problem solving improper fractions

Improper Fractions

An Improper Fraction has a top number larger than (or equal to) the bottom number. It is usually " top-heavy "

More Examples:

See how the top number is bigger than (or equal to) the bottom number? That makes it an Improper Fraction, (but there is nothing wrong about Improper Fractions ).

Three Types of Fractions

There are three types of fraction:

A Fraction (such as 7 / 4 ) has two numbers:

Numerator Denominator

The top number (the Numerator) is the number of parts we have . The bottom number (the Denominator) is the number of parts the whole is divided into .

Example: 7 / 4 means:

• We have 7 parts
• Each part is a quarter ( 1 / 4 ) of a whole

So we can define the three types of fractions like this:

Improper Fraction

So an improper fraction is a fraction where the top number (numerator) is greater than or equal to the bottom number (denominator): it is top-heavy .

Can be Equal

What about when the numerator equals the denominator? Such as 4 4 ?

Well it is the same as a whole, but it is written as a fraction, so most people agree it is a type of improper fraction.

Improper Fractions or Mixed Fractions

We can use either an improper fraction or a mixed fraction to show the same amount.

For example 1 3 4 = 7 4 , as shown here:

Converting Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, follow these steps:

• Divide the numerator by the denominator.
• Write down the whole number answer
• Then write down any remainder above the denominator.

Example: Convert 11 4 to a mixed fraction.

Write down the 2 and then write down the remainder (3) above the denominator (4).

That example can be written like this:

Example: Convert 10 3 to a mixed fraction.

Converting mixed fractions to improper fractions.

To convert a mixed fraction to an improper fraction, follow these steps:

• Multiply the whole number part by the fraction's denominator.
• Add that to the numerator
• Then write the result on top of the denominator.

Example: Convert 3 2 5 to an improper fraction.

Multiply the whole number part by the denominator:

Add that to the numerator:

Then write that result above the denominator:

We can do the numerator in one go:

Example: Convert 2 1 9 to an improper fraction.

Are improper fractions bad .

NO, they aren't bad!

For mathematics they are actually better than mixed fractions. Because mixed fractions can be confusing when we write them in a formula: should the two parts be added or multiplied?

But, for everyday use , people understand mixed fractions better.

Example: It is easier to say "I ate 2 1 4 sausages", than "I ate 9 4 sausages"

We Recommend:

• For Mathematics: Improper Fractions
• For Everyday Use: Mixed Fractions

• Math Article

Improper Fractions

Improper fractions , with the name, signifying that the fractions are not done in a proper manner for any number, object or element. In Maths, a fraction is a part of a whole. Fractions have two parts, numerator and denominator. If  ⅓ is a fraction, then 1 is the numerator, and 3 is the denominator. An improper fraction has a numerator greater than the denominator. For example, 3/2 is an improper fraction, but 2/3 is a proper fraction , whose  denominator is greater than the numerator.  

In this article, you will find all the topics related to improper fractions. Also, we perform various arithmetic operations on these fractions, such as addition, division, multiplication, etc.

Table of contents:

What are Improper Fractions?

An improper fraction is defined as a fraction whose numerator is greater than the denominator. Suppose, x/y is an improper fraction, such that x > y. It is therefore, the improper fraction is always greater than one.

Examples of Improper Fractions are:

How to Simplify Improper Fractions?

We have understood what an improper fraction is. Now, let us discuss here how to simplify such fractions here.

• Step 1: Determine if the given fraction is improper or not.
• Step 2: Now interpret the denominator and check for how many parts it is dividing the numerator.
• Step 3: Check the common factors of numerator and denominator
• Step 4: Cancel the like terms both from numerator and denominator.

The resulted fraction is the simplified fraction. Let us understand these steps with the help of examples.

Example: Simplify the fraction 24/10.

Solution: Since we know 24>10, thus 24/10 is an improper fraction.

Let us factorise the numerator and denominator.

24 = 2 x 2 x 2 x 3

Since we can see 2 is the common factor for 24 and 10, so after cancelling 2 from the numerator and denominator, we get;

(2 x 2 x 3)/5 = 12/5

Improper Fractions to Mixed Fractions

To convert an improper fraction to a mixed fraction, divide the denominator by the numerator. The quotient becomes the whole number, the remainder will be the numerator, and the divisor becomes the denominator.

Let us consider an example. Convert 17/4 to a mixed fraction.

When we divide 17 by 4, the quotient is 4 and the remainder is 1. So the mixed fraction is 4  1/4.

Although an improper fraction can be converted to a mixed fraction, there are situations where expressing a fraction as an improper fraction minimizes a lot of confusion, especially when the fractions are expressed in calculations.

For example, consider 3 + 6  2/3.

Is it 3 + 6  +  2/3 or 3 + 6  × 2/3 ?

This confusion can be removed by writing it as  3 + 20/3.

Mixed to Improper Fraction

Let us say 3 1 / 4 is a mixed fraction to be converted into an improper fraction. Multiply 4 by 3 and add with the numerator 1, we get; 4 x 3 + 1 = 13 Hence, the required numerator of the improper fraction is 13 while the denominator remains the same, i.e., 4 Therefore, the required fraction is 13/4

Another Method: Write the mixed number as a sum of the whole number and proper fraction, such that; 3 + ¼ Now rationalise the denominators and write the two parts with a common denominator. 12/4 + ¼ Now add the fractions 13/4

Addition of Improper Fractions

The addition of improper fractions can have two scenarios. If the denominators of all the fractions are equal, we can add all the numerators and keep the same denominator.

For example, (17/4) + (9/4) + (5/4)

Since the denominator of all three fractions is equal, we add all the numerators by keeping the common denominator as 4.

Hence, we get the following:

(17/4) + (9/4) + (5/4) = (17 + 9 + 5) / 4

(17/4) + (9/4) + (5/4) = 31/4.

The other case is that denominators of fractions are not equal. Here, the process is slightly different and involves the calculation of the least common multiple of the denominators.

You can learn the calculation of the least common multiples here.

Consider the example, (15/3) + (3/4) + (5/2)

Here, the denominators of the fractions are different. (i.e., 3, 4, 2), and take the LCM of 2, 3, and 4.

Hence, to add the given fractions, follow the below steps:

• The least common multiple of 3, 4, and 2 is 12.
• Now the fraction we get by adding all these fractions will have 12 as the denominator.
• Divide the LCM by each of the denominators and multiply the quotient by the numerators.
• The numerator of the new fraction will be the sum of all the numbers obtained in the previous step.
• Now, in the first fraction, the denominator is 3. LCM 12 divided by 3 is 4. The numerator is 15. 15 x 4 is 60. Similarly, for the other two fractions, the numbers are 9, and 30.

Therefore, (15/3) + (3/4) + (5/2) = (60/12) + (9/12) + (30/12)

(15/3) + (3/4) + (5/2) = (60 + 9 + 30) / 12

(15/3) + (3/4) + (5/2) = 99/12.

Now, the sum of fractions can be simplified as:

(15/3) + (3/4) + (5/2) = 33/4

convert into mixed fractions, if required.

Hence, the mixed fraction is:

(15/3) + (3/4) + (5/2) = 8  1/4.

Subtraction of Improper Fractions

In a similar manner, as we added the improper fractions, we can also subtract them. 

• The first step is to check if denominators are the same or not
• The second is to rationalise the denominators 
• The last step is to subtract the given fractions and simply, if required

Let us see an example:

Subtract 5/2 – 7/3.

LCM (2,3) = 6

(5/2 x 3/3) – (7/3 x 2/2) 

= 15/6 – 14/6

= (15-14)/6

Related Articles

Solved examples of improper fractions.

Multiply 3 by 9/7.

Given: 3 x 9/7 

3 x 9/7  = 27/7

Convert the improper fraction 5/4 into a mixed fraction.

To convert an improper fraction into a mixed fraction, first, divide the number 5 by 4.

