep the denominators the same.

\cfrac{16}{10} can be simplified to 1 \cfrac{6}{10} and then further simplified to 1 \cfrac{3}{5}.

Therefore, the correct answer is “Killian used 1 \cfrac{3}{5} gallons of paint in all.”

6. Evan pours \cfrac{4}{5} of a liter of orange juice evenly among some cups.

He put \cfrac{1}{10} of a liter into each cup. How many cups did Evan fill?

Evan filled \cfrac{2}{25} cups.

Evan filled 8 cups.

Evan filled \cfrac{9}{10} cups.

Evan filled 7 cups.

To find the number of cups Evan filled, you need to divide the total amount of orange juice by the amount being poured into each cup. To divide fractions, you mulitply the first fraction (the dividend) by the reciprocal of the second fraction (the divisor).

\cfrac{40}{5} can be simplifed to 8.

Therefore, the correct answer is “Evan filled 8 cups.”

Fraction word problems FAQs

Fraction word problems are math word problems involving fractions that require students to use problem-solving skills within the context of a real-world situation. Fraction word problems may involve addition, subtraction, multiplication, or division of fractions.

To solve fraction word problems, first you need to determine the operation. Then you can write an equation and solve the equation based on the arithmetic rules for that operation.

Fraction word problems and decimal word problems are similar because they both involve solving math problems within real-world contexts. Both types of problems require understanding the problem, determining the operation needed to solve it (addition, subtraction, multiplication, division), and solving it based on the arithmetic rules for that operation.

The next lessons are

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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?

To keep learning, try Smartick’s free trial.

Learn More:

• Understand What a Fraction Is and When It Is Used
• Fraction Word Problems: Addition, Subtraction, and Mixed Numbers
• Learn and Practice How to Subtract or Add Fractions
• Learn How to Subtract Fractions
• Review and Practice the Two Methods of Dividing Fractions
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40 Comments

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Hi Letlhogonolo,

Thank you very much for your comment. If you want to learn more content like this and practice elementary school math, just sign up at Smartick . You have a free trial period with no strings attached. If you have any additional questions or doubts you can write to my colleagues of the pedagogical team at [email protected] .

Best regards!

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

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Fraction Word Problems Worksheet

Related Topics: More Lessons on Fractions More Fraction Worksheets

Printable “Add/Subtract Fractions” worksheets: Add Like Fractions Add Unlike Fractions Subtract Like Fractions Subtract Unlike Fractions Subtract Fraction from Whole Number Add/Subtract Fraction Word Problems

Solving fraction word problems involves understanding the problem, identifying what is given, determining what needs to be found, and then applying appropriate fraction operations and problem-solving strategies.

• Read the problem carefully to understand what it is asking. Identify key information and what needs to be found.
• Determine what information is given in the problem and what is known. Pay attention to quantities, fractions, operations, and any other relevant details.
• Identify what the problem is asking you to find. Is it a fraction, a sum or difference of fractions, a fraction of a quantity, etc.?
• Decide on a strategy to solve the problem. This may involve using fraction operations such as addition, subtraction, multiplication, or division, or using visual models like fraction bars or number lines.
• Apply the chosen strategy to solve the problem step by step. Perform any necessary fraction operations, simplify fractions if needed, and convert between mixed numbers and improper fractions as necessary.
• After solving the problem, check your answer to ensure it makes sense and matches the problem’s requirements. Re-read the problem to verify that you have answered the question correctly.
• Clearly state your answer in the context of the problem. Use proper units and labels if applicable.

Printable Fraction Word Problems Answers on the second page.) Fraction Word Problems Worksheet #1 Fraction Word Problems Worksheet #2

Online Fraction Word Problems

A group of people were standing in line. 3 / 8 of the people were boys and 1 / 4 of the people were girls. How much of the group was made up of boys and girls? Answer : How much of the group was not boys or girls? Answer:

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Math Interventions

• Introduction
• Subitizing Interventions
• Counting Interventions: Whole Numbers Less Than 30
• Counting Interventions: Whole Numbers Greater Than 30 (Place Value)
• Counting Interventions: Fractions

Tools for Counting

Counting fractions, deepening conceptual knowledge of fractions, interpreting fractions, representing fractions, comparing fractions, response to error: counting fractions, feedback during the lesson, strategies to try after the lesson.

• Counting Interventions: Decimals
• Composing and Decomposing Numbers Interventions
• Rounding Interventions
• Number Sense Lesson Plans
• Addition and Subtraction Facts
• Multiplication and Division Facts
• Computational Fluency Lesson Plans
• Understanding the Problem Interventions
• Planning and Executing a Solution Interventions
• Monitoring Progress & Reflecting on a Solution Interventions
• Problem-Solving Process Interventions
• Problem-Solving Lesson Plans
• Identifying Essential Variables Interventions
• Direct Models Interventions
• Counting On/Back Interventions
• Deriving Interventions
• Interpreting the Results Interventions
• Mathematical Modeling Lesson Plans
• Math Rules and Concepts Interventions
• Math Rules and Concepts Lesson Plans

Another way to support your student's understanding of quantity is by teaching him to count. Knowing how to count includes the following: knowing how to verbally count, understanding cardinality (that numbers represent quantities), and having the ability to interpret, represent, and compare quantities. Counting ability develops in the following developmental stages:

• Counting Whole Numbers from 0-30
• Counting Whole Numbers above 30  (place value)
• Counting Fractions
• Counting Decimals

Before you learn about interventions that support one of these stages, you'll learn about tools you can use to support student mastery. 

The intervention pages to come refer to the following tools: Counters, Drawing and Labeling Objects, Ten-Frames, Number Lines, Base-Ten Blocks, and Area Models. Use this page as a resource as you learn more about each of the tools in this activity. 

Tool # 1: Counters 

Counters are the concrete objects that students can manipulate in order to count or build quantities. Counters can come in many different forms: gummy bears, pebbles, buttons, Unifix cubes, and so on. When students are first learning to count, counters are the best tool to use, since the student can physically hold and move this object. 

Wikimedia. (2012). Cubes. Creative Commons Attribution Share alike 3.0. Unported. Retrieved at https://commons.wikimedia.org/wiki/File:Multilink_cubes.JPG

As students become better at counting or building groups of objects using counters, you can link the counters to visual representations of each quantity. Forbringer and Fuchs (2014, pp. 92-93) explain why and how to do so: 

Sometimes students initially model a concept or procedure with counters and then are asked to perform the skill using visual or symbolic representation but without practice that explicitly connects the various forms of presentation. While normally achieving students may be able to successfully transition from one form of representation to another without the need for scaffolded support, students who have difficulty with number sense often struggle to connect the various forms of mathematical representation (Hecht, Vogi, & Torgesen, 2007), and so benefit when these connections are made explicit. For example, if students initially used M&Ms to count, they could use pencil, crayon, or chalk to draw a model of their M&Ms. Recording the written numeral on their drawing helps students connect the three-dimensional objects with pictorial and symbolic representations.

Drawing and labeling objects is a great tool to use if your students does not need to physically hold an object to count it. 

Forbringer, L., & Fuchs, W. (2014).   RtI in Math: Evidence-Based Interventions for Struggling Students.   Hoboken: Routledge Ltd. Wilson, A. (2017). M and Ms image. New York, NY: Relay Graduate School.

Wilson, A. (2017). Circles image. New York, NY: Relay Graduate School.

Tool # 4: Ten-Frames

One of the best tools to help students connect three-dimensional concrete representation to two-dimensional visual representation is the ten-frame. A ten-frame consists of an empty 2 x 5 grid onto which students place counters... When [students] are first learning to model numbers with frames, students should lay the frame horizontally so that there are five boxes in the top row. They begin by placing the first counter in the upper-left corner and progress from left to right across the top row, then move to the bottom row and continue placing counters from left to right, just as the eyes move when reading. Placing counters on the ten-frame in this set order helps students organize their counting and develop a mental model of each quantity.

Click  here  for a ten-frame template.

Forbringer, L., & Fuchs, W. (2014).   RTI in Math: Evidence-Based Interventions for Struggling Students.   Hoboken: Routledge Ltd.

Tool # 5: Hundred Chart   Another graphic organizer that a student can use as he learns to count is a hundred chart. A hundred chart lists all the numbers from 1-100, in order, in rows of ten. This tool helps a student visualize each quantity (for example, 23 represents all of the space on the 100 chart up to the number 23). It can also help a student recognize symbolic representations of numbers as he learns how to count (e.g.,, the student can touch each number and count up to 31 to find out that 31 is written "31") .

Tool # 6: Number Lines

A number line is another tool you can use to support your student's ability to represent a number visually or recognize symbolic representations of numbers . Teach your student that the distance from zero to a given number on the number line represents the size of that number. Or, teach the student to touch each number as he counts to find out what the symbolic representation of a certain number looks like.

Note:   A common misunderstanding when using a number line is thinking that the number of ticks on the line represents the size of the number (when actually the quantity represented is the distance between the number and zero) . Teach your student to focus on the spaces between ticks and not the ticks themselves, when identifying a quantity. You can do this by helping him build a number line, placing equal-sized units (e.g., blocks or squares) next to each other. Or, you might incorporate movement to draw your student's attention to the spaces. Forbringer and Fuchs (2014, pp. 99-100) write "...students can walk along a large number line taped on the floor, counting their steps as they move. The large muscle motion helps them focus on counting space. They can then progress to demonstrating bunny hops... on a small number line taped on the desk." 

