Fractions operations

Here you will learn about fractions operations, including how to add, subtract, multiply and divide with fractions.

Students will first learn about fractions operations as part of number and operations in fractions in elementary school. They will continue to build on this knowledge in the number system in 6th grade and 7th grade.

What are fractions operations?

Fractions operations are when you add, subtract, multiply or divide with fractions.

For example,

Adding and subtracting fractions

Adding and subtracting fractions means finding the sum or the difference of two or more fractions.

In order to do this, the fractions must have a common denominator (bottom number).

The numerator shows the number of parts out of the whole and the denominator shows how many equal parts the whole is divided into.

To add or subtract the numerators (top numbers) and keep the denominators the same.

For example,

The equation is taking \, \cfrac{1}{8} \, away from \, \cfrac{4}{8} \, .

Since the denominators are the same, the parts are the same size.

You subtract to see how many parts are left: 4-1 = 3.

There are 3 parts and the size is still eighths, so the denominator stays the same.

When fractions have unlike denominators, create equivalent fractions with common denominators to solve.

For example,

The parts are NOT the same size, since the denominators are different.

Use equivalent fractions to create a common denominator of 10.

Multiply the numerator and denominator of \, \cfrac{2}{5} \, by 2.

\cfrac{2 \, \times \, 2}{5 \, \times \, 2}=\cfrac{4}{10}

Add to find how many parts there are in all: 2 + 4 = 6.

There are 6 parts and the size is still tenths, so the denominator stays the same.

The sum could also be written as the equivalent fraction \, \cfrac{3}{5} \, .

Step-by-step guide: Adding fractions

Step-by-step guide: Subtracting fractions

Step-by-step guide: Adding and subtracting fractions

Multiplying and dividing fractions

Multiplying and dividing fractions means using multiplication and division to calculate with fractions. Fraction multiplication and division can be solved using models or an algorithm.

For example,

Using models:

In the model, \, \cfrac{2}{3} \, is yellow and \, \cfrac{1}{2} \, is blue.

The product is where the fractions overlap in green.

The model shows \, \cfrac{2}{3} \, of \, \cfrac{1}{2}, \, so \, \cfrac{1}{2} \times \cfrac{2}{3} = \cfrac{2}{6} \, .

Using the algorithm:

To multiply fractions, you multiply the numerators together, and multiply the denominators together:

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

You can also divide fractions with a model or an algorithm.

For example,

Using models:

Think of this equation as how many \, \cfrac{1}{4} \, fit into \, \cfrac{1}{2} \, .

In the model, \, \cfrac{1}{2} \, is orange and \, \cfrac{1}{4} \, is yellow.

To divide into equal groups, use the equivalent fraction \, \cfrac{2}{4} \, .

The quotient is the final fraction formed when \, \cfrac{2}{4} \, is put into a group of \, \cfrac{1}{4} \, .

Two groups of \cfrac{1}{4} can be made, so \cfrac{1}{2} \div \cfrac{1}{4}=2.

Using the algorithm:

KEEP the first fraction, FLIP the second fraction, CHANGE to multiplication.

\cfrac{1}{2} \div \cfrac{1}{4}

Keep the dividend (first fraction): \, \cfrac{1}{2}

Take the reciprocal of the divisor (flip the second fraction): \, \cfrac{1}{4} \rightarrow \cfrac{4}{1}

Change to multiplication: \, \cfrac{1}{2} \times \cfrac{4}{1}

Multiply the fractions: \, \cfrac{1}{2} \times \cfrac{4}{1}=\cfrac{4}{2} \, which simplifies to 2.

\cfrac{1}{2} \div \cfrac{1}{4}=2

Since \, \cfrac{1}{2} \, is larger than \, \cfrac{1}{4} \, , the answer makes sense.

A larger number divided by a smaller number, will have a quotient of greater than 1.

Notice that it is not necessary to create a common denominator to multiply and divide fractions when using the algorithm, like it is to add and subtract fractions.

Step-by-step guide: Multiplying fractions

Step-by-step guide: Dividing fractions

Step-by-step guide: Multiplying and dividing fractions

The algorithm for dividing fractions involves using the reciprocal.

When two numbers are multiplied by something other than 1, and have a product of 1, they are reciprocals.

This is also known as the multiplicative inverse.

For example,

The reciprocal of all numbers can be found by writing the number as a fraction and then flipping it so that the numerator becomes the denominator and the denominator becomes the numerator.

Step-by-step guide: Reciprocal

Step-by-step guide: Multiplicative inverse

What are fractions operations?

Common Core State Standards

How does this relate to 4th grade math, 5th grade math, and 6th grade math?

• Grade 4 – Number and Operations – Fractions (4.NF.B.3a)
  Understand addition and subtraction of fractions as joining and separating parts referring to the same whole.

• Grade 4 – Number and Operations – Fractions (4.NF.B.3c)
  Add and subtract mixed numbers with like denominators, for example, by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.

• Grade 4 – Number and Operations – Fractions (4.NF.B.4b)
  ​​Understand a multiple of \, \cfrac{a}{b} \, as a multiple of \, \cfrac{1}{b} \, , and use this understanding to multiply a fraction by a whole number.

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

• Grade 5 – Number and Operations – Fractions (5.NF.B.4b)
  Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

• Grade 6 – Number System (6.NS.A.1)
  Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

How to use fractions operations

There are a lot of ways to use fractions operations. For more specific step-by-step guides, check out the fraction pages linked in the “What are fractions operations?” section above or read through the examples below.

Fractions operations examples

Example 1: adding fractions with like denominators

Solve \, \cfrac{5}{8}+\cfrac{7}{8} \, .

1. Add or subtract the numerators (top numbers).

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

2Write your answer as a fraction.