If we divide 5 by 4, we get the quotient as 1 and the remainder as 1.

Hence, keep the quotient as the whole number part and the remainder as the fraction’s numerator part and the denominator as 4.

Hence, 5/4 written in the mixed fraction is 1  1/4.

i.e., 5/4 = 1  1/4.

Convert the mixed number 4  2/3 into an improper fraction.

Given: Mixed fraction = 4  2/3

To convert the mixed fraction into an improper fraction, follow the below steps:

Step 1: Multiply 3 by 4, and we get 12.

Step 2: Now, add 12 and 2, we get 14

I,e 3(4) + 2 =14

Now, put the number 14 in the fraction’s numerator and keep the number 3 as the denominator.

So, 14/3 is the required fraction.

Write 3 1/4 as an improper fraction.

Given that 3 1/4 is a mixed number.

Now, multiply denominator 4 by the whole number 3.

Now, add 12 to the numerator.

12 + 1 = 13

Hence, 13 is the new numerator of the improper fraction, whereas the denominator remains the same.

3 1 / 4  = 13/4

Video Lesson

Practice Questions

Convert the given improper fractions into mixed numbers.

Convert into improper fractions.

• 4 1 / 2 = ?
• 6 1 / 9 = ?

Frequently Asked Questions on Improper Fraction

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Improper Fraction Worksheets

Welcome to our Improper Fraction Worksheets page. Here you will find a wide range of free printable Fraction Worksheets which will help your child understand and practice how to convert improper fractions to mixed numbers.

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Improper fractions are fractions with an absolute value greater than 1.

We have more information about what proper and improper fractions are with some examples below.

There are also a range of sheets to help develop the skill of converting improper fractions - both easier sheets with visual models and harder sheets without any visual aids.

Using these sheets will help your child to:

• understand what improper fractions are;
• convert improper fractions to mixed numbers;
• convert mixed numbers to improper fractions.

Want to test yourself to see how well you have understood this skill?

• Try our NEW quick quiz at the bottom of this page.

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What are proper, improper and mixed fractions.

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Convert Improper Fractions Online Quiz

A fraction is a number with the form: \[ {n \over d} \]

• If the numerator is smaller than the denominator (if n < d), then the fraction is called a proper fraction , and its value is less than 1.
• If the numerator is greater than the denominator (if n > d), then the fraction is called an improper fraction , and its value is greater than 1.
• If the numerator is equal to the denominator (if n = d), then the fraction's value is equal to 1.
• A mixed fraction (or mixed number) is a number that has a whole number part to it, and a fraction part to it.
• Please note: if the value of n or d is negative, then we compare the absolute values of the numerator and denominator.

Types of Fractions Examples

\[{3 \over 7} \; , \; {2 \over 10} \; and \; {17 \over 26} \] are all proper fractions.

\[ {31 \over 5} \; , \; {15 \over 10} \; , \; -{7 \over 4} \; and \; {37 \over 23} \] are all improper fractions.

\[ 3{1 \over 5} \; , \; 2 {5 \over 8} \; and \; 4 {1 \over 2} \] are all mixed numbers or mixed fractions.

Improper Fractions Support

Here you will find the support page on how to find equivalent fractions if you get stuck or want some support.

• Convert Improper Fractions Support

Converting Improper Fraction Worksheets

Here you will find a selection of Fraction worksheets designed to help your child to understand and practice converting improper fractions.

The sheets are carefully graded so that the easiest sheets come first, and the most difficult sheet is the last one.

We have split the sheets into two sections with the first section looking at improper fractions visually.

The sheets in the second sections are more abstract (with no visual models) and tricky!

These sheets are aimed at students at 4th and 5th grade, with the more abstract worksheets for 5th graders.

Converting Improper Fraction VISUAL Worksheets

Improper fraction worksheets to mixed numbers visual fractions.

These sheets are all about using circles to create visual fractions to aid understanding of improper fractions and mixed numbers.

• Improper Fractions to Mixed Numbers Sheet 1
• PDF version
• Improper Fractions to Mixed Numbers Sheet 2

Mixed Numbers to Improper Fraction Worksheets Visual Fractions

• Mixed Numbers to Improper Fractions Sheet 1
• Mixed Numbers to Improper Fractions Sheet 2

Convert Improper Fraction Worksheets - Visual Fractions

These sheets involve both converting improper fractions to mixed numbers, and also converting mixed numbers to improper fractions.

• Convert Improper Fractions Sheet 1
• Convert Improper Fractions Sheet 2
• Convert Improper Fractions Sheet 3
• Convert Improper Fractions Sheet 4
• Convert Improper Fractions Sheet 5

Converting Improper Fraction Trickier Worksheets

Convert improper fractions to mixed fractions trickier sheets.

These improper fraction worksheets are for students who are more confident working in a more abstract way.

• Convert Improper Fractions to Mixed Fractions 1
• Sheet 1 Answers
• Converting Improper Fractions to Mixed Fractions 2
• Sheet 2 Answers

Convert Mixed Numbers to Improper Fractions Trickier sheets

• Convert Mixed Numbers to Improper Fractions 1
• Convert Mixed Numbers to Improper Fractions 2

Convert Mixed Numbers to Improper Fractions Walkthrough Video

This short video walkthrough shows several problems from our Convert Mixed Numbers to Improper Fractions Sheet 1 being solved and has been produced by the West Explains Best math channel.

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More Recommended Math Resources

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• Improper Fraction Calculator

If you just want a calculator to do the hard work for you, then try our Improper Fraction Calculator.

The calculator will convert any mixed number to an improper fraction, or an improper fraction into a mixed number.

The great thing is that it will also show you all the working out - job done!

What is a Mixed Number

This support page introduces mixed numbers with some examples and an interactive teaching app.

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Simplifying Fractions

Take a look at our Simplifying Fractions Practice Zone or try our worksheets for finding the simplest form for a range of fractions.

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You can print out your results or benchmark your scores against future achievements.

Good for practising equivalent fractions as well as converting to simplest form.

Great for using with a group of children as well as individually.

• Simplify Fractions Practice Zone
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Proper Fractions, Improper Fractions and Mixed Numbers (Mixed Fractions)

These lessons, with videos, examples, and solutions, help students learn about Improper Fractions.

Related Pages Mixed Numbers & Improper Fractions More Lessons for Arithmetic Math Worksheets

What are Proper Fractions? Proper Fractions are fractions where the numerator is smaller than the denominator, for example:

What are Mixed Numbers (Mixed Fractions)? Mixed Numbers contain a whole number and a proper fraction. A mixed number can also be expressed as an improper fraction, for example:

The following diagrams show examples of proper fraction, improper fraction, and mixed number. Scroll down the page for more examples and solutions on proper fractions, improper fractions and mixed numbers.

This fun video will illustrate the different types of fractions

How to convert between improper fractions and mixed numbers? Convert from Improper Fraction to Mixed Number An improper fraction can be converted into a mixed number - first divide the numerator by the denominator. The resultant becomes the whole number, and the remainder becomes the numerator of the new fraction. The denominator of the new fraction is the same as the original denominator. If there is no remainder, then there is no fraction - the result is simply a whole number.

Convert from Mixed Number to Improper Fraction This following video will illustrate how to convert from a mixed number to an improper fraction

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Helping with Math

Improper Fractions

Introduction.

Numbers that represent a portion of a whole are called fractions . Let us use the square below as an example.

Since it has been cut into 4 equal parts, each piece represents one out of four. Mathematically, each piece is described as $\frac{1}{4}$. The number at the top is referred to as the numerator, and its bottom number is the denominator. $\frac{1}{4}$ is an example of a proper fraction because the denominator is greater than the numerator. 

Otherwise, we will have an improper fraction when the numerator exceeds the denominator. Let us suppose that we instead have two squares and that each square is divided into four equal parts, as illustrated below.