Forbringer, L., & Fuchs, W. (2014).   RtI in Math: Evidence-Based Interventions for Struggling Students.  Hoboken, NJ: Routledge Ltd. Wikimedia. (2007). Number Lines. Creative Commons Attribution Share alike 3.0. Unported. Retrieved from https://commons.wikimedia.org/wiki/File:NumberLineIntegers.svg.  

Tool # 7: Base-Ten Blocks & Place Value Organization Mats

Thomspon. (2017). Base ten blocks. Retrieved at http://thompsongrade6.weebly.com/uploads/1/3/5/7/13570936/8575679_orig.jpg

Tool # 8: Area Models

There are numerous concrete objects that students can manipulate in order to identify or build fractional parts of a whole. These include: fraction tiles, pattern blocks, legos, or strips of paper. Each of these tools can be used to visually model the number of equal parts a whole is divided into (as represented by the denominator). The student can then identify the number of equal parts represented by a particular fraction (as reflected in the numerator). 

Image inspired by: Walle, J., Lovin, L., Karp, K., & Bay-Williams, J. (2014). Teaching Mathematics for Understanding. In  Teaching student-centered mathematics: Developmentally appropriate instructions for grades Pre-K-2 (V1) (2nd ed., Pearson new international ed., Vol. 1, p. 245) . Upper Saddle River, NJ: Pearson Education.

Knowing how to count fractions starts with a conceptual understanding of what a fraction represents. Before reading about interventions to support your student's ability to count fractions, read how a kindergarten and first-grade teacher, Ms. Murphy, exposes her students to the concept of fractions. As you read, consider: How is teaching a student to count fractions different than teaching a student to count whole numbers? Why is building this conceptual understanding so important before starting to intervene?

Conceptual Understanding of Fractions_0.pdf Note: An  Equal Sharing problem is one that allows a total number of items to be distributed to a given number of groups or people (Empson & Levi, 2011), such as the brownie example used in this reading. 

As you've read, Ms. Murphy's students are able to grasp the concept of fractions without understanding the terminology, and without being able to read or write fractions themselves. Therefore, intervening to support a student's ability to count fractions (and, later, decimals) is different than intervening to support a student's ability to count whole numbers because a teacher will not start by teaching verbal, or rote, counting. Instead, she will develop a student's conceptual understanding of fractions, and then teach a student how to interpret, represent, and compare fractions. This page includes strategies that you can use to support your students in this area. As you read, consider which of these interventions best aligns with your student's strengths and needs in the whole learner domains.

Explicit Instruction

Once your student has developed a conceptual understanding of what a fraction represents, you can explicitly teach what a fraction is. This sounds like:

• Explain the Skill/Concept.  Define fraction and explain activity. ( "A fraction is a number that represents parts of a whole." "Today we are going to use fractions to describe numbers smaller than one whole" )
• Model Skill with Examples.  Think aloud about how to identify fractions.  ( "I'm looking at this circle and I see that it's divided into four parts, or fourths. I notice that one of the fourths is blue and the rest of the fourths are yellow. If I wanted to describe the blue section of this circle, I could use a fraction. I know that the blue section is one fourth of the whole circle. I would write it like this: 1/4.
• Model Skill with Non-Examples.  Think aloud about how not to identify fractions. ( "Let's look at the yellow section of the circle. I notice that the yellow part of the circle is also less than the whole circle. It must be 1/4 too. No? Why not? How many pieces are yellow? 3. Oh, so then it's not one fourth, it's three fourths. I can write it like this: 3/4 )
• Practice the Skill.  Engage in one or more of the activities below to practice the skill with your student, providing feedback as necessary. ( "Now you try. I'm going to show you  how to... " )

Once a student has developed a conceptual understanding of what a fraction means, you can support his understanding of the terminology by implementing following interventions:

Activity A: Teach the Terms If your student is struggling to conceptualize fractions, it's important that he becomes familiar with the terminology. Therefore, this intervention is a simple one: teach the student to describe the equal pieces that make up the whole (Empson & Levi, 2011). The teacher should present a visual representation of a whole and ask a question, "How many of these pieces fit into the whole?" The student should then look at a visual representation of a whole, count the number of smaller pieces that fit into the whole. Then, you should tell the student the term they would use to describe each fractional piece.Initially, encourage the student to write the name of the fraction in words instead of in symbolic notation.

Teach the Terms in Action Teacher: Let's look at this brownie. It's divided into parts. How many of these parts fit into the whole brownie?

Teacher: Oh, so the brownie is divided into fourths. Into what?

Student: Fourths

Teacher: Since four of these little pieces make up the whole brownie, each little piece is called a fourth. What is this piece called?

Student: A fourth.

Teacher. Yes, one fourth. You can write that 'one fourth' or '1 fourth.'

Activity B: Teach the Symbols As your student develops understanding of fractional terms, you should introduce the standard symbolic representation of a fraction (a/b). According to Empson and Levi (2011), symbolic notation is intuitive to students who understand fractional terms. They write:

For example, consider how the fraction symbol 2/3 can refer to an amount of a candy bar. The numerator, 2, refers to the number of pieces in the share. The denominator, 3, refers to the size of the piece relative to the whole. To represent 2/3, young children would write "2 thirds." When students begin to independently use correct fraction terminology to describe shares, you can introduce the standard fraction symbol (a/b). 

Teach the Symbols in Action

Teacher: Let's look at this brownie. It's divided into parts. How many of these parts fit into the whole brownie?

Teacher: Since four of these little pieces make up the whole brownie, what would we call two pieces of this brownie?

Student: Two fourths.

Teacher. Yes, two fourths. Show me how you could write that?

[Student writes 'two fourths']

Teacher: Another way to write two fourths is 2/4. Can you write your fraction the way that I did? 

[Student writes '2/4']

Teacher: This means, were are talking about two out of four pieces that make up the whole brownie. 

As a student learns to interpret fractions, he will follow a similar progression as he did for interpreting whole numbers. At first, the student might use more concrete objects to interpret fractions. Then, he'll develop the ability to interpret fractions using more abstract representations (such as drawing a circle cut into 5 pieces to represent a cookie cut into 5 equally-sized pieces). In order to support a student's ability to interpret fractions, you should support his understanding with the following interventions: 

Activity C : Identifying Fractional Parts of Concrete Objects If your student is struggling to understand fractions, specifically the concept that wholes are composed of many equal sized parts, teach him using concrete objects (e.g., a chocolate bar or a group of cubes) (Empson & Levi, 2011). Teach the student to start by examining the whole and then ask the student to describe how much each person would get if the object was divided in a certain way.

Identifying Fractional Parts of Concrete Objects in Action Teacher: Here is a chocolate bar. Let's pretend that I wanted to split the chocolate bar with you. I'm going to divide the chocolate bar right here, so that we each get the same amount. How much chocolate do I get?

Student: One piece

Teacher: Yes. How many of these pieces make up the whole chocolate bar?

Student: Two

Teacher: Yes, so I get one half of a chocolate bar. How much do I get?

Student: One half of a chocolate bar.

Teacher: Yes. How much chocolate do you get?

Student: One half.

Teacher: One half of what?

Activity D : Identifying Fractional Parts using Pictorals

Once a student has learned how to identify fractional parts of concrete objects, you should move on to teaching the student to describe fractional parts using pictorals (Empson & Levi, 2011). In this intervention, the teacher draws or references a fraction bar, and asks a student to identify the fraction as she shades different parts.

Wilson, A. Fraction bar. New York, NY: Relay Graduate School.

Identifying Fractional Parts using Pictorals in Action Teacher: Here is a fraction bar. How many equal parts is it divided into? 

Student: Three

Teacher: Yes. So if I asked you to color in the entire fraction bar, how many parts would you color?

Teacher: Yes. What if I asked you to color in one part of the fraction bar? How much is that? 

Student: One out of three. 

Teacher: Yes. We can say one out of three, or 1/3. What if I asked you to color in two parts?

Student: 2 out of 3, or 2/3. 

Activity E: Identifying Fractions on a Number Line If your student is able to name the fractional parts of a rectangle, you'll want to teach him to extend this understanding to identify fractional parts on a number line (Empson & Levi, 2011). In this intervention, the teacher gives the student a number line from zero to one. Then, he draws the student's attention to the number of parts that make up the whole distance between zero and one.

Identify Fractions on a Number Line in Action Teacher: This is a number line from zero to one . To day, we are going to talk about what to call numbers that fall between zero and one on the number line. Each of the spaces between these tally marks represent parts of one. How many spaces fit into the distance between zero and one?

Student: Three.

Teacher: So, what can we call each of these parts?

Student: Thirds.

Teacher: Yes, I can label them on the number line like this: one third, two thirds OR like this: 1/3, 2/3.

Once a student is able to interpret fractions, he can start to represent (or build) fractions. You should support his understanding with the following interventions:  

Activity F: Representing Fractional Parts using Concrete Objects If your student is able to identify fractions with objects, but needs additional support to build fractions on his own, teach him Representing Fractional Parts using Objects. This strategy is similar to Identifying Fractional Parts of Concrete Objects but instead of having the student simply identify the fraction, the teacher asks the student to build (or represent) the fraction with objects (Empson & Levi, 2011). For example, the teacher might give the student a chocolate bar (such as a Hershey's bar that has 12 equal parts) ask the student to make 8/16. The teacher can then ask challenge the student to identify another fraction that amount makes (6/12 is the same as 1/2 of the bar).

