Mixed and improper fractions: Mixed numbers and improper fractions review (article)
Posted onProper and Improper Fractions and Mixed Numbers (Video)
TranscriptFAQsPractice
Hi, and welcome to this review of fractions and mixed numbers!
Before we dive in, let’s review the basic parts of a fraction. Remember, a fraction simply represents a part of a whole. It has a numerator and a denominator, which tells us what the “part” is and what the “whole” is. Let’s look at the fraction \(\frac{3}{4}\) as an example. We can see the 3 is our numerator and the 4 is our denominator. So the fraction \(\frac{3}{4}\) is really saying 3 parts out of 4 parts total. It can also be helpful to visualize \(\frac{3}{4}\) as simply
\(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\)
It’s very important to remember that a denominator of 4 does not represent the value of 4. A denominator of 4 represents the value of 1 that is divided up into 4 equal parts, or fourths. This type of fraction represents a value less than one whole. \(\frac{3}{4}\) is not quite 1. If we had \(\frac{4}{4}\) that would be equivalent to one, but we only have 3 out of 4 parts.
We see and use fractions that are less than one all the time in our daily lives, whether it’s for things like recipes or keeping track of time. Recipes often call for amounts such as “1/2 tsp salt,” and we often keep track of time in terms of quarter hours, like “a quarter past three” for 3:15. Though we observe this type of fraction very frequently in our daily lives, it is not the only type of fraction.
Consider the following scenario. You are ordering pizza for a big celebration. There will be a lot of hungry guests at this celebration, so you order 3 pizzas. Each pizza is cut into 6 slices. This means that each pizza has 6 equal parts, and as a fraction, 6 would be considered our “whole,” or our denominator.
If your first guest eats 2 slices we would represent this as the fraction \(\frac{2}{6}\). Two parts, out of 6 parts total.
But what if that first guest was really hungry and grabbed 7 slices? Again, each pizza was cut into 6 equal slices, so 6 remains as our “whole,” or denominator.
But this time our “part” is 7: \(\frac{7}{6}\). In this scenario, our numerator is larger than our denominator.
Fractions with a numerator larger than their denominator are referred to as improper fractions. Essentially, improper fractions equal a value that is more than one. One whole pizza would be represented by \(\frac{6}{6}\), or “six sixths.” \(\frac{7}{6}\) represents “seven sixths,” which is more than one pizza. This could be visualized as \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}=7\). It can also be written in another form called a mixed number. An improper fraction and a mixed number will represent the same amount but simply be written in a different form.
For example, the improper fraction \(\frac{7}{6}\) could also be written as the mixed number \(1\frac{1}{6}\). Mixed numbers and improper fractions show the same amount, but as a mixed number, the “parts” are collected and consolidated into as many groups of 1 whole as possible.
For example, \(\frac{4}{4}\) would be grouped together as 1. \(\frac{7}{7}\) would also be grouped together as 1. Any value where the numerator is equivalent to the denominator would be expressed simply as 1.
In our pizza example, the guest took 7 slices from a group of pizzas that were sliced into sixths. We said that this could be expressed as the improper fraction \(\frac{7}{6}\), or visualized as \(\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}\). As a mixed number, we would group 6 of these “sixths” in order to form \(1\frac{6}{6}\), or 1 whole. By grouping \(\frac{6}{6}\) together, we can see that \(\frac{1}{6}\) is left over, on its own. We would write our mixed number as \(1\frac{1}{6}\).
Converting Improper Fractions to Mixed Numbers
Let’s try a few more examples. Let’s write the following improper fractions as mixed numbers: \(\frac{4}{3}\) and \(\frac{3}{2}\).
\(\frac{4}{3}\) can be visualized as \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\).
We know that \(\frac{3}{3}\) is equal to 1, so let’s group 3 of these “thirds” together. We are now left with \(1\frac{1}{3}\) as our mixed number.
\(\frac{3}{2}\) can be visualized as \(\frac{1}{2}+\frac{1}{2}+\frac{1}{2}\). We then know that \(\frac{2}{2}\) makes one whole. And we’re left with \(\frac{1}{2}\) left over. So \(\frac{3}{2}\) as a mixed number is \(1\frac{1}{2}\).
Let’s try one more example: \(\frac{7}{4}\)
\(\frac{7}{4}\) is the same as \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\). Now we know that \(\frac{4}{4}\) is grouped as one whole. So these \(\frac{4}{4}\) pulled over, to equal 1. And we’re left with \(\frac{3}{4}\). \(\frac{7}{4}\) written as a mixed number is \(1\frac{3}{4}\).
This process will take place in reverse in order to convert a mixed number to an improper fraction.