There are 7 yellow parts out of 8 equal parts, which can be expressed as $\frac{7}{8}$. This number is an example of an improper fraction.

The topic of improper fractions will be explored in this article, along with methods for converting them to mixed fractions, simplifying them, and finding solutions to problems involving them.

What is an Improper Fraction?

When the numerator exceeds the denominator, the fraction is called improper. Suppose $\frac{a}{b}$ is an improper fraction; thus, a must be greater than b. Examples of improper fractions are $\frac{4}{3}$, $\frac{7}{4}$, $\frac{8}{5}$, and $\frac{11}{2}$. 

A fraction has two main types, proper fractions and improper fractions. We have an improper fraction when the numerator is higher than the denominator; otherwise, it is a proper fraction.

Simplifying Improper Fractions

Simplifying improper fractions is the same thing as reducing the fractions. We do this by dividing the denominator and the numerator by their greatest common factor . 

Let us say, for example; we have the improper fraction $\frac{18}{12}$. The numerator in the fraction is 18, while the denominator is 12. These are the factors of 18 and 12.

Factors of 18: { 2, 3, 6, 9 }

Factors of 12: { 2, 3, 4, 6 }

From the list above, the common factors of 18 and 12 are 2, 3, and 6. To simplify the $\frac{18}{12}$, we will use 6 as the divisor since it is the greatest common factor. Hence, we have,

$\frac{18÷6}{12÷6}$ = $\frac{3}{2}$

Therefore, $\frac{18÷6}{12÷6}$ is $\frac{3}{2}$ when simplified.

Improper Fraction vs Mixed Number

When the numerator exceeds the denominator, the fraction is called an improper fraction. A mixed number is a way of writing fractions that are greater than one. A whole number and a proper fraction make up a mixed number.

Examples of improper fractions are:

Examples of mixed numbers are:

Improper Fraction Conversions

We can easily convert improper fractions to mixed numbers or decimals by following simple steps.

Converting Improper Fractions to Mixed Numbers

We shall divide the numerator by the denominator to make an improper fraction into a mixed number. To complete the mixed number, the quotient is the whole number part; the numerator is the remainder, while the divisor is the denominator.

For example, let us say that we must convert the improper fraction $\frac{23}{7}$ to a mixed number. When we divide 23 by 7, we will get 3 with the remainder of 2. We can form 3 equal groups of 7 with the remainder of 2. The image below illustrates the division process.

To form the equivalent mixed number, we will use the quotient ( 3 ) as the whole number part, and the fractional part is $\frac{2}{7}$. Notice that we use the same denominator.

$\frac{23}{7}$ =3$\frac{2}{7}$

To check the conversion, we must multiply the denominator by the whole number part and then add the answer to the numerator. The denominator stays the same. Hence, we have, 

( 7 × 3 ) + 2 = 21 + 2 = 23 

Therefore, the improper fraction $\frac{23}{7}$ is equal to 3$\frac{2}{7}$ in mixed number.

For another example, let us convert $\frac{9}{2}$ to a mixed number. When we divide 9 by 2, we will have 4 with the remainder of 1. The figure below shows the division process.

( 2 × 4 ) + 1 = 8 + 1 = 9

  Therefore, the improper fraction $\frac{9}{2}$ is equal to 4$\frac{1}{2}$ in mixed number.

Converting Improper Fractions to Decimals

To convert improper fractions to decimals, we must divide the numerator by the denominator. The process works the same way in converting improper fractions to mixed numbers. 

For example, let us say that we must convert the improper fraction $\frac{11}{2}$ to decimals. 

Therefore, $\frac{11}{2}$ is equal to 5.5 in decimals.

Addition and Subtraction of Improper Fractions

When adding or subtracting improper fractions, it is important to identify if you are adding or subtracting like or unlike terms. Like fractions have the same denominators; otherwise, they are unlike fractions. For example, $\frac{4}{3}$ and $\frac{5}{3}$ are like improper fractions, while $\frac{5}{4}$ and $\frac{7}{6}$ are unlike improper fractions.

Adding Like Improper Fractions

To add like improper fractions, we must add the numerators of the given fractions and then copy the same denominator. Simplify the answer when necessary. 

Let us say, for example, let us add the like improper fractions $\frac{7}{4}$ and $\frac{9}{4}$. Hence we have,

$\frac{7}{4}$+$\frac{9}{4}$=$\frac{16}{4}$

Since $\frac{16}{4}$ can still be simplified, the sum of $\frac{7}{4}$ and $\frac{9}{4}$ is equal to 16.

For another example, let us say we have $\frac{11}{3}$+$\frac{14}{3}$. Since they have the same denominators, it is easy to get the sum of these fractions by adding the numerators and copying the same denominator.

$\frac{11}{3}$+$\frac{14}{3}$=$\frac{25}{3}$

Thus, the sum of $\frac{11}{3}$ and $\frac{14}{3}$ is equal to $\frac{25}{3}$.

Adding Unlike Improper Fractions

To add unlike improper fractions, we must first make the given improper fractions into proper fractions, then proceed to the process of adding like fractions. Identifying the least common multiple ( LCM ) of the denominators in converting the improper fraction to proper fractions is important. Then, simplify the fraction when needed.

Let us say, for example, we have the $\frac{5}{2}$+$\frac{13}{6}$.

Let us first make the given fractions to like improper fractions by identifying the LCM of the denominators.

Multiples of 2 are: { 2, 4, 6, 10, 8, 12, 14, 16, 18, 20, … }

Multiples of 6 are: { 6, 12, 24, 18, 36 30, 40, … }

From the list, 6, 12, and 18 are the common multiples, but we have to get the least common multiple ( LCM ), which is 6.

Thus, to make the given improper fractions into proper fractions, we must identify the numbers to multiply to these fractions, making the denominators equal to 6.

Hence, we have,

$\frac{5}{2}$ × $\frac{3}{3}$ = $\frac{15}{6}$

$\frac{13}{6}$  × $\frac{1}{1}$ = $\frac{13}{6}$

We shall then add the fractions $\frac{15}{6}$ and $\frac{13}{6}$ and add like fractions.

$\frac{15}{6}$+$\frac{13}{6}$=$\frac{28}{6}$

Since $\frac{28}{6}$ can still be simplified since 28 and 6 are divisible by 2, we have $\frac{28÷2}{6÷2}$=$\frac{14}{3}$.

Therefore $\frac{5}{2}$+$\frac{13}{6}$=$\frac{14}{3}$.

Subtracting Like Improper Fractions

To subtract like improper fractions, we must subtract the numerators of the given fractions and then copy the same denominator. Simplify the answer when necessary. 

Let us say, for example, we subtract the like improper fractions $\frac{12}{5}$ and $\frac{9}{5}$. Hence we have,

$\frac{12}{5}$-$\frac{9}{5}$=$\frac{3}{5}$

Therefore, the difference between $\frac{12}{5}$ and $\frac{9}{5}$ is equal to $\frac{3}{5}$.

For another example, let us say we have $\frac{10}{7}-\frac{8}{7}$. Since they have the same denominators, it is easy to get the sum of these fractions by subtracting the numerators and copying the same denominator.

$\frac{10}{7}$-$\frac{8}{7}$=$\frac{2}{7}$

Thus, the difference between $\frac{10}{7}$ and $\frac{8}{7}$ is equal to $\frac{2}{7}$.

Subtracting Unlike Improper Fractions

To subtract, unlike improper fractions, we must first make the given improper fractions into proper fractions, then proceed to the process of subtracting like fractions. Finding the denominators’ least common multiple (LCM) is crucial for converting an improper fraction to a proper fraction. Then, simplify the fraction when needed.

Let us say, for example, we have the $\frac{15}{4}-\frac{9}{5}$.

Multiples of 4 are: { 4, 8, 16, 12, 20, 24, 28, 32, 36, 40, … }

Multiples of 5 are: { 5, 15, 10, 20, 25, 30, 35, 40, … }

From the list, 20 and 40 are the common multiples of 4 and 5. but we have to get the least common multiple ( LCM ), which is 20.