Represent Fractions with Objects in Action Teacher: Here is a chocolate bar. As you can see, it has 12 equal parts. I am going to show you how I use this bar to represent fractions. For example, if I wanted to show 6/12, I could break off 6 pieces, like this (teacher breaks bar in half). Now, I have 6/12. As I look at this piece, I also notice that I just broke the bar in half, so I also am representing the fraction of 1/2. So, that tells me that 6/12 is actually the same as 1/2. Now, I'm going to ask you to represent a fraction using this Hershey bar.

Activity G: Representing Fractional Parts using Pictorals If your student has mastered interpreting fractions with pictorals, but needs additional support to build the fractions on his own, teach him Representing Fractional Parts using Pictorals (Empson & Levi, 2011). In this intervention, the teacher tells the student a fraction and asks the student to build it, using the model. 

Represent Fractional Parts using Pictorals  Teacher: I'm going to show you how I can build 1/3 by drawing a fraction bar.

[Teacher draws a fraction bar and draws two lines, splitting the fraction bar into 3 equal pieces.]

Teacher: Ok, so I now have 3 parts total. Now, I need to represent 1 of them. I'm going to shade in this segment. (Teacher shades in segment.) Now, I have 1/3. Now, it's going to be your turn. I'll give you a piece of paper to represent the fraction 2/8.

Activity H : Represent Fractions on a Number Line If your student is able to interpret fractions on a number line, but needs continued support to build them on his own, teach him to Represent Fractions on a Number Line (Empson & Levi, 2011). In this intervention, the teacher gives the student a blank number line and asks him to show a fraction (such as 1/3). 

Represent Fractions on a Number Line in Action Teacher: This is a number line. I can use it to represent a fraction. Let's say I had to represent the fraction 1/3. First, I would need to designate those marks on the number line by drawing tally marks. So, I know that I need to separate my unit interval into three equal parts, so that means I'll draw two tally marks. Now, I have to count them. 1/3, 2/3, 3/3, which is one. So, the space between 0 and this first tally mark is 1/3.

Activity I: Write the Fraction  If your student has been able to successfully interpret fractions, it's time for him to learn to Write the Fraction (Stein et al., 2006).This strategy helps a student's ability to represent fractions using numbers. In this intervention, the student will convert a fraction written as words (two-thirds) to its corresponding fraction (2/3).

Write the Fraction in Action Teacher: You are going to convert a fraction written in words into numbers. Watch as I do this one: two-thirds. Hmmm....Well, 2 is the first number. Let me think: two-thirds. If I have two-thirds of something, that I know I have two parts out of three. That means that two must be my numerator, or the number of parts I have out of the whole. So, I'll write two on the top part of the fraction bar. Three represents how many pieces are in the whole, and I know that number goes on the bottom of the fraction bar as my denominator. So, two-thirds written as a fraction is 2/3. Now, you try!

Activity J: What Stays the Same Once your student is able to write fractions, he is ready to write equivalent fractions (Empson & Levi, 2011). In this intervention, the teacher shows the student how to write equivalent fractions by visualizing an object being split multiple times, and writing a fraction every time the object is split. By using a visual (such as imagining splitting a chocolate bar or cutting a cake), the student can understand how different fractions can represent the same amount.

What Stays the Same in Action Teacher: I'm going to show you different ways to write 1/3. The new fractions I write will look different, but they will still represent the same amount. In my head, I'm going to picture a cake split in thirds. Can you see that cake, split into thirds?

Student: Yes.

Teacher: If I have 1/3 of the cake, I have one piece. Now, I could split all of those pieces of cake again. So, in my mind, I'm going to cut each of those pieces in half. How many pieces of cake will I have now?

Student: 6.

Teacher: That's right. The denominator is now 6. But, if I still have 1/3 of the case, I have more than 1 piece out of the 6. I now have how many pieces of cake ?

Student: 2.

Teacher: So, fraction is equivalent to, or the same size as 1/3?

Student: 2/6.

In order to support a student's ability to compare fractions, you should support his understanding with the following interventions:  

Activity K: Compare the Fraction

Once a student is able to match, draw, and write fractions, he is ready to Compare the Fraction (Stein et al., 2006). In this intervention, the student determines whether a fraction is more than, equal to, or less than 1. Once a student can solve these types of problems with fluency, he can then practice comparing two fractions to identify which is greater. Empson and Levi (2011) write that when asked to compare two unit fractions such as 1/5 and 1/6, "children...often mistakenly conclude that 1/6 is bigger than 1/5 because 6 is bigger than 5. They make be thinking in terms of the number of total parts that is created when a whole is partitioned, or they may be simply looked only at the denominators." Therefore, interventions that support a student's ability to compare fractions are important to help a student better understand the relationship between the numerator and denominator. 

Compare the Fraction in Action Teacher: You are going to be reading a fraction and identifying if it is greater or less than one. Watch as I complete this problem: 2/3 is ______ 1. Well, I see that my numerator is 2, so I have 2 parts. I see that my denominator is 3, so there are 3 parts total. If I have 2 parts out a whole that is made up of 3 parts, do I have the whole thing? If my numerator were the same as my denominator, or 3/3, I would have the whole thing. I also know that 3/3 would equal 1. Do I have more than the whole thing? If my numerator were greater than my denominator, such as 4/3, it would be greater than 1. Hmm... I think maybe I have less than the whole thing. Let me check. If I have 2 parts of 3 parts total, I have less than the whole thing, since my numerator is smaller than my denominator,  so 2/3 must be less than one. Now, you try.

Activity L: Ordering Fractions Once a student can state whether a given fraction is greater than or less than one, he is ready to practice ordering fractions. In this intervention (Empson & Levi, 2011), the student learns principles that help him put fractions in order. For example, the teacher may teach the student one (or more) of the following principles of order:

• The more people sharing, the smaller each share will be.
• When numerators are equal, the bigger the denominator, the smaller the value of the fraction. 
• Imagine each fraction as a whole with parts missing. Then, think which are bigger. 
• Use a benchmark fraction, such as 1/4 or 1/2, to think about whether each fraction is bigger or smaller than the benchmark . 

At first, a student might only order fractions with one principle in mind. However, over time, a student should be able to use all four principles to put fractions in the correct order. 

Ordering Fractions in Action Teacher: We are going to order fractions, and I want you to decide which is greater based on the following principle: The more people sharing, the smaller each share will be. Let's think about this. Hmm...the bottom number, or the denominator, tells me how many sharing make up the whole , so if my denominator is small, what size are the shares? (They are big) What happens as my denominator gets bigger? (The shares get smaller)

Teacher: So if the top number, or numerator, is the same, we can look at the size of the denominators to figure out how big the shares are. My first problem is :Which is greater? 2/7 or 2/11. Well, there are 7 people sharing in the first, and 11 in the second. If there are 11 shares, each piece will be smaller than if there are 7 shares. The numerator is the same. So, 2/7 is greater than 2/11. Now, you try. 

Empson, S., & Levi, S. (2011). Extending children's mathematics: Fractions and decimals. Portsmouth, NH: Heineman.  Stein, M., Kinder, D., Silbert, J., & Carnine, D.W. (2006). Designing effective mathematics instruction. Upper Saddle River, NJ: Pearson.

Think about the following scenario, which takes place after a teacher has explicitly taught a student strategies for counting fractions.       Teacher: "Can you write the fraction two-fourths?"      Student writes 24 

In such a case, what might you do? 

When you are planning your lessons, you should anticipate that your student will make errors throughout. Here are a series of prompts that you can use to respond to errors. Keep in mind that all students are different, and that some students might respond better to some types of feedback than others.

If your student struggles to meet your objective, there are various techniques that you might try to adjust the activity to your student's needs. 

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• Mahindra Jain
• Tapas Mazumdar
• Munem Shahriar
• Gene Keun Chung
• Mohammad Farhat
• Abdulrahman El Shafei
• Blan Morrison

We understand how to count the integers and how the order of operations work for them. But what happens if the items that we're working with are not in whole parts? If someone ate one slice of your pizza, how much do you have left? We certainly do not have 1 pizza, but neither do we have 0 pizzas.

Fractions are numbers defined by division , used to represent any number of equal parts of a whole. They are real numbers of the form \(\frac { p }{ q },\) where \(p\) and \(q\) are integers. The number of parts is given by the number at the top (above the horizontal bar of the fraction), called the numerator ; the number of pieces that make up the whole, which tells us their size, is described by the denominator , the number at the bottom (below the horizontal bar). Thus, in the fraction \( \frac{2}{3} \) (read as "two thirds" and written as "two-thirds"), 2 is the numerator and 3 is the denominator. This tells us that there are two parts, and each of them is one-third of a whole.

Let us explore the concept of fractions, and understand how it is used to solve various problems.

Visual Representation

Classification, multiplying fractions, adding fractions, fraction arithmetic, complex fractions, fractions - word problems, fractions - problem solving.