Converting Mixed Numbers to Improper Fractions
For example, if we started with the mixed number \(1\frac{3}{4}\) and we wanted to convert it to an equivalent improper fraction, we would take a look at the whole number, in this case, it is 1.
This whole number is really representing our denominator that’s in the fraction. In this case, it’s 4, so the 1 is equal to \(\frac{4}{4}\). When we combine these four fourths with the \(\frac{3}{4}\), we end up with seven fourths total, or \(\frac{7}{4}\).
That’s all there is to it! I hope that this video was helpful. Thanks for watching, and happy studying!
Frequently Asked Questions
Q
What are examples of a proper fraction?
A
A proper fraction is a fraction that has no whole number part and its numerator is smaller than its denominator. Some examples of proper fractions are \(\frac{1}{4}\),\(\frac{7}{9}\),\(\frac{12}{13}\),\(\frac{23}{25}\), and \(\frac{17}{76}\).
Q
What is a proper and an improper fraction?
A
A proper fraction is a fraction that has no whole number part and its numerator is smaller than its denominator. An improper fraction is a fraction that has a larger numerator than denominator and it represents a number greater than one.
Proper Fraction Examples: \(\frac{1}{2}\), \(\frac{1}{6}\), \(\frac{2}{5}\), \(\frac{13}{14}\), \(\frac{7}{11}\)
Improper Fraction Examples: \(\frac{16}{11}\), \(\frac{12}{7}\), \(\frac{6}{4}\), \(\frac{3}{2}\), \(\frac{8}{3}\)
Q
What is an improper fraction example?
A
\(\frac{17}{11}\) is an example of an improper fraction because its numerator is greater than its denominator, which means it represents a value greater than one.
Q
What is a mixed number example?
A
A mixed number is a number that consists of a whole number part and a proper fractional part. \(4\frac{1}{3}\) is an example of a mixed number because it has a whole number part \((4)\) and a proper fractional part \((\frac{1}{3})\).
Q
How do you turn an improper fraction into a mixed number?
A
To turn an improper fraction into a mixed number, figure out how many times the denominator can fit into the numerator and then how much of the numerator is left over.
Then, the number of times the denominator fits into the numerator becomes the whole number part of the mixed number, and the number left over is the numerator of the fractional part over the original denominator.
Ex. Convert \(\frac{17}{4}\) to a mixed number.
1) How many times can 4 fit into 17? 4 because \(4×4=16\) – this becomes the whole number part
2) How much is left over in the numerator? 1 because \(17-16=1\) – this becomes the numerator of the fractional part
3) \(\frac{17}{4}=4\frac{1}{4}\)
Q
How do you turn a mixed number into an improper fraction?
A
To turn a mixed number into an improper fraction, multiply the whole number part by the denominator and add the numerator. This becomes the new numerator over the original denominator.
Ex. Convert \(3\frac{5}{7}\) to an improper fraction.
\(3\frac{5}{7}=\frac{3\times7+5}{7}=\frac{21+5}{7}=\frac{26}{7}\)
Practice Questions
Question #1:
Which list of fractions contains amounts that are all more than one whole?
\(\frac{5}{3},\frac{7}{5},\text{ and }\frac{2}{2}\)
\(\frac{3}{5},\frac{7}{5},\text{ and }\frac{3}{2}\)
\(\frac{5}{3},\frac{7}{5},\text{ and }\frac{3}{2}\)
\(\frac{5}{5},\frac{7}{7},\text{ and }\frac{3}{3}\)
Show Answer
Answer:
The correct answer is C: \(\frac{5}{3},\frac{7}{5},\text{ and }\frac{3}{2}\).
When a fraction has a numerator larger than its denominator, the total value is more than one whole. Fractions like \(\frac{3}{3}, \frac{5}{5}, \text{ and }\frac{7}{7}\) are all equal to one whole. Fractions with numerators less than their denominators represent values that are less than one whole. \(\frac{5}{3}, \frac{7}{5}, \text{ and }\frac{3}{2}\) are all improper fractions representing total values larger than one whole.
Hide Answer
Question #2:
Express the mixed number \(2\frac{1}{3}\) as an improper fraction.
\(\frac{8}{3}\)
\(\frac{5}{3}\)
\(\frac{7}{3}\)
\(\frac{4}{3}\)
Show Answer
Answer:
The correct answer is C: \(\frac{7}{3}\).
\(2\frac{1}{3}\) is equivalent to \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\). This represents seven groups of \(\frac{1}{3}\), or as an improper fraction, \(\frac{7}{3}\).
Hide Answer
Question #3:
\(\frac{5}{3}\) is equivalent to what mixed number?