There are 12 parts, and the size is still eighths, so the denominator stays the same.

\cfrac{12}{8} \, is an improper fraction and converts to the mixed number 1 \, \cfrac{4}{8} \, .

\cfrac{5}{8}+\cfrac{7}{8}=1 \cfrac{4}{8}

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

Example 2: subtracting fractions with unlike denominators

Solve \cfrac{6}{10}-\cfrac{1}{3} \, .

Add or subtract the numerators (top numbers).

Now use the equivalent fractions to solve: \, \cfrac{18}{30}-\cfrac{10}{30} \, .

Since the denominators are the same, the parts are the same size. You subtract to see how many parts are left: 18-10 = 8.

Example 3: multiplying a mixed number by a fraction with the algorithm

Solve 1 \, \cfrac{11}{12} \times \cfrac{3}{4} \, .

Convert whole numbers and mixed numbers to improper fractions.

Convert the mixed number to an improper fraction.

Example 4: dividing a fraction by a fraction

Divide the numbers \, \cfrac{1}{12} \div \cfrac{1}{4} \, .

If possible, simplify or convert to a mixed number.

\cfrac{4}{12}=\cfrac{1}{3}

This can also be solved with a model.

You can think of this equation as how many \, \cfrac{1}{4} \, fit into \, \cfrac{1}{12} \, .

In the model, \, \cfrac{1}{12} \, is yellow and \, \cfrac{1}{4} \, is orange.

To divide into equal groups, the fractional pieces need to be the same size.

Use \, \cfrac{1}{12} \, and \, \cfrac{3}{12} \, to solve.

The quotient is the final fraction formed when \, \cfrac{1}{12} \, is put into groups of \, \cfrac{3}{12} \, .

One out of the three parts are filled, so \, \cfrac{1}{12} \div \cfrac{3}{12}=\cfrac{1}{3} \, .

Example 5: adding mixed numbers with unlike denominators

There are 2 \, \cfrac{1}{3} \, pounds of red apples and 4 \, \cfrac{1}{6} \, pounds of green apples.

How many pounds of apples are there in all?

Add or subtract the whole numbers.

Create common denominators (bottom numbers).

Since \, \cfrac{1}{3} \, and \, \cfrac{1}{6} \, do not have like denominators, the parts are NOT the same size.

Use equivalent fractions to create a common denominator.

A common denominator of 6 can be used.

Multiply the numerator and denominator of \, \cfrac{1}{3} \, by 2 to create an equivalent fraction.

\cfrac{1}{3}=\cfrac{1 \, \times \, 2}{3 \, \times \, 2}=\cfrac{2}{6} \quad and \quad \cfrac{1}{6}

Add or subtract the fractions.

Since the denominators are the same, the parts are the same size. You add to see how many parts there are in total: 2 + 1 = 3.

There are 3 parts, and the size is still sixths, so the denominator stays the same.

Write your answer as a mixed number.

Add the whole number and fraction together.

You can also write this answer as the equivalent mixed number 6 \, \cfrac{1}{2} \, .

There are 6 \, \cfrac{1}{2} \, pounds of apples in all.

Example 6: word problem dividing with fractions

Each seed needs \, \cfrac{1}{5} \, cup of soil. How many seeds can be planted with 11 cups of soil?

Teaching tips for fractions operations

• Fraction work in lower grades emphasizes understanding through models, including area models and number lines. To support students in upper grades, always have digital or physical models available for students to use as they work with fractions operations.

• Fraction worksheets can be useful when students are developing understanding around basic operations with fractions. However, when students have successful strategies and can flexibly operate, make the practice more engaging by using math games or real world projects that allow students to use fractions in a variety of situations.

• Highlight patterns within and between the operations as students are learning and encourage them to look for patterns on their own. This will help students make sense of the algorithms used to operate with fractions and minimize conceptual errors.

• Let students find reciprocal numbers on their own by consistently asking questions such as, “What number multiplied by 7 will have a product of 1 ?” Each time this is discussed, write these equations on an anchor chart and students will begin to see a pattern over time.
  
  Although worksheets can serve a purpose and help with skill and test prep practice, having students discover and make sense of mathematical concepts is more meaningful for building long lasting understanding.

Easy mistakes to make

• Adding or subtracting the denominators of fractions
  The denominator indicates the size of the parts or whole group. This does not change when a fraction is being added or subtracted, so keep the denominator the same and only add or subtract the numerators.
  
  For example,
  
  

  

• Adding or subtracting unlike denominators
  To add or subtract, fractions need to have like denominators. Use equivalent fractions to create a common denominator before adding or subtracting any fractions that have unlike denominators.
  
  For example,
  
  

  

• Finding a common denominator when multiplying or dividing fractions
  To use the algorithm, you do not need to find a common denominator to multiply or divide fractions. Using a common denominator won’t automatically give you the wrong answer, but it creates unnecessary calculations and leaves more room for error.
  
  For example,
  
  

  

• Using the reciprocal when multiplying
  When multiplying fractions, using the reciprocal (flipping the second fraction) changes the value of the equation and will lead to a different answer. The reciprocal is used in the algorithm for division.
  
  For example,
  
  

  

• Forgetting how to find the reciprocal of a whole number
  Whole numbers can be written as an improper fraction and then the numerator and denominator of the improper fraction can be flipped to find the reciprocal of the whole number.
  
  For example,
  16 can be written as \, \cfrac{16}{1} \, and the reciprocal is \, \cfrac{1}{16} \, .

The next lessons are

• Decimals
• Exponents
• Algebraic expression
• Percent
• Converting fractions decimals and percents
• Interpret fractions as division
• Fraction word problems

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