The part shaded blue below is a visual representation of two-thirds:

two thirds filled in

Proceeding step by step, we can see why \( \frac{2}{3} \) looks like the image above:

Begin with an entire object or unit: \(1\). unit

Divide it according to the denominator \((\)here we are dividing by \(3): \frac{1}{3}\). divide

Multiply that result according to the numerator \((\)here we multiply by \(2): \frac{2}{3}\). result

A proper fraction is one with the numerator less than the denominator, e.g. \( \frac{4}{5}. \)

An improper fraction is one with the numerator greater than the denominator, e.g. \( \frac{7}{4}. \)

A mixed number is written as a whole number part followed by a fractional part. The fractional part of a mixed number is always a proper fraction, e.g. \( 4\frac{1}{3}. \)

To multiply fractions, we follow these steps:

• Multiply the numerators.
• Multiply the denominators.
• Simplify the fraction by dividing throughout by their GCD.

Evaluate \( \frac{ 2}{3} \times \frac{ 9}{ 4 } \). Multiplying the numerators and denominators, we get \[ \dfrac{2}{3} \times \dfrac{9}{4} = \dfrac{ 2 \times 9 } { 3 \times 4} = \dfrac{ 18 } { 12 } . \] Since \( \gcd(12, 18) = 6 \), the simplified fraction is \[ \dfrac{18}{12} = \dfrac{ 18 \div 6 } { 12 \div 6 } = \dfrac{3}{2}. \] Thus \( \frac{2}{3} \times \frac{9}{4} = \frac{3}{2} \). \( _\square \)

If the fraction is a mixed fraction, then we must first convert it into an improper fraction before multiplying.

Evaluate \( 1 \frac{2}{3} \times 2 \frac{ 3}{4} \). Converting the mixed fractions into improper fractions, we have \[ 1 \dfrac{2}{3} = \dfrac{ 1 \times 3 + 2 } { 3} = \dfrac{5}{3}, \quad 2 \dfrac{3}{4} = \dfrac{ 2 \times 4 + 3 } { 4} = \dfrac{11}{4}. \] Hence, \[ 1 \dfrac{2}{3} \times 2 \dfrac{3}{4} = \dfrac{5}{3} \times \dfrac{ 11}{4} = \dfrac{5 \times 11 } { 3 \times 4 } = \dfrac{55}{12} . \] Since the greatest common divisor of \(55\) and \(12\) is \(1,\) the final answer is \(\frac{55}{12} \). \( _\square\) Note that \[ 1 \dfrac{2}{3} \times 2 \dfrac{3} {4} \neq ( 1 \times 2) \dfrac{ 2 \times 3 } { 3 \times 4} = 2 \dfrac{1}{2}.\]

Simplify \[\dfrac{4}{5} \times 3 \dfrac{3}{2}. \] Converting the mixed fraction into an improper fraction, we have \( 3 \frac{3}{2} = \frac{3 \times 2 + 3}{2} = \frac{9}{2}.\) Hence, \[ \dfrac{4}{5} \times 3 \dfrac{3}{2} = \dfrac{4}{5} \times \dfrac{9}{2} =\dfrac{4 \times 9}{5 \times 2}= \dfrac{36}{10}= \dfrac{18}{5},\] where we simplified the fraction by dividing both the numerator and denominator by \( \gcd(36, 10) = 2 .\) \( _\square \)

Evaluate \( \frac{2}{3} \times 1 \frac{3}{4} \times \frac{6}{5} .\) Converting the mixed fraction into an improper fraction, we have \( 1 \frac{3}{4} = \frac{7}{4}. \) Hence, \[ \dfrac{2}{3} \times \dfrac{7}{4} \times \dfrac{6}{5} = \dfrac{2 \times 7 \times 6}{3 \times 4 \times 5} = \dfrac{84}{60} = \dfrac{7}{5} ,\] where we simplified the fraction by dividing both the numerator and denominator by \( \gcd(84, 60) =12 .\) \( _\square \)

To add fractions, we follow these steps:

• Convert to a common denominator using the lowest common multiple .
• Add the numerators.
• Simplify the fraction by dividing throughout by the greatest common divisor .

Evaluate \( \frac{2}{3} + \frac{4}{5} \). Since the denominators are different, we have to find a common denominator. The lowest common multiple of 3 and 5 is 15, so we have \( \frac{2}{3} = \frac{ 2 \times 5 } { 3 \times 5 } = \frac{10}{15} \) and \( \frac{4}{5} = \frac{4 \times 3 } { 5 \times 3 } = \frac{ 12}{15} \). Then, we add the numerators to get \[ \frac{10}{15} + \frac{12}{15} = \frac{22}{15} . \] Since \( \gcd(22, 15) = 1 \), this fraction is already simplified. Thus, \( \frac{2}{3} + \frac{4}{5} = \frac{22}{15} = 1 \frac{7}{15} \). \(_\square\)

The procedure is similar for mixed fractions, but we must convert to an improper fraction first.

Alternatively, it can be simpler to add the integers and the fractional parts separately.

Evaluate \( 1 \frac{2}{3} + 2 \frac{5}{6} \). Converting to improper fractions, we have \( 1 \frac{2}{3} = \frac{1 \times 3 + 2 } { 3} = \frac{5}{3} \) and \( 2 \frac{5}{6} = \frac{2 \times 6 + 5 } { 6} = \frac{17}{6} \). Since the denominators are different, we make a common denominator of \(6\). This gives us \[ \frac{5}{3} + \frac{17}{6} = \frac{ 5 \times 2 } { 3 \times 2 } + \frac{17}{6} = \frac{27} { 6} \] Finally, we simplify the fraction. Since \( \gcd(27, 6) = 3 \), hence we divide throughout by 3 to get \( \frac{27 \div 3 } { 6 \div 3} = \frac{9}{2} \). Thus, \( 1 \frac{2}{3} + 2 \frac{5}{6} = \frac{9}{2} = 4 \frac{1}{2} \). \(_\square\)

Evaluate \(2 \frac{3}{4} + 1\frac{1}{2}.\) Converting each term to improper fractions, we have \( 2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{3} \) and \( 1\frac{1}{2} = \frac{1\times2+1}{2}=\frac{3}{2}.\) Since the denominators are different, we make a common denominator of \(6.\) This gives us \[ 2 \frac{3}{4} + 1\frac{1}{2} = \frac{11}{3} + \frac{3}{2} = \frac{11 \times 2}{3 \times 2} + \frac{3 \times 3 }{2 \times 3} = \frac{22 + 9}{6} = \frac{31}{6} . \ _\square \]

Evaluate \( 3\frac{1}{2} + 2\frac{2}{3} + 1\frac{3}{4}.\) Converting each term to improper fractions, we have \( 3\frac{1}{2} = \frac{3\times2+1}{2} = \frac{7}{2}, 2\frac{2}{3} = \frac{2\times3 + 2}{3} = \frac{8}{3}\) and \( 1\frac{3}{4} = \frac{1\times4+3}{4}=\frac{7}{4}.\) Since the denominators are different, we make a common denominator of \(12.\) This gives us \[ \begin{align} 3\frac{1}{2} + 2\frac{2}{3} + 1\frac{3}{4} &= \frac{7}{2} + \frac{8}{3} + \frac{7}{4} \\ &= \frac{7\times6}{2\times6} + \frac{8\times4}{3\times4}+\frac{7\times3}{4\times3} \\ &= \frac{42+32+21}{12} \\ &= \frac{95}{12}. \end{align} \] Since \( \gcd(95,12) = 1 ,\) this fraction is already simplified. Thus, \( 3\frac{1}{2} + 2\frac{2}{3} + 1\frac{3}{4} = \frac{95}{12} = 7\frac{11}{12}.\) \( _ \square\)

Which of the following fractions, when added to the sum of the above numbers, makes the result a whole number?

Given a sequence of operations on proper fractions, possibly including multiplication, division, addition, and subtraction, we first figure out the order to carry out the sequence of operations by following the usual rules for order of operations .

What is \(\frac{1}{4} + \frac{2}{3} - \frac{3}{5}?\) Gathering the fractions using the least common multiple of the denominators, we have \[\begin{align} \frac{1}{4} + \frac{2}{3} - \frac{3}{5} &= \frac{15}{60} + \frac{40}{60} - \frac{36}{60}\\\\ &= \frac{15 + 40 - 36}{60}\\ & = \frac{19}{60}. \end{align}\] Since the greatest common divisor of \(19\) and \(60\) is 1, the final answer is \(\frac{19}{60} \). \( _\square\)

What is \( \frac{1}{3} + \frac{2}{5} \div \frac{6}{5}?\) Following order of operations, we must divide first, so we have \( \frac{1}{3} + \left( \frac{2}{5} \div \frac{6}{5} \right) = \frac{1}{3} + \frac{1}{3} = \frac{2}{3} \). Thus the answer is \( \frac{2}{3} \). \( _\square \)

What is \( \frac{20}{3} -\frac{1}{9} \div \left[ \frac{3}{7} \times \left( - \frac{1}{4} \right) \times \left( -\frac{2}{3} \right)^2 \right]?\) Following order of operations, we first evaluate the quantity in the parenthesis, which gives \[\begin{align} \frac{20}{3} -\frac{1}{9} \div \left[ \frac{3}{7} \times \left( - \frac{1}{4} \right) \times \left( -\frac{2}{3} \right)^2 \right] &= \frac{20}{3} -\frac{1}{9} \div \left[ \frac{3}{7} \times \left( - \frac{1}{4} \right) \times \left( \frac{4}{9} \right) \right]\\ &= \frac{20}{3} -\frac{1}{9} \div \left[ \frac{3}{7} \times \left( - \frac{1}{9} \right) \right]\\ &= \frac{20}{3} + \frac{1}{9} \div \left[ \frac{3}{63} \right]\\ &= \frac{20}{3} + \left( \frac{1}{9} \times \frac{63}{3} \right)\\ &= \frac{20}{3} + \left( \frac{7}{3} \right)\\ &= \frac{27}{3}\\ & = 9.\ _\square \end{align}\]

Find the value of \( a + b \) in the expression below, where \( a \) and \( b \) are positive integers with no common factors:

\[ \frac{1}{6} + \frac{1}{10} = \frac {a}{b}. \]

Main Article: Complex Fractions

A complex fraction is a fraction with another fraction in the numerator or denominator (or both). Below is an example of a complex fraction:

\[ \frac{1-\frac{1}{x}}{1+\frac{1}{x}}. \]

Complex fractions should not be confused with fractions involving complex numbers .