\(1\frac{2}{3}\)
\(1\frac{3}{4}\)
\(2\frac{1}{3}\)
\(2\frac{2}{3}\)
Show Answer
Answer:
The correct answer is A: \(1\frac{2}{3}\).
\(\frac{5}{3}\) can be thought of as \(\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}+\frac{1}{3}\). We can combine three groups of \(\frac{1}{3}\) in order to create \(\frac{3}{3}\), or one whole. There are still two thirds left, so our mixed number is \(1\frac{2}{3}\).
Hide Answer
Question #4:
Mr. Jones orders sub sandwiches for his son’s basketball team. He orders eight subs and requests that they are each cut into three equal pieces. If his son eats two slices from one sub, what fraction of the total amount of food is left?
\(\frac{7}{11}\)
\(\frac{11}{28}\)
\(\frac{11}{15}\)
\(\frac{11}{12}\)
Show Answer
Answer:
The correct answer is D: \(\frac{11}{12}\).
Eight subs, cut into three pieces each, creates twenty-four smaller sandwiches. Mr. Jones’ son eats two of these pieces, or \(\frac{2}{24}\) from the total amount of food. This means that twenty-two pieces are left, or \(\frac{22}{24}\).
If the numerator and denominator are both divided by two, the simplified fraction becomes \(\frac{11}{12}\). \(\frac{11}{12}\) of the total amount of food is still remaining.
Hide Answer
Question #5:
Kristina plans on drinking \(\frac{1}{4}\) quart of water for every mile she runs. If she runs six miles, how much water will she drink?
\(1\frac{1}{4}\) quarts of water
\(1\frac{1}{2}\) quarts of water
\(2\frac{1}{2}\) quarts of water
\(\frac{1}{2}\) quarts of water
Show Answer
Answer:
The correct answer is B: \(1\frac{1}{2}\) quarts of water.
If Kristina drinks \(\frac{1}{4}\) quart of water for every mile, she is essentially drinking \(\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}+\frac{1}{4}\) quarts of water. The sum of this list of fractions is \(1\frac{2}{4}\) or in simplest form, \(1\frac{1}{2}\).
Hide Answer
Return to Basic Arithmetic Videos
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Improper Fractions and Mixed Numbers Video
Meet our gang of fraction friends: proper fractions, improper fractions, and mixed numbers.
Learn how to convert improper fractions to and from mixed numbers. Warning: even though some of these numbers may be tiny, they’ve got big personalities.
…and others are a bit fractured.
As far as fractions go, it’s a mixed bag.
Some fractions are proper…
…some are improper…
…and then there are those that are just a little mixed up.
Proper fractions have a numerator that’s smaller than the denominator.
Which means they’re always less than 1.
Proper fractions also donate to charities and volunteer in soup kitchens.
They know they’re not greater than anyone else around them.
Improper fractions, on the other hand, are a little full of themselves.
They’ve got big heads.
.. well, big numerators, anyway.
In improper fractions, the numerator is bigger than the denominator…
…which means they’re greater than 1.
Since fractions aren’t supposed to be greater than 1, we call these guys improper.
They’re a bit on the rude side.
But if they get on our nerves…
… we can change them into mixed numbers.
Mixed numbers are combinations of whole numbers and fractions…
… so they may be a bit confused at times.
Maybe they’re going through an identity crisis or something, we don’t really know.
We can change improper fractions into mixed numbers
by dividing the numerator by the denominator.
That’s what fractions are, right? Division.
We’ll end up with a number and a remainder.
The number is the whole number in front,
and the remainder goes in the numerator of the fraction that follows it.
If we’re sick of being mixed up by mixed numbers…
… we can change them back into improper fractions, too.
Since we divided to change them, let’s multiply to change them back.
First, we multiply the denominator by the whole number.
Once we find that product, we add it to the numerator of our mixed number’s fraction.
Then, we put that number all over the original denominator…
…to get our improper fraction.
So in the end, mixed numbers and improper fractions
can express the same number 2 different ways.
It’s up to you to decide which is better: improper… or mixed.
Mixed fraction. Operations with mixed fractions
A mixed fraction is called , which has an integer and a fractional part.
They are written as \(a\)\(\frac{m}{n}\), where — \(a\) integer ,\(\frac{m}{n}\) — regular fraction .
For example : \(2\)\(\frac{3}{5}\) where \(2\) is the integer part, \(\frac{3}{5}\) is the fractional part ( proper fraction ).
\(17\)\(\frac{17}{18}\) here \(17\) is the integer part, \(\frac{17}{18}\) is the fractional part (proper fraction).
In fact, such fractions are the sum of an integer and a fraction, that is, there is a plus sign