Thus, to make the given improper fractions into proper fractions, we must identify the numbers to multiply to these fractions, which will make the denominators equal to 20.

$\frac{15}{4}$ × $\frac{5}{5}$ = $\frac{75}{20}$

$\frac{9}{5}$  × $\frac{4}{4}$ = $\frac{36}{20}$

We shall then subtract the fractions $\frac{45}{20}$ and $\frac{36}{20}$ and proceed to add like fractions.

$\frac{75}{20}$ – $\frac{36}{20}$ = $\frac{39}{20}$

Therefore $\frac{15}{4}$ – $\frac{9}{5}$ = $\frac{39}{20}$.

Multiplying Improper Fractions

When multiplying improper fractions, we must obtain the product of the numerators of the provided fractions over the product of the denominators. And then simply the answer when necessary.

Let us say, for example; we have $\frac{7}{5}$ x $\frac{9}{8}$.

To get the product, we must multiply the numerators 7 and 9 and then multiply the denominators 5 and 8. Thus, 

$\frac{7}{5}$ x $\frac{9}{8}$ =6340

Since 63 and 40 do not have common factors, $\frac{63}{40}$ is simplified already.

Thus, $\frac{7}{5}$ x $\frac{9}{8}$ is equal to $\frac{63}{40}$.

More Examples

Which of the following are improper fractions?

When the denominator exceeds the numerator, we have a proper fraction; otherwise, it is improper. Hence, the improper fractions above are as follows:

In each of the improper fractions, we have, 

7 > 2 ; 11 > 7; 9 > 4 12 > 5 ; 5 > 3 ; 21 > 4

Convert the following improper fractions to mixed numbers.

( a ) $\frac{7}{3}$ ( b ) $\frac{16}{5}$ ( c ) $\frac{27}{2}$ ( d ) $\frac{39}{4}$ ( e ) $\frac{50}{11}$

The improper fractions can be converted into mixed numbers by simply dividing the numerator by the denominator. To complete the mixed number, the quotient is the whole number part, while the numerator is the remainder. The denominator is the divisor or the same denominator of the given improper fraction.

( a ) $\frac{7}{3}$

When we divide 7 by 3, we will have 2 as the quotient and 1 as the remainder. Therefore, the improper fraction $\frac{7}{3}$ is equal to 2$\frac{1}{3}$ in mixed number.

( b ) $\frac{16}{5}$

When we divide 16 by 5, we will have 3 as the quotient and 1 as the remainder. Therefore, the improper fraction $\frac{16}{5}$ is equal to 3$\frac{1}{5}$ in mixed number.

( c ) $\frac{27}{2}$

When we divide 27 by 2, we will have 13 as the quotient and 1 as the remainder. Therefore, the improper fraction $\frac{27}{2}$ is equal to 13$\frac{1}{2}$ in mixed numbers.

( d ) $\frac{39}{4}$

When we divide 39 by 4, we will have 9 as the quotient and 3 as the remainder. Therefore, the improper fraction $\frac{39}{4}$ is equal to 9$\frac{3}{4}$ in mixed numbers.

( e ) $\frac{50}{11}$

When we divide 50 by 11, we will have 4 as the quotient and 6 as the remainder. Therefore, the improper fraction $\frac{50}{11}$ is equal to 4$\frac{6}{11}$ in mixed numbers.

Write the following mixed numbers to improper fractions.

( a ) 2$\frac{3}{5}$ ( b ) 3$\frac{9}{11}$ ( c ) 7$\frac{4}{7}$ ( d ) 5$\frac{5}{9}$ ( e ) 12$\frac{1}{4}$

When we convert a mixed number into an improper fraction, we must multiply the denominator by the whole number part and then add the result to the numerator. The improper fraction and the mixed number have the same denominator.

( a ) 2$\frac{3}{5}$

We must multiply the denominator, 5, by the whole number part, 2. The product of 5 and 2 is10, then add the numerator, 3, which results in 13. Hence, we have ( 5 × 2 ) + 3 = 13. The answer gives the numerator of the improper fraction and then uses the same denominator.

2$\frac{3}{5}$= $\frac{(2×5)+3}{5}$=$\frac{13}{5}$

Therefore, the mixed number 2$\frac{3}{5}$ is equal to $\frac{13}{5}$ in improper fraction.

( b ) 3$\frac{9}{11}$

We must multiply the denominator, 11, by the whole number part, 3. The product of 3 and 11 is 33, then add the numerator 9, which results in 42. Hence, we have ( 11 × 3 ) + 9 = 42. The answer forms the numerator of the improper fraction and then copies the same denominator.

3$\frac{9}{11}$= $\frac{(11×3) +9}{11}$=$\frac{42}{11}$

Therefore, the mixed number 3$\frac{9}{11}$ is equal to $\frac{42}{11}$ in improper fraction.

( c ) 7$\frac{4}{7}$

We must multiply the denominator, 7, by the whole number part, 7. The product of 7 and 7 is 49, then add the numerator 4, which results in 53. Hence, we have ( 7 × 7 ) + 4 = 53. The answer forms the numerator of the improper fraction and then copies the same denominator.

7$\frac{4}{7}$= $\frac{(7×7) +4}{7}$=$\frac{53}{7}$

Therefore, the mixed number 7$\frac{4}{7}$ is equal to $\frac{53}{7}$ in improper fraction.

( d ) 5$\frac{5}{9}$

We must multiply the denominator, 9, by the whole number part, 5. The product of 5 and 9 is 45, then add the numerator, 5, which results in 50. Hence, we have ( 9 × 5 ) + 5 = 50. The answer forms the numerator of the improper fraction and then copies the same denominator.

5$\frac{5}{9}$= $\frac{(9×5) +5}{9}$=$\frac{50}{9}$

Therefore, the mixed number 5$\frac{5}{9}$ is equal to $\frac{50}{9}$ in improper fraction.

( e ) 12$\frac{1}{4}$

We must multiply the denominator, 4, by the whole number part, 12. The product of 4 and 12 is 48, then add the numerator 1, which results in 49. Hence, we have ( 4 × 12 ) + 1 = 49. The answer forms the numerator of the improper fraction and then copies the same denominator.

12$\frac{1}{4}$= $\frac{(4×12) +1}{4}$=$\frac{49}{4}$

Therefore, the mixed number 12$\frac{1}{4}$ is equal to $\frac{49}{4}$ in improper fraction.

Perform the indicated operations and simplify.

( a ) $\frac{5}{3}$+$\frac{8}{3}$

( b ) $\frac{16}{5}$-$\frac{9}{5}$

( c ) $\frac{7}{4}$ x $\frac{5}{3}$

( c ) $\frac{5}{2}$ x $\frac{8}{3}$

( a ) $\frac{4}{3}$+$\frac{8}{3}$=$\frac{12}{3}$=4

( b ) $\frac{16}{5}$ – $\frac{9}{5}$=$\frac{7}{5}$

( c ) $\frac{7}{4}$ x $\frac{5}{3}$ = $\frac{35}{12}$

( c ) $\frac{5}{2}$ x $\frac{8}{3}$ = $\frac{40}{6}$ = $\frac{20}{3}$

Numbers that represent a portion of a whole are called fractions .

When the numerator exceeds the denominator, the fraction is called an improper fraction. Suppose $\frac{a}{b}$ is an improper fraction; thus, a must be greater than b. 

A mixed number is a way of writing fractions that are greater than one. A whole number and a proper fraction make up a mixed number.

To create a mixed number from an improper fraction , divide the numerator by the denominator. To form the mixed equivalent mixed number, the quotient is the whole number part; the numerator is the remainder, while the divisor is the denominator.

Simplifying improper fractions is the same thing as reducing the fractions. We do this by dividing the denominator and the numerator by their greatest common factor. 