Simplify \[ \frac{\hspace{2mm} \frac23 \hspace{2mm} }{\hspace{2mm} \frac27\hspace{2mm} }.\] We can multiply both the numerator and the denominator by 21, which is the LCM of \(3\) and \(7.\) Then \[ \frac{\frac{2}{3}\times 21}{\frac{2}{7}\times 21} = \frac{2 \times 7}{2 \times 3} = \frac{14}{6} = \frac{7}{3} = 2 \frac{1}{3}. \ _ \square \]

To be able to solve word problems with fractions, you must firstly be able to solve regular word problems by translating common language to math. The SAT Translating Word Problems wiki gives a great insight of how this can be done. Give it a look if you're not familiar with word problems.

Fractions can be of great use in more realistic scenarios in which part of a total must be calculated. Chemical solutions, for example, use a great deal of fraction to calculate parts of a total, and just like problems that deal with solutions, any other problem which must differentiate or do any other process with parts of a total can usually be solved and simplified by the correct use of fractions.

Fractions can be given directly in a word problem \(\big(\)e.g. \( \frac{7}{10} \) of the human body is composed of water\(\big)\) or by common language, with no numbers representing it. In that case you must be able to search and identify keywords which refer to fractions, which are usually the same ones that refer to division since they're technically the same operation.

The final math exam will be 1 hour long. The teacher said that the whole test could be read in 5 minutes, each question answered in 2 minutes, and the work reviewed with the time that's left. If the test has 20 questions, then what fraction of the time to do the test can be used to review it? Considering the problem has been read, we must identify what's been asked by it. The problem is requiring us to get the fraction of time that can be used to review from the total time the test has to be done. So here we need two things: the time to review the test (numerator) from the total test time (denominator): \[ \frac{\text{Time to review the test}}{\text{Test time}}. \] The time to review the test is not given, so we must figure that out. The total test time is 1 hour. Each hour has 60 minutes. Because everything else done in the test is given in minutes, we must have both units in the same measure to be able to correctly compare them. So the hours must be converted to minutes: \[\begin{align} \text{Test time} &= 1 \text{hour} \\\\ 1 \text{ hour} &= 60 \text{ minutes} \\\\ \text{Test time} &= 60 \text{ minutes} \\\\ \frac{\text{Time to review the test}}{\text{Test time}} &= \frac{\text{Time to review the test}}{60 \text{ minutes}}. \end{align} \] Now we must find the time to review the test. The time to review is the time that's left after everything else is done since, according to the problem, "the work [is] reviewed with the time that's left." To find the time that's left, we must find the time that's used. 5 minutes of the test are going to be used to read it and 2 minutes to read each question. There are 20 questions, so \( 2 \text{ minutes} \times 20 \) minutes are going to be used to do the questions. Out of the total of 1 hour, that's the time that'll be left: \[\begin{align} \text{Time to review the test} &= 60 \text{ minutes} - 5 \text{ minutes} - 2 \text{ minutes} \times 20 \\\\ &= 55 \text{ minutes} - 40 \text{ minutes} \\\\ &= 15 \text{ minutes}. \end{align}\] Now we have the numerator and denominator: \[ \frac{\text{Time to review the test}}{\text{Test time}}= \frac{15 \text{ minutes}}{60 \text{ minutes}} = \frac{1}{4}. \] So the answer is \( \frac{1}{4} \). \(_\square\)

Problems with basic addition and subtraction tasks are relatively simple to analyze and identify the underlying procedures to solve for it. Other problems, however, may require a deeper contextual understanding, so further correlations between given data may be identified.

Jed buys some oranges. He sells \(\frac{3}{5}\) of these oranges. Of the oranges he has left, \(\frac{1}{4}\) are bad. Jed throws these away. He now has 24 oranges left. How many oranges did Jed buy?

How many \( \frac{1}{7}\)'s are there in \( 10 \frac{2}{5}?\) Measuring one number by another is just division, so the question is equivalent to asking for \( 10\frac{2}{5} \div \frac{1}{7} \). Converting the mixed number to an improper fraction and performing the operation, we get \[ \begin{align} 10\frac{2}{5} \div \frac{1}{7} &= \frac{52}{5} \div \frac{1}{7} \\ &= \frac{52}{5} \times 7 \\ &= \frac{364}{5}.\ _\square \end{align} \]

If \(k\) is a positive integer satisfying

\[ \dfrac{k}{0.5} + \dfrac{k}{0.2} + \dfrac{k}{0.25} =297,\]

find the value of \(k^2\).

• Converting Decimals and Fractions
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Home / Count in fractions | Mastery Cards

Count in fractions | Mastery Cards

Problem solving and reasoning cards allowing children to solve complex problems in various contexts for counting in fractions.

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Singapore math support and resources

Solving Fraction Word Problems Through Visualization

Through visualization, algebra-type word problems can be solved without formally introducing algebra. Here is a typical example of a non-routine word problem from a 4th grade Singapore Math workbook involving fractions:

⅓ of Amy’s money is equal to ½ of Bob’s money. Amy has $40 more than Bob. How much does Amy have?

A good visualization tool to solve such a problem is the bar model. For example,

Without explicitly representing any values with variables, students can easily visualize the problem and are able to understand there are 5 equal units involved. The extra unit that Amy has is exactly $40. So, since Amy has 3 of these units, she has 3 x $40 = $120.

Recognizing problems visually also makes them think algebraically. Without realizing it, students would be working with algebra through interesting word problems.

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For more related resources, please refer to our Bar Models page.

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Home / United States / Math Classes / 5th Grade Math / Problem Solving using Fractions

Problem Solving using Fractions

Fractions are numbers that exist between whole numbers. We get fractions when we divide whole numbers into equal parts. Here we will learn to solve some real-life problems using fractions. ...Read More Read Less

Table of Contents

What are Fractions?

Types of fractions.

• Fractions with like and unlike denominators
• Operations on fractions
• Fractions can be multiplied by using
• Let’s take a look at a few examples

Solved Examples

• Frequently Asked Questions

Equal parts of a whole or a collection of things are represented by fractions . In other words a fraction is a part or a portion of the whole. When we divide something into equal pieces, each part becomes a fraction of the whole.

For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction  \(\frac{1}{2}\) . 

Similarly, if it is divided into 4 equal parts, then each part is one fourth of the whole, that can be represented by the fraction \(\frac{1}{4}\) .

Proper fractions

A fraction in which the numerator is less than the denominator value is called a  proper fraction.

For example ,  \(\frac{3}{4}\) ,  \(\frac{5}{7}\) ,  \(\frac{3}{8}\)   are proper fractions.

Improper fractions 

A fraction with the numerator higher than or equal to the denominator is called an improper fraction .

Eg \(\frac{9}{4}\) ,  \(\frac{8}{8}\) ,  \(\frac{9}{4}\)   are examples of improper fractions.

Mixed fractions

A mixed number or a mixed fraction is a type of fraction which is a combination of both a whole number and a proper fraction.

We express improper fractions as mixed numbers.

For example ,  5\(\frac{1}{3}\) ,  1\(\frac{4}{9}\) ,  13\(\frac{7}{8}\)   are mixed fractions.

Unit fraction

A unit fraction is a fraction with a numerator equal to one. If a whole or a collection is divided into equal parts, then exactly 1 part of the total parts represents a unit fraction .

Fractions with Like and Unlike Denominators

Like fractions are those in which two or more fractions have the same denominator, whereas unlike fractions are those in which the denominators of two or more fractions are different.

For example,  

\(\frac{1}{4}\)  and  \(\frac{3}{4}\)  are like fractions as they both have the same denominator, that is, 4.

\(\frac{1}{3}\)  and  \(\frac{1}{4}\)   are unlike fractions as they both have a different denominator.

Operations on Fractions

We can perform addition, subtraction, multiplication and division operations on fractions.

Fractions with unlike denominators can be added or subtracted using equivalent fractions. Equivalent fractions can be obtained by finding a common denominator. And a common denominator is obtained either by determining a common multiple of the denominators or by calculating the product of the denominators.

There is another method to add or subtract mixed numbers, that is, solve the fractional and whole number parts separately, and then, find their sum to get the final answer.

Fractions can be Multiplied by Using:

Division operations on fractions can be performed using a tape diagram and area model. Also, when a fraction is divided by another fraction then we can solve it by multiplying the dividend with the reciprocal of the divisor. 

Let’s Take a Look at a Few Examples

Addition and subtraction using common denominator

( \(\frac{1}{6} ~+ ~\frac{2}{5}\) )

We apply the method of equivalent fractions. For this we need a common denominator, or a common multiple of the two denominators 6 and 5, that is, 30.

\(\frac{1}{6} ~+ ~\frac{2}{5}\)

= \(\frac{5~+~12}{30}\)  

=  \(\frac{17}{30}\) 

( \(\frac{5}{2}~-~\frac{1}{6}\) )

= \(\frac{12~-~5}{30}\)

= \(\frac{7}{30}\)

Examples of Multiplication and Division

Multiplication:

(\(\frac{1}{6}~\times~\frac{2}{5}\))

= (\(\frac{1~\times~2}{6~\times~5}\))                                       [Multiplying numerator of fractions and multiplying denominator of fractions]

=  \(\frac{2}{30}\)

(\(\frac{2}{5}~÷~\frac{1}{6}\))

= (\(\frac{2 ~\times~ 5}{6~\times~ 1}\))                                     [Multiplying dividend with the reciprocal of divisor]

= (\(\frac{2 ~\times~ 6}{5 ~\times~ 1}\))

= \(\frac{12}{5}\)

Example 1: Solve \(\frac{7}{8}\) + \(\frac{2}{3}\)

Let’s add \(\frac{7}{8}\)  and  \(\frac{2}{3}\)   using equivalent fractions. For this we need to find a common denominator or a common multiple of the two denominators 8 and 3, which is, 24.

\(\frac{7}{8}\) + \(\frac{2}{3}\)

= \(\frac{21~+~16}{24}\)    

= \(\frac{37}{24}\)

Example 2: Solve \(\frac{11}{13}\) – \(\frac{12}{17}\)

Solution:  

Let’s subtract  \(\frac{12}{17}\) from \(\frac{11}{13}\)   using equivalent fractions. For this we need a common denominator or a common multiple of the two denominators 13 and 17, that is, 221.

\(\frac{11}{13}\) – \(\frac{12}{17}\)

= \(\frac{187~-~156}{221}\)

= \(\frac{31}{221}\)

Example 3: Solve \(\frac{15}{13} ~\times~\frac{18}{17}\)

Multiply the numerators and multiply the denominators of the 2 fractions.

\(\frac{15}{13}~\times~\frac{18}{17}\)

= \(\frac{15~~\times~18}{13~~\times~~17}\)

= \(\frac{270}{221}\)

Example 4: Solve \(\frac{25}{33}~\div~\frac{41}{45}\)

Divide by multiplying the dividend with the reciprocal of the divisor.

\(\frac{25}{33}~\div~\frac{41}{45}\)

= \(\frac{25}{33}~\times~\frac{41}{45}\)                            [Multiply with reciprocal of the divisor \(\frac{41}{45}\) , that is, \(\frac{45}{41}\)  ]

= \(\frac{25~\times~45}{33~\times~41}\)

= \(\frac{1125}{1353}\)

Example 5: 

Sam was left with   \(\frac{7}{8}\)  slices of chocolate cake and    \(\frac{3}{7}\)  slices of vanilla cake after he shared the rest with his friends. Find out the total number of slices of cake he had with him. Sam shared   \(\frac{10}{11}\)  slices from the total number he had with his parents. What is the number of slices he has remaining?

To find the total number of slices of cake he had after sharing we need to add the slices of each cake he had,

=   \(\frac{7}{8}\) +   \(\frac{3}{7}\)   

=   \(\frac{49~+~24}{56}\)

=   \(\frac{73}{56}\)

To find out the remaining number of slices Sam has   \(\frac{10}{11}\)  slices need to be deducted from the total number,

= \(\frac{73}{56}~-~\frac{10}{11}\)

=   \(\frac{803~-~560}{616}\)

=   \(\frac{243}{616}\)

Hence, after sharing the cake with his friends, Sam has  \(\frac{73}{56}\) slices of cake, and after sharing with his parents he had  \(\frac{243}{616}\)  slices of cake left with him.

Example 6: Tiffany squeezed oranges to make orange juice for her juice stand. She was able to get 25 ml from one orange. How many oranges does she need to squeeze to fill a jar of   \(\frac{15}{8}\) liters? Each cup that she sells carries 200 ml and she sells each cup for 64 cents. How much money does she make at her juice stand?

First  \(\frac{15}{8}\) l needs to be converted to milliliters.

\(\frac{15}{8}\)l into milliliters =  \(\frac{15}{8}\) x 1000 = 1875 ml

To find the number of oranges, divide the total required quantity by the quantity of juice that one orange can give.

The number of oranges required for 1875 m l of juice =  \(\frac{1875}{25}\) ml = 75 oranges

To find the number of cups she sells, the total quantity of juice is to be divided by the quantity of juice that 1 cup has

=  \(\frac{1875}{200}~=~9\frac{3}{8}\) cups

We know that, the number of cups cannot be a fraction, it has to be a whole number. Also each cup must have 200ml. Hence with the quantity of juice she has she can sell 9 cups,   \(\frac{3}{8}\) th  of a cup cannot be sold alone.

Money made on selling 9 cups = 9 x 64 = 576 cents

Hence she makes 576 cents from her juice stand.

What is a mixed fraction?

A mixed fraction is a number that has a whole number and a fractional part. It is used to represent values between whole numbers.

How will you add fractions with unlike denominators?

When adding fractions with unlike denominators, take the common multiple of the denominators of both the fractions and then convert them into equivalent fractions. 

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Introducing operations with fractions in mathematics can sometimes be confusing for students and frustrating for teachers.

Providing some context within which students can conceptualise fractions can really help solidify these abstract concepts in students’ minds, especially when adding and subtracting fractions to amounts that are greater than a whole.

In this ‘low threshold, high ceiling’ task, students take the role of the owner of a cake wholesaler, baking and supplying cakes to local café businesses. As café owners order their weekly cakes by the slice, students are required to add unit fractions together to calculate total cake orders. They then solve problems associated with subtracting fractional remainders, using equivalent fractions and converting between improper fractions and mixed numerals.

This downloadable ‘rich task’ lesson resource is designed for teachers and students in Years 5 to 7. It is mapped against Australian Curriculum (Mathematics), with an emphasis on problem solving and reasoning in operating with fractions .

The task comes with a comprehensive grading rubric to assist teachers who wish to use the project for formal assessment purposes.

This task was developed in consultation with Greta Public School (NSW), a CHOOSE MATHS Schools Outreach partner school.

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PROBLEM OF THE WEEK ARCHIVE

Every week, we release a new multi-part problem related to a holiday, season, special event or cool STEM topic...completely for free!

The solution for each Problem of the Week (POTW) is posted here the following week (included in the downloadable POTW file).

Do you have a STEM background and love writing math problems?

Volunteer to write a Problem of the Week! You can feature your STEM career or area of study...or write a just-for-fun POTW about a holiday or event! Click the volunteer button below to learn more about how you can support MATHCOUNTS resource creation.

Counting Problems With Solutions

Counting problems are presented along with their detailed solutions and detailed explanations.

Counting Principle

Solution to problem 1.

• Using the counting principle used in the introduction above, the number of all possible computer systems that can be bought is given by N = 4 × 2 × 4 × 3 = 96

Solution to Problem 2

• Using the counting principle, the total number of possible telephone numbers is given by N = 1 × 1 × 9 × 10 × 10 × 10 × 10 × 10 × 10 = 9,000,000

Solution to Problem 3

• The total number N of different ways that the students can select his 3 books is given by N = 6 × 3 × 4 = 72

Solution to Problem 4

• The total number N of different ways that someone can go from city A to city C, passing by city B is N = 3 × 2 = 6

Solution to Problem 5

• The total number N of different ways that this man can wear one of his suits, one of his shirts and a pair of his shoes is N = 3 × 4 × 5 = 60

Solution to Problem 6

• The number N of ID cards is given by N = 10 × 9 × 8 × 7 × 6 = 30,240

Solution to Problem 7

• 26 (all letters in the alphabet) choices are possible for each of the 3 letters to be used to form the licence number. 10 choices (0,1,2,3,4,5,6,7,8,9) are possible for each of the 4 digits. The total number of licence numbers is given by N = 26 × 26 × 26 × 10 × 10 × 10 × 10 = 175,760,000

Solution to Problem 8

• a) 1 choice for the first digit. 4 choices for the last 3 digits that form the 4 digit number since repetition is allowed. Hence the number N of numbers that we may form is given by N = 1 × 4 × 4 × 4 = 64
• b) 1 choice for the first digit. 3 choices for the second digit of the number to be formed since repetition is not allowed. 2 choices for the third digit of the number to be formed. 1 choice for the fourth digit of the number to be formed. Hence the number N of numbers that we may form is given by N = 1 × 3 × 2 × 1 = 6
• c) For the number to be formed to be divisible by two, the last digit must be 2, hence one choice for this digit. 4 choices for each of the other digits since repetition is allowed. Hence the number N of numbers that we may form is given by N = 4 × 4 × 4 × 1 = 64
• d) For the number to be formed to be divisible by two, the last digit must be 2, hence one choice for this digit. 3 choices for the first digit, 2 choices for the second digit and 1 choice for the third digit that form the number. Hence the number N of numbers that we may form is given by N = 3 × 2 × 1 × 1 = 6

Solution to Problem 9

• The first time the coin is tossed, 2 different outcomes are possible (heads,tails). The second time the coin is tossed, another 2 different outcomes are possible and the third time the coin is tossed, another 2 different outcomes are possible. Hence the total number of possible outcomes is equal to N = 2 × 2 × 2 = 8

Solution to Problem 10

• Six possible outcomes for the first die (1,2,3,4,5,6) and 6 other possible outcomes for the second die. The total number of different outcomes is N = 6 × 6 = 36

Solution to Problem 11

• Two possible outcomes for the coin (heads,tails) and 6 possible outcomes (1,2,3,4,5,6) for the die. The total number of different outcomes is N = 2 × 6 = 12

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Counting is one of the most important early math skills for kids to develop. Without being able to count numbers or objects, they cannot progress to more advanced math. Our suite of free educational games gives children the solid – and enjoyable – grounding they need in this skill!