Frequently Asked Questions on Improper Fractions ( FAQs )

How do proper and improper fractions differ from one another.

Numbers that represent a portion of a whole are called fractions. Fractions are written with a numerator, a fraction bar, and a denominator. When the denominator is greater than the numerator, we have a proper fraction; otherwise, it is improper.

Examples of proper fractions ( Numerator <  Denominator )

 Examples of improper fractions ( Numerator > Denominator )

How do you convert mixed numbers to improper fractions?

A mixed number has two parts: the whole number and the proper fraction. To make a mixed number into an improper fraction, we must multiply the denominator by the whole number part, then add the answer to the numerator. The denominator stays the same.

For example, let us say that we must convert the mixed number 5$\frac{2}{3}$ to an improper fraction. We will have,

( 3× 5 ) + 2  = 15 + 2 = 17 , this is the numerator of the improper fraction

Therefore, the mixed number 5$\frac{2}{3}$ is equal to $\frac{17}{3}$. 

How do like and unlike fractions

 differ from one another? 

If two fractions have the same denominator, they are said to be like; otherwise, they are unlike.

Examples of like fractions ( same denominators )

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

$\frac{9}{7}$ and $\frac{10}{7}$ are like fractions.

 Examples of unlike fractions ( different denominators )

$\frac{11}{3}$ and $\frac{11}{2}$ are unlike fractions.

$\frac{12}{5}$ and $\frac{16}{9}$ are unlike fractions.

How do we simplify improper fractions?

When simplifying improper fractions is the same thing as reducing the fractions. We do this by dividing the denominator and the numerator by their greatest common factor. 

Let us say, for example; we have the improper fraction $\frac{50}{10}$. Since the greatest common factor of 10 and 50 is 10, we have,

$\frac{50÷10}{10÷10}$ = $\frac{5}{1}$ = 5

Therefore, the simplified form of  $\frac{50}{10}$ is 5.

How do we add and subtract improper fractions?

When adding unlike improper fractions, it is important to make the fractions proper using the least common multiple ( LCM ) of the denominators. Then, proceed to the process of adding like fractions.

How to multiply improper fractions?

Let us say, for example; we have $\frac{10}{2}$ x $\frac{5}{3}$.

To get the product, we must multiply the numerators 10 and 5, then multiply the denominators 2 and 3. Thus, 

$\frac{10}{2}$ × $\frac{5}{3}$ = $\frac{50}{6}$

Since 2 is the greatest common factor of 50 and 6,  $\frac{50}{6}$ can be simplified as $\frac{25}{3}$.Thus, $\frac{10}{2}$ × $\frac{5}{3}$ is equal to $\frac{25}{3}$.

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Improper Fractions

Fractions were first used in Ancient Egypt around 1600 BC, which makes the concept quite old. Fractions are of different types. When the numerator is smaller than the denominator i.e. 1/2, it is a proper fraction. However, when the denominator is smaller than or equal to the numerator, then it is an improper fraction. An example of this is 16/15.

An improper fraction is one type of fraction in which the numerator is equal to or greater than the denominator. Its value is always one or greater than one. Improper fractions are usually written in mixed number form in a simplified manner, as mixed fractions are easier to comprehend.

What is an Improper Fraction?

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. For example, 5/2 and 8/5, are improper fractions. Every fraction has two parts, numerator, and denominator. In mathematics, there are two main types of fractions based on the values of numerator and denominator, and those are proper fractions and improper fractions.

Improper Fraction Definition

A fraction whose numerator is greater than or larger than the denominator is defined as an improper fraction such as 7/3 and 12/5. Improper fractions are easier to solve using addition and subtraction compared to the type of fractions such as mixed fractions.

Improper Fraction and Mixed Fraction

An improper fraction is a fraction whose numerator is greater than or equal to its denominator. For example, 9/4, 4/3 are improper fractions. Numerically, they are always equal to or greater than 1. On the other hand, a mixed fraction is a fraction that is written as a combination of a natural number and a proper fraction. It is a simplified form of an improper fraction. For example, \( 21\dfrac{4}{5}, 16\dfrac{2}{3}\) are mixed fractions. Numerically, a mixed fraction is always greater than 1. Also, any mixed fraction can be written as an improper fraction.

Generally, in real life, mixed fractions are easier to interpret and compare, as compared to improper fractions. We can easily convert any improper fraction to a mixed number or mixed fraction to an improper fraction by following some basic steps that you will study in later sections on this page.

Converting Improper Fractions to Decimals

Fractions and decimals are two ways to represent numbers. Improper fractions can be converted to decimals easily by dividing the numerator with the denominator . Let’s look at a quick and easy example of how we can convert improper fractions to decimals . Let’s consider the following example.

Example: Convert the given improper fraction to a decimal: 10/4.

The first step is to divide 10 by 4. When we do so, we will get 10 ÷ 4 = 2.5.

Here, 10/4 is an improper fraction and 2.5 is a decimal. Similarly, the value of 3/2 is 1.5 in decimals and 5/2 is 2.5 in decimals. When we convert an improper fraction to a decimal, the decimal value is always greater than 1.

Converting Improper Fractions to Mixed Numbers

The denominator of the mixed fraction form of an improper fraction is always the same as that of the original fraction. Mixed numbers are considered as the simplified form of improper fractions, that is why it is important to learn this conversion. For converting an improper fraction to a mixed number, we have to follow the below-listed steps:

• Step 1- Divide the numerator with the denominator.
• Step 2- You will get values of quotient and remainder.
• Step 3- Arrange those values of the quotient, remainder, and divisor in the following manner to express a fraction as a mixed number: \(Quotient\dfrac{Remainder}{Divisor}\).

Let’s look at a quick and easy example of how we can convert improper fractions to mixed numbers. Let’s say you have an improper fraction, 13/4. The first step is to divide 13 by 4. We get 3 as the quotient with a remainder of 1. Next, we will place 1 as the numerator, 4 as the denominator, and 3 as the whole number . Thus, we get the mixed fraction: \( 3\dfrac{1}{4}\).

Similarly, let’s solve another example. Here, we have an improper fraction: 9/2. On dividing 9 by 2, we get 4 as the quotient with a remainder of 1. Again, we will repeat the same process. We will place 1 as the numerator, 2 as the denominator, and 4 as the whole number. Thus, we get the mixed fraction: \(4\dfrac{1}{2}\).

How to Solve Improper Fractions?

Solving improper fractions mean performing arithmetic operations on them and simplifying the value of the answer so obtained. There are mainly four arithmetic operators in mathematics and those are addition, subtraction, multiplication, and division. Solving an improper fraction is the same as solving any other proper fraction, the only difference is that, here, we have to simplify the answer and write it in mixed numbers.

Let's solve the improper fraction: 4/3 + 7/3.

Step 1: We have the same denominator for both the fractions. Therefore, we will directly add the numerators 4 and 7. We get 11. Thus, on adding improper fractions, we get 11/3.

Step 2: Simplifying the improper fraction (dividing 11 by 3), we will get 3 as a whole, 2 as a numerator, and 3 as the denominator.

The answer is \(3\dfrac{2}{3}\).

Related Articles on Improper Fractions

Check the following topics related to the concept of improper fractions.

• Improper Fraction to Mixed Number
• Mixed Number to Improper Fraction
• Improper Fraction to Mixed Number Calculator
• Types of Fractions

Improper Fraction Examples

Example 1: Identify improper fractions out of the following: 12/5, 3, 1/9, 4/2, 4/5.

Improper fractions are the ones in which the numerator is either equal to or greater than the denominator. Out of the given fractions, 12/5, 3, and 4/2 are improper fractions. As, in 12/5, 12 > 5, 3 can be written as 3/1 where 3 > 1, and 4/2 can be simplified as 2/1 where 2 > 1.