With Math Games, pupils can practice:

• Counting objects and grouped objects
• Understanding concepts such as fewer, equal and more
• Skip-counting, and working with number patterns and sequences
• Writing and comparing numbers

Our online games are tons of fun and can be played on cellphones, tablets or computers. We offer printable worksheets and a digital textbook too. Choose a topic to explore our resources for yourself!

The Basic Counting Principle

When there are m ways to do one thing, and n ways to do another, then there are m×n ways of doing both .

Example: you have 3 shirts and 4 pants.

That means 3 × 4 = 12 different outfits.

Example: There are 6 flavors of ice-cream, and 3 different cones.

That means 6 × 3 = 18 different single-scoop ice-creams you could order.

It also works when you have more than 2 choices:

Example: You are buying a new car.

How many total choices?

You can see in this "tree" diagram:

You can count the choices, or just do the simple calculation:

Total Choices = 2 × 5 × 3 = 30

Independent or Dependent?

But it only works when all choices are independent of each other.

If one choice affects another choice (i.e. depends on another choice), then a simple multiplication is not right.

Example: You are buying a new car ... but ...

the salesman says "You can't choose black for the hatchback" ... well then things change!

You now have only 27 choices.

Because your choices are not independent of each other.

But you can still make your life easier with this calculation:

Choices = 5 × 3 + 4 × 3 = 15 + 12 = 27

Many AoPS Community members and online school students have been participants at National MATHCOUNTS, including many Nationals Countdown Round participants in the past decade. MATHCOUNTS is a large national mathematics competition and mathematics coaching program that has served millions of middle school students since 1984. Sponsored by the CNA Foundation , National Society of Professional Engineers , the National Council of Teachers of Mathematics , and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three years, but cannot compete beyond their eighth grade year.

• 1 MATHCOUNTS Curriculum
• 2 Past State Team Winners
• 3.1 Sprint Round
• 3.2 Target Round
• 3.3 Team Round
• 3.4 Countdown Round
• 3.5 Chapter and State Competitions
• 3.6 National Competition
• 3.7 Ciphering Round
• 3.8 Masters Round
• 3.9 Scoring and Ranking
• 4.1 School Competition
• 4.2 Chapter Competition
• 4.3 State Competition
• 4.4.1 National Competition Sites
• 5 What comes after MATHCOUNTS?
• 6 See also...

MATHCOUNTS Curriculum

MATHCOUNTS curriculum includes arithmetic , algebra , counting , geometry , number theory , probability , and statistics . The focus of MATHCOUNTS curriculum is in developing mathematical problem solving skills.

Before 1990, MATHCOUNTS chose particular areas of mathematics to highlight each year before changing the focus of the competition more broadly to problem solving.

Past State Team Winners

• 1984: Virginia
• 1985: Florida
• 1986: California
• 1987: New York
• 1988: New York
• 1989: North Carolina
• 1991: Alabama
• 1992: California
• 1993: Kansas
• 1994: Pennsylvania
• 1995: Indiana
• 1996: Wisconsin
• 1997: Massachusetts
• 1998: Idaho
• 1999: Massachusetts
• 2000: California
• 2001: Virginia
• 2002: California
• 2003: California
• 2004: Illinois
• 2005: Texas
• 2006: Virginia
• 2007: Texas
• 2008: Texas
• 2009: Texas
• 2010: California
• 2011: California
• 2012: Massachusetts
• 2013: Massachusetts
• 2014: California
• 2015: Indiana
• 2016: Texas
• 2017: Texas
• 2018: Texas
• 2019: Massachusetts
• 2020: CANCELLED
• 2021: New Jersey
• 2022: New Jersey
• 2023: Texas
• 2024: Texas

MATHCOUNTS Competition Structure

Sprint round.

30 problems are given all at once. Students have 40 minutes to complete the Sprint Round. This round is very fast-paced and requires speed and accuracy as well. The first 20 problems are usually the easiest problems in the competition, and the last 10 problems can be as hard as some of the Team Round questions. No calculators are allowed during this round.

Target Round

8 problems are given 2 at a time. Students have 6 minutes to complete each set of two problems. Students may not go back to previous rounds (or forwards to future rounds) even if they finish before time is called. Calculators are allowed for the Target Round. Usually comprised of one "confidence booster" and one hard problem.

10 problems in 20 minutes for a team of 4 students. These problems typically include some of the most difficult problems of the competition. Use of a calculator is allowed (and required for some questions).

Countdown Round

High scoring individuals compete head-to-head until a champion is crowned. People compete from off a screen taking 45 seconds or less to finish the problem. The Countdown round is run differently in various different chapter, state, and national competitions. In the national competitions, it is the round that determines the champion. Calculators are not allowed, but scratch paper will be provided.

Chapter and State Competitions

In the chapter and state competitions, the countdown round is not mandatory. However, if it is deemed official by the chapter or state, the following format must be used:

• The 10th place written finisher competes against the 9th place written finisher. A problem is displayed, and both competitors have 45 seconds to answer the question, and the first competitor to correctly answer the question receives one point. The person who gets the most correct out of three questions (not necessarily two out of three) is the winner.
• The winner of the first round goes up against the 8th place finisher.
• The winner of the second round goes up against the 7th place finisher.

This process is continued until the countdown round reaches the top four written competitors. Starting then, the first person to get three questions correct wins (as opposed to the best-out-of-three rule).

If the countdown round is unofficial, any format may be used, and if it is unofficial it will not determine your placement towards the final results and may be determined by another format. Single-elimination bracket-style tournaments are common.

National Competition

At the national competition, there are some structural changes to the countdown round. The top 12 (not the top 10) written finishers make it to the countdown round, and the format is changed from a ladder competition to a single-elimination tournament where the top four written competitors get a bye. This setup makes it far more likely for a 12th place finisher to become champion, and it makes it less likely for a first-place written finisher to become champion, equalizing the field. But even then, a 12th place written competitor will have less of a chance to become champion than the top 4, because the top 4 get a bye. Until the semi-finals, the scoring is best out of five advances.

In the first round and the second round, the person to correctly answer the most out of 5 questions wins. However, at the semifinals, the rules slightly change— the first person to correctly answer four questions wins.

Ciphering Round

In some states (most notably Florida), there is an optional ciphering round. Very similar to countdown (in both difficulty and layout), a team sends up a representative to go against all representatives from the other teams. A problem is shown on a screen and students work fast to answer the problem. The students give their answer and after 45 seconds the answer is shown and the answers are checked to see if they are right. The fastest correct answer gets five points, the next fastest gets 4, etc. There are 4 questions per individual and teams send up 4 people. A perfect score is then 80. Often times the questions take clever reading skills. For example, one question was "How much dirt is in a 3 ft by 3 ft by 4 ft hole?" The answer was 0 because there is no dirt in a hole.

Masters Round

Top students give in-depth explanations to challenging problems. This round was optional at the state level competition and mandatory at the national competition (up to 2011). At nationals the top two on the written and countdown participate. In 2012, it was replaced by the Reel Math Challenge (now called the Math Video Challenge).

Scoring and Ranking

A team's score is the individual scores of its members divided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a relatively low team score and a phenomenal individual score, as the team score is only roughly 30% of the total team score. Note that when there are less than four members the score will become less.

MATHCOUNTS Competition Levels

School competition.

Students vie for the chance to make their school teams. Problems at this level are generally the easiest and most basic in curriculum.

Chapter Competition

Chapter competitions serve as a selection filter for state competitions. A few states don't need to host chapter competitions due to a small population size.

State Competition

The top 4 students in each state form the state team for the national competition. The coach of the top school team at the state level is invited to coach the state team at the national competition. Interestingly, the coach of a state team is not necessarily the coach of any of the state's team members. State competition competitors may be determined from the Chapter competition, based on the population of that particular state.

National Competition Sites

For many years, the National MATHCOUNTS competition was held in Washington, D.C. More recently, the competition has changed venues often.