Example 2: There are five-fourth liters of plain milk in the refrigerator. Convert the given quantity into a mixed number.

To convert the given improper fraction 5/4 to a mixed number, we have to divide 5 by 4. By dividing, we will get 1 as the quotient, and 1 as the remainder. Therefore, the answer is \(1\dfrac{1}{4}\).

Example 3: What is 4 1/3 as an improper fraction?

To convert the given mixed number to an improper fraction, we have to first multiply the denominator and the whole number and then we need to add the numerator to the sum. ⇒ \(4\dfrac{1}{3}\) = (4 × 3 + 1)/3 ⇒ (12 + 1)/3 ⇒ 13/3 Therefore, \(4\dfrac{1}{3}\) is equivalent to 13/3.

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Practice Questions on Improper Fractions

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FAQs on Improper Fractions

Fractions in which the numerator is equal to or less than the denominator are known as improper fractions in math. 3/2, 6/5, 18/11 are a few examples of improper fractions.

What is Proper Fraction and Improper Fraction?

Proper fractions are those in which the numerator is less than the denominator, while improper fractions are opposite of proper fractions where numerator ≥ denominator.

Are Whole Numbers Examples of Improper Fraction?

Yes, whole numbers are examples of improper fractions, as we can write any whole number in the form of a fraction in which the numerator is greater than the denominator. For example, 3= 3/1, 5= 5/1, etc.

How can we Add Improper Fractions?

The addition of two like improper fractions is done by adding the values of their numerator and writing the common denominator as the denominator of the sum obtained. While, in the case of two unlike improper fractions, we first take the LCM of the denominators and convert them into like fractions. Then, we add those two fractions. Let's take an example of addition of improper fractions 5/4 and 7/4.

• Step 1: We have the same denominator in these fractions 5/4 + 7/4. Therefore, we will simply add the numerators 5 and 7. We get 12 as the sum. Thus, on adding, we have 12/4.
• Step 2: By simplifying the improper fraction (dividing 12 by 4), we get 3 as the answer.

How do we Simplify an Improper Fraction?

Simplification of improper fractions means finding the lowest value of the fraction by dividing the numerator with the denominator. Let us take an example to understand how to simplify an improper fraction. Consider the fraction 100/5. By dividing 100 by 5, we get 20 as the answer. Thus, 100/5 = 20 in the simplified form.

How to Subtract Improper Fractions?

Subtraction of two like improper fractions is done by subtracting the values of their numerator and writing the common denominator as the denominator of the difference obtained. While, in the case of two unlike improper fractions, we first take the LCM of the denominators and convert them into like fractions. Then, we subtract those two fractions.

How to Convert Improper Fraction to Mixed Numbers?

Improper fractions are converted to mixed numbers by dividing the numerator with the denominator and write the answer in this form: \(Quotient\dfrac{Remainder}{Divisor}\).

How to Multiply Improper Fractions?

The numerator and denominator of an improper fraction are multiplied with the numerator and denominator of the other fraction to find the product. After multiplying, we simplify the fraction so obtained. In this way, we multiply two improper fractions.

Improper Fractions – Definition, Conversion, Examples, FAQs

What is an improper fraction, fraction conversion, solved examples on improper fractions, practice problems on improper fractions, frequently asked questions on improper fractions.

Fractions are often referred to as numbers between numbers. Fractions are numerical values that represent a part or a portion of a whole. For example, look at the pizza below.

This pizza has been cut into 4 equal parts. So each piece of the pizza represents 1 out of 4 equal parts. So mathematically, we can represent each piece as  $\frac{1}{4}$. This number is called a fraction . 

The number below the bar that represents the total number of equal parts that the whole is divided into is called the denominator .

And the number on the top that represents the number of equal parts we are considering is called the numerator . 

Before discussing improper fraction, we first need to understand proper fractions.

Proper fractions: Fractions for which the numerator is less than the denominator are called proper fractions . For example, if you order a pizza and cut it into 4 equal parts, and then eat one slice out of it, the remaining pizza can be expressed as $\frac{3}{4}$ of the whole pizza. Here, since 3 is less than 4, the fraction is a proper fraction.

Improper fractions: Improper fractions are defined as fractions for which the numerator is greater than the denominator. Let us take an example of improper fraction, imagine you order a pizza that has 4 slices. Your friends eat all 4 slices. And you realize you didn’t get any. You order another pizza. After eating 1 slice from it, you realize you are done eating. So, how much of the pizza did your friends and you have in all?

Your friends first had all 4 slices of 1 pizza, and then you had 1 slice of the same size from the second pizza.

So, the total pizza eaten is $\frac{5}{4}$ slices of pizza. And that’s an improper fraction with a numerator greater than the denominator. 

Mixed numbers: Mixed numbers are just another way of writing fractions greater than a whole. A mixed number has a whole part and a fractional part, which is actually a proper fraction.

Let’s take the same example from above. You and your friends ate one whole pizza and one-fourth of a pizza. As a mixed fraction, we can write this as $1$ and $\frac{1}{4}$ of a pizza or $1\frac{1}{4}$ of a pizza.

Improper Fraction to Mixed Number

An improper fraction can be converted to a mixed fraction and a mixed fraction can be converted to an improper fraction.

Let’s say we want to convert the fraction $\frac{7}{2}$ into a mixed number. 

Step 1: Find the quotient and the remainder by dividing the numerator with the denominator.

Step 2: The quotient will be the number of wholes for the mixed number. The remainder will be the numerator of the fractional part and the denominator will remain the same.

So, the mixed number form of $\frac{7}{2}$will be:

Mixed Number to Improper Fraction

Any mixed number can be converted into an improper fraction.

Let’s say we want to convert the mixed number $1\frac{3}{4}$ into an improper fraction. 

Step 1: Multiply the denominator with the whole. And then add the numerator to the product.

Step 2: Keep the denominator same and solve for the numerator.

Therefore, the improper fraction for the mixed number $1\frac{3}{4}$ will be $\frac{7}{4}$. 

Related Worksheets

Example 1. Identify the improper fractions:

$\frac{1}{5}$, $2\frac{7}{5}$, $\frac{3}{2}$, $\frac{1}{3}$, $\frac{5}{4}$, $6 \frac{1}{6}$

Solution: $\frac{2}{3}$ and $\frac{5}{4}$ are improper fractions as the value of the numerator is greater than the value of the denominator for both of these fractions.

Example 2. Write $4\frac{2}{7}$ as an improper fraction.

Solution: Converting $4\frac{2}{7}$ into an improper fraction,

$4 \frac{2}{7}$ = $\frac{(7\times4) + 2}{7}$ = $\frac{30}{7}$

Example 3. Write $2\frac{9}{6}$ as an improper fraction.

Solution: While dividing 29 by 6, the quotient and remainder are 4 and 5 respectively. So  $\frac{29}{6}$ written as a mixed number is $ 4 \frac{5}{6}$ .

Converting $3\frac{1}{5}$ to an improper fraction:

What are Improper Fractions? Meaning, Definition, Examples

Attend this quiz & Test your knowledge.

What type of fraction is $\frac{11}{4}$?

Write the mixed number $3\frac{1}{5}$ as an improper fraction., identify the improper fraction., convert $\frac{5}{2}$ into a mixed fraction..

Which are easier for calculations: improper fractions or mixed fractions?

Improper fractions are easy for arithmetic operations of addition , subtraction , multiplication , and division . Mixed fractions are easier to understand and represent in figures.

Why are fractions important?

Every quantity cannot be expressed in whole numbers . Fractions are employed to indicate a certain part or portion of any item. They are a fundamental concept in computation and applied in algebra , ratios, and several other concepts in mathematics and other fields.

How are decimals related to fractions?

Fractions and decimals are both ways to represent quantities or numbers. Decimals are nothing but a different way to represent fractions whose denominator is 10 or powers of 10. For example, the fraction $\frac{15}{10}$ in decimal form is 1.5.