• The 2024 competition was held in Washington, D.C.
• The 2023 competition was held in Orlando, Florida.
• The 2022 competition was held in Washington, D.C.
• The 2021 competition was an online event.
• The 2020 competition was canceled due to the COVID-19 pandemic, but was due to be held in Orlando, Florida.
• The 2019 competition was held in Orlando, Florida.
• The 2018 competition was held in Washington, D.C.
• The 2017 competition was held in Orlando, Florida.
• The 2016 competition was held in Washington, D.C.
• The 2015 competition was held in Boston, Massachusetts.
• The 2014 competition was held in Orlando, Florida.
• The 2013 competition was held in Washington, D.C.
• The 2012 competition was held in Orlando, Florida.
• The 2011 competition was held in Washington, D.C.
• The 2009 and 2010 competitions were held in Orlando, Florida.
• The 2008 competition was held in Denver, Colorado.
• The 2007 competition was held in Fort Worth, Texas.
• The 2006 competition was held in Arlington, Virginia.
• The 2005 competition was held in Detroit, Michigan.
• The 2004 competition was held in Washington, D.C.
• The 2002 and 2003 competitions were held in Chicago, Illinois.

What comes after MATHCOUNTS?

Give the following competitions a try and take a look at the List of United States high school mathematics competitions .

• American Mathematics Competitions
• American Regions Math League
• Mandelbrot Competition
• Mu Alpha Theta

See also...

• List of national MATHCOUNTS teams
• Mathematics competition resources
• Math contest books
• List of United States middle school mathematics competitions
• List of United States high school mathematics competitions
• 2006 MATHCOUNTS Countdown Video
• Mathematics competitions
• Introductory mathematics competitions

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Counting Activities

Teach your child all about counting with amazing educational resources for children. These online counting learning resources break down the topic into smaller parts for better conceptual understanding and grasp. Get started now to make counting practice a smooth, easy and fun process for your child!

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Counting Objects Within 5

Count the Dots from 1 to 3 Game

Count the dots from 1 to 3 to begin the exciting journey of becoming a math wizard.

Count to Tell How Many up to 3 Game

Ask your little one to count to tell 'how many' up to 3.

Groups of Animals Worksheet

Focus on core math skills with this fun worksheet by working with groups of animals.

Count in Patterns Worksheet

Solidify your math skills by practicing to count in patterns.

Counting Objects Within 10

Hop and Count from 1 to 10 Game

Kids must hop and count from 1 to 10 to practice counting.

Match Number and Pattern Game

Have your own math-themed party by learning how to match numbers and patterns.

Count using Pictures Worksheet

Put your skills to the test by practicing to count using pictures.

Count and Match the Objects Worksheet

Make math practice a joyride by counting and matching the objects.

Counting Objects Within 20

Number Sequence from 1 to 20 Game

Enjoy the marvel of mathematics by practicing the number sequence from 1 to 20.

Count Up to 15 Using Ten-Frames Game

Enjoy the marvel of math-multiverse by exploring how to count up to 15 using ten-frames.

Count upto 20 using Objects Worksheet

Help your child revise number sense by counting upto 20 using objects.

Count upto using 10-frames Worksheet

Focus on core math skills with this fun worksheet by counting using 10-frames.

All Counting Resources

Match Numbers up to 3 Game

Enjoy the marvel of math-multiverse by exploring how to match numbers up to 3.

Count the Dots from 5 to 8 Game

Kids must count the dots from 5 to 8 to learn counting.

Count using 5-frames Worksheet

Boost your ability to count using 5-frames by printing this playful worksheet.

Counting using Objects Worksheet

Make math practice a joyride by counting using objects.

Count Up to 20 Using Ten-Frames Game

Enjoy the marvel of math-multiverse by exploring how to count up to 20 using ten-frames.

Counting Sequence from 1 to 5 Game

Help your little one practice the counting sequence from 1 to 5.

10 and some more Worksheet

Focus on core math skills with this fun worksheet by solving to find 10 and some more.

Count using Fingers Worksheet

Focus on core math skills with this fun worksheet by practicing to count using fingers.

Count the Dots from 5 to 10 Game

Enhance your counting skills by counting the dots from 5 to 10.

Count upto 20 Objects in Array Game

Dive deep into the world of counting by helping your child count upto 20 objects in an array.

Counting Objects up to 10 Worksheet

Learn number sense at the speed of lightning by counting objects up to 10.

Counting Using 10-frames Worksheet

Put your skills to the test by counting using 10-frames.

Recognize Patterns Game

Have your own math-themed party by learning how to recognize patterns.

Subitize to Count within 10 Game

Enter the madness of math-multiverse by exploring how to subitize to count within 10.

Draw More Worksheet

Help your child draw more in order to revise number sense.

Count using Objects Worksheet

Learners must count using objects to enhance their math skills.

Count Out Objects upto 20 Game

Add more arrows to your child’s math quiver by helping them count out objects upto 20.

Count the Dots from 1 to 5 Game

Teach your little one to count the dots from 1 to 5.

Teen Numbers Worksheet

Kids must practice teen numbers by printing this playful worksheet.

Draw More to Match the Number Worksheet

Focus on core math skills with this fun worksheet by drawing more to match the number.

Count Objects to 10 Game

Let your child see the world through math-colored shades with our 'Count Objects to 10' game!

Count to 20 Game

Ask your little one to count to 20 to play this game.

How many Objects Worksheet

Pack your math practice time with fun by identifying how many objects.

Count and Color Fruits within 20 Worksheet

An engaging worksheet that helps kids practice counting up to 20 by identifying and coloring different fruits.

Subitize to Count within 5 Game

Learn to solve math problems by subitizing to count within 5.

Count Objects in Rectangular Arrays Game

Shine bright in the math world by learning how to count objects in rectangular arrays.

Color Fruits (Within 3) Worksheet

Engaging worksheet where students enhance counting skills by coloring and counting depicted fruits.

Counting in a Group Worksheet

Use this printable worksheet to count in a group to strengthen your math skills.

Count Objects to 5 Game

Kids must count objects to 5 to practice counting.

Count All and Match the Number Game

Kids must count all and match the number to practice counting.

Count and Color Shapes within 20 Worksheet

Engaging worksheet to improve counting skills by identifying and coloring shapes within 20.

Color Fruits (Within 5) Worksheet

An engaging worksheet tasking students to count and color various fruits within a 5 range.

Count Objects in Linear Arrangement Game

Add more arrows to your child’s math quiver by helping them count objects in linear arrangement.

Count to Tell How Many from 6 to 10 Game

Ask your little one to count to tell 'how many' from 6 to 10.

Making Groups of Objects Worksheet

Learn number sense at the speed of lightning by making groups of objects.

Count and Match the Number within 20 Worksheet

Sharpen your counting skills with this engaging worksheet on matching numbers within 20.

Your one stop solution for all grade learning needs.

Reading & Math for K-5

• Kindergarten
• Learning numbers
• Comparing numbers
• Place Value
• Roman numerals
• Subtraction
• Multiplication
• Order of operations
• Drills & practice
• Measurement
• Factoring & prime factors
• Proportions
• Shape & geometry
• Data & graphing
• Word problems
• Children's stories
• Leveled Stories
• Sentences & passages
• Context clues
• Cause & effect
• Compare & contrast
• Fact vs. fiction
• Fact vs. opinion
• Main idea & details
• Story elements
• Conclusions & inferences
• Sounds & phonics
• Words & vocabulary
• Reading comprehension
• Early writing
• Numbers & counting
• Simple math
• Social skills
• Other activities
• Dolch sight words
• Fry sight words
• Multiple meaning words
• Prefixes & suffixes
• Vocabulary cards
• Other parts of speech
• Punctuation
• Capitalization
• Narrative writing
• Opinion writing
• Informative writing
• Cursive alphabet
• Cursive letters
• Cursive letter joins
• Cursive words
• Cursive sentences
• Cursive passages
• Grammar & Writing

Breadcrumbs

• Math by topic

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Counting Worksheets

Counting worksheets for kindergarten through grade 2.

Our first counting worksheets concentrate on counting objects ; latter exercises focus on sequential counting including skip counting and counting backwards.

Choose your grade / topic:

Kindergarten counting worksheets.

Grade 1 number charts &  counting worksheets

Grade 1 number patterns

Grade 2 skip counting worksheets

Grade 3 skip counting worksheets.

Topics include:

• Counting objects and circling the correct number (1-20)
• Counting and coloring objects (1-5)
• Counting objects and writing numbers (1-5, 1-10)
• Reading numbers and circling objects
• Count objects up to 20 and write the number
• Count by ones up to 10 (missing numbers)
• What number comes next? Before?  (1-20)
• Count backwards 10-1
• Skip count by 2's or 10's

Grade 1 number charts and counting worksheets

• Counting objects (1-20)
• Number charts 1-100
• Number charts: counting by 2s (even, odd)
• Number charts: counting by 3's, 4's, 5's, 10's
• Counting backwards 100-1
• Counting backwards by 2's, 5's

Grade 1 number patterns worksheets

• Counting patterns: fill in the missing numbers by skip counting (ascending /descending, up to 50 or 100)
• Extending number patterns (ascending / descending, up to 100)
• Identifying number patters & fill in the missing numbers
• Input - Output charts (numbers up to 100)
• Skip count by 10 starting from 10
• Skip count by 10 starting from 1-10
• Skip count by 10 starting from 1-100
• Skip count by 20, 25, 50, 100
• Skip count by 2's (even, odd)
• Skip count by 3's, 4's, 5's, 6's, 7's, 8's, 9's
• Skip count backwards by 2's, 5's, 10's
• Skip count by 100 starting from random numbers
• Skip count by 150, 200 or 250

Related topics

Learning numbers worksheets

Comparing numbers worksheets

Sample Counting Worksheet

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