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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.

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Fraction Word Problem Worksheets

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

Represent and Simplify the Fractions: Type 1

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

• Download the set

Represent and Simplify the Fractions: Type 2

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

Adding Fractions Word Problems Worksheets

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

(15 Worksheets)

Subtracting Fractions Word Problems Worksheets

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

Multiplying Fractions Word Problems Worksheets

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

Fraction Division Word Problems Worksheets

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

Related Worksheets

» Decimal Word Problems

» Ratio Word Problems

» Division Word Problems

» Math Word Problems

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4.9: Solving Equations with Fractions

• Last updated
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• Page ID 24084

• David Arnold
• College of the Redwoods

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Undoing Subtraction

We can still add the same amount to both sides of an equation without changing the solution.

Solve for x : \(x - \frac{5}{6} = \frac{1}{3}\).

To “undo” subtracting 5/6, add 5/6 to both sides of the equation and simplify.

\[ \begin{aligned} x - \frac{5}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Original equation.}} \\ x - \frac{5}{6} + \frac{5}{6} = \frac{1}{3} + \frac{5}{6} ~ & \textcolor{red}{ \text{ Add } \frac{5}{6} \text{ to both sides.}} \\ x = \frac{1 \cdot 2}{3 \cdot 2} + \frac{5}{6} ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 6.}} \\ x = \frac{2}{6} + \frac{5}{6} ~ & \textcolor{red}{ \text{ Simplify.}} \\ x = \frac{7}{6} ~ & \textcolor{red}{ \text{ Add.}} \end{aligned}\nonumber \]

It is perfectly acceptable to leave your answer as an improper fraction. If you desire, or if you are instructed to do so, you can change your answer to a mixed fraction (7 divided by 6 is 1 with a remainder of 1). That is \(x = 1 \frac{1}{6}\).

Checking the Solution

Substitute 7/6 for x in the original equation and simplify.

\[ \begin{aligned} x - \frac{5}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Original equation.}} \\ \frac{7}{6} - \frac{5}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Substitute 7/6 for } x.} \\ \frac{2}{6} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Subtract.}} \\ \frac{1}{3} = \frac{1}{3} ~ & \textcolor{red}{ \text{ Reduce.}} \end{aligned}\nonumber \]

Because the last statement is true, we conclude that 7/6 is a solution of the equation x − 5/6 = 1/3.

Undoing Addition

You can still subtract the same amount from both sides of an equation without changing the solution.

Solve for x : \(x + \frac{2}{3} = - \frac{3}{5}\).

To “undo” adding 2/3, subtract 2/3 from both sides of the equation and simplify.

\[ \begin{aligned} x + \frac{2}{3} = - \frac{3}{5} ~ & \textcolor{red}{ \text{ Original equation.}} \\ x + \frac{2}{3} - \frac{2}{3} = - \frac{3}{5} - \frac{2}{3} ~ & \textcolor{red}{ \text{ Subtract } \frac{2}{3} \text{ from both sides.}} \\ x = - \frac{3 \cdot 3}{5 \cdot 3} - \frac{2 \cdot 5}{3 \cdot 5} ~ & \textcolor{red}{ \text{ Equivalent fractions, LCD = 15.}} \\ x = - \frac{9}{15} - \frac{10}{15} ~ & \textcolor{red}{ \text{ Simplify.}} \\ x = - \frac{19}{15} ~ & \textcolor{red}{ \text{ Subtract.}} \end{aligned}\nonumber \]

Readers are encouraged to check this solution in the original equation.

Solve for x : \(x + \frac{3}{4} = - \frac{1}{2}\)

Undoing Multiplication

We “undo” multiplication by dividing. For example, to solve the equation 2 x = 6, we would divide both sides of the equation by 2. In similar fashion, we could divide both sides of the equation

\[ \frac{3}{5} x = \frac{4}{10}\nonumber \]

by 3/5. However, it is more efficient to take advantage of reciprocals. For convenience, we remind readers of the Multiplicative Inverse Property .

Multiplicative Inverse Property

Let a / b be any fraction. The number b / a is called the multiplicative inverse or reciprocal of a / b . The product of reciprocals is 1.

\[ \frac{a}{b} \cdot \frac{b}{a} = 1.\nonumber \]

Let’s put our knowledge of reciprocals to work.

Solve for x : \(\frac{3}{5}x = \frac{4}{10}\).

To “undo” multiplying by 3/5, multiply both sides by the reciprocal 5/3 and simplify.

\[ \begin{aligned} \frac{3}{5} x = \frac{4}{10} ~ & \textcolor{red}{ \text{ Original equation.}} \\ \frac{5}{3} \left( \frac{3}{5} x \right) = \frac{5}{3} \left( \frac{4}{10} \right) & ~ \textcolor{red}{ \text{ Multiply both sides by 5/3.}} \\ \left( \frac{5}{3} \cdot \frac{3}{5} \right) x = \frac{20}{30} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, use the associative property to regroup.} \\ \text{ On the right, multiply.} \end{array}} \\ 1x = \frac{2}{3} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, } \frac{5}{3} \cdot \frac{3}{5} = 1. \\ \text{ On the right, reduce: } \frac{20}{30} = \frac{2}{3}. \end{array}} \\ x = \frac{2}{3} ~ & \textcolor{red}{ \text{ On the left, } 1x = x.} \end{aligned}\nonumber \]

Substitute 2/3 for x in the original equation and simplify.

\[ \begin{aligned} \frac{3}{5} x = \frac{4}{10} ~ & \textcolor{red}{ \text{ Original equation.}} \\ \frac{3}{5} \left( \frac{2}{3} \right) = \frac{4}{10} ~ & \textcolor{red}{ \text{ Substitute 2/3 for }x.} \\ \frac{6}{15} = \frac{4}{10} ~ & \textcolor{red}{ \text{ Multiply numerators; multiply denominators.}} \\ \frac{2}{5} = \frac{2}{5} ~ & \textcolor{red}{ \text{ Reduce both sides to lowest terms.}} \end{aligned}\nonumber \]

Because this last statement is true, we conclude that 2/3 is a solution of the equation (3/5) x = 4/10.

Solve for y : \( \frac{2}{3} y = \frac{4}{5}\)

Solve for x : \(- \frac{8}{9} x = \frac{5}{18}\).

To “undo” multiplying by −8/9, multiply both sides by the reciprocal −9/8 and simplify.

\[ \begin{aligned} - \frac{8}{9} x = \frac{5}{18} ~ & \textcolor{red}{ \text{ Original equation.}} \\ - \frac{9}{8} \left( - \frac{8}{9} x \right) = - \frac{9}{8} \left( \frac{5}{18} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by } -9/8.} \\ \left[ - \frac{9}{8} \cdot \left( - \frac{8}{9} \right) \right] x = - \frac{3 \cdot 3}{2 \cdot 2 \cdot 2} \cdot \frac{5}{2 \cdot 3 \cdot 3} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, use the associative property to regroup.} \\ \text{ On the right, prime factor.} \end{array}} 1x = \frac{ \cancel{3} \cdot \cancel{3}}{2 \cdot 2 \cdot 2} \cdot \frac{5}{2 \cdot \cancel{3} \cdot \cancel{3}} ~ & \textcolor{red}{ \begin{array}{l} \text{ On the left, } - \frac{9}{8} \cdot \left( - \frac{8}{9} \right) = 1. \\ \text{ On the right, cancel common factors.} \end{array}} \\ x = - \frac{5}{16} ~ & \textcolor{red}{ \text{ On the left, } 1x = x. \text{ Multiply on the right.}} \end{aligned}\nonumber \]

Solve for z: \(− \frac{2}{7} z = \frac{4}{21}\)

Clearing Fractions from the Equation

Although the technique demonstrated in the previous examples is a solid mathematical technique, working with fractions in an equation is not always the most efficient use of your time.

To clear all fractions from an equation, multiply both sides of the equation by the least common denominator of the fractions that appear in the equation.

Let’s put this idea to work.

In Example 1, we were asked to solve the following equation for x :

\[x - \frac{5}{6} = \frac{1}{3}.\nonumber \]

Take a moment to review the solution technique in Example 1. We will now solve this equation by first clearing all fractions from the equation.

Multiply both sides of the equation by the least common denominator for the fractions appearing in the equation.

\[ \begin{aligned} x - \frac{5}{6}= \frac{1}{3} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 6 \left( x - \frac{5}{6} \right) = 6 \left( \frac{1}{3} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 6.}} \\ 6x - 6 \left( \frac{5}{6} \right) = 6 \left( \frac{1}{3} \right) ~ & \textcolor{red}{ \text{ Distribute the 6.}} \\ 6x-5 = 2 ~ & \textcolor{red}{ \text{ On each side, multiply first.}} \\ ~ & \textcolor{red}{6 \left( \frac{5}{6} \right) = 5 \text{ and } 6 \left( \frac{1}{3} \right) = 2.} \end{aligned}\nonumber \]

Note that the equation is now entirely clear of fractions, making it a much simpler equation to solve.

\[ \begin{aligned} 6x - 5 + 5 = 2 + 5 ~ & \textcolor{red}{ \text{ Add 5 to both sides.}} \\ 6x = 7 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \\ \frac{6x}{6} = \frac{7}{6} ~ & \textcolor{red}{ \text{ Divide both sides by 6.}} \\ x = \frac{7}{6} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]

Note that this is the same solution found in Example 1.

Solve for t : \(t - \frac{2}{7} = - \frac{1}{4}\)

In Example 4, we were asked to solve the following equation for x .

\[- \frac{8}{9}x = \frac{5}{18}\nonumber \]

Take a moment to review the solution in Example 4. We will now solve this equation by first clearing all fractions from the equation.

Multiply both sides of the equation by the least common denominator for the fractions that appear in the equation.

\[ \begin{aligned} - \frac{8}{9} x = \frac{5}{18} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 18 \left( - \frac{8}{9} x \right) = 18 \left( \frac{5}{18} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 18.}} \\ -16x=5 ~ & \textcolor{red}{ \text{ On each side, cancel and multiply.}} \\ ~ & \textcolor{red}{ 18 \left( - \frac{8}{9} \right) = -16 \text{ and } 18 \left( \frac{5}{18} \right) = 5.} \end{aligned}\nonumber \]

Note that the equation is now entirely free of fractions. Continuing,

\[ \begin{aligned} \frac{-16x}{-16} = \frac{5}{-16} ~ & \textcolor{red}{ \text{ Divide both sides by } -16.} \\ x = - \frac{5}{16} ~ & \textcolor{red}{ \text{ Simplify.}} \end{aligned}\nonumber \]

Note that this is the same as the solution found in Example 4.

Solve for u :

\[ - \frac{7}{9} u = \frac{14}{27}\nonumber \]

Solve for x : \(\frac{2}{3}x + \frac{3}{4} = \frac{1}{2}\).

\[ \begin{aligned} \frac{2}{3} x + \frac{3}{4} = \frac{1}{2} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 12 \left( \frac{2}{3} x + \frac{3}{4} = \right) = 12 \left( \frac{1}{2} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 12.}} \\ 12 \left( \frac{2}{3}x \right) + 12 \left( \frac{3}{4} \right) = 12 \left( \frac{1}{2} \right) ~ & \textcolor{red}{ \text{ On the left, distribute 12.}} \\ 8x + 9 = 6 ~ & \textcolor{red}{ \text{ Multiply: } 12 \left( \frac{2}{3} x \right) = 8x, ~ 12 \left( \frac{3}{4} \right) = 9,} \\ ~ & \textcolor{red}{ \text{ and } 12 \left( \frac{1}{2} \right) = 6.} \end{aligned}\nonumber \]

Note that the equation is now entirely free of fractions. We need to isolate the terms containing x on one side of the equation.

\[ \begin{aligned} 8x + 9 - 9 = 6 - 9 ~ & \textcolor{red}{ \text{ Subtract 9 from both sides.}} \\ 8x = - 3 ~ & \textcolor{red}{ \text{ Simplify both sides.}} \\ \frac{8x}{8} = \frac{-3}{8} ~ & \textcolor{red}{ \text{ Divide both sides by 8.}} \\ x = - \frac{3}{8} ~ & \textcolor{red}{ \text{ Simplify both sides.}} \end{aligned}\nonumber \]

Solve for r : \(\frac{3}{4} r + \frac{2}{3} = \frac{1}{2}\)

Solve for x : \( \frac{2}{3} - \frac{3x}{4} = \frac{x}{2} - \frac{1}{8}.\)

Multiply both sides of the equation by the least common denominator for the fractions in the equation.

\[ \begin{aligned} \frac{2}{3} - \frac{3x}{4} = \frac{x}{2} - \frac{1}{8} ~ & \textcolor{red}{ \text{ Original equation.}} \\ 24 \left( \frac{2}{3} - \frac{3x}{4} \right) = 24 \left( \frac{x}{2} - \frac{1}{8} \right) ~ & \textcolor{red}{ \text{ Multiply both sides by 24.}} \\ 24 \left( \frac{2}{3} \right) - 24 \left( \frac{3x}{4} \right) = 24 \left( \frac{x}{2} \right) - 24 \left( \frac{1}{8} \right) ~ & \textcolor{red}{ \text{ On both sides, distribute 24.}} \\ 16 - 18x = 12x - 3 ~ & \textcolor{red}{ \text{ Left: } 24 \left( \frac{2}{3} \right) = 16, ~ 24 \left( \frac{3x}{4} \right) = 18x.} \\ ~ & \textcolor{red}{ \text{ Right: } 24 \left( \frac{x}{2} \right) = 12x, ~ 24 \left( \frac{1}{8} \right) = 3.} \end{aligned}\nonumber \]

\[ \begin{aligned} 16 - 18x - 12x = 12x - 3 - 12x ~ & \textcolor{red}{ \text{ Subtract } 12x \text{ from both sides.}} \\ 16 - 30x = -3 ~ & \textcolor{red}{ \begin{aligned} \text{ Left: } -18x - 12x = -30x. \\ \text{ Right: } 12x - 12x = 0. \end{aligned}} \\ 16 - 30x - 16 = -3 - 16 ~ & \textcolor{red}{ \text{ Subtract 16 from both sides.}} \\ -30x = -19 ~ & \textcolor{red}{ \begin{aligned} \text{ Left: } 16-16=0. \\ \text{ Right: } -3 - 16 = -19. \end{aligned}} \\ \frac{-30x}{-30} = \frac{-19}{-30} ~ & \textcolor{red}{ \text{ Divide both sides by } -30.} \\ x = \frac{19}{30} ~ & \textcolor{red}{ \text{ Simplify both sides.}} \end{aligned}\nonumber \]

Solve for s : \( \frac{3}{2} - \frac{2s}{5} = \frac{s}{3} - \frac{1}{5}\).

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improper fraction/mixed number word problems

Subject: Mathematics

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terible word porblems becuse i dont like it

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I was unfamiliar with the low, moderate, and high achiever acronym, but Google works.

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Published 2013 Revised 2019

Exploring Fractions

• The first group  gives you some starting points to explore with your class, which are applicable to a wide range of ages.  The tasks in this first group will build on children's current understanding of fractions and will help them get to grips with the concept of the part-whole relationship. 
• The second group of tasks  focuses on the progression of ideas associated with fractions, through a problem-solving lens.  So, the tasks in this second group are curriculum-linked but crucially also offer opportunities for learners to develop their problem-solving and reasoning skills.   

• are applicable to a range of ages;
• provide contexts in which to explore the part-whole relationship in depth;
• offer opportunities to develop conceptual understanding through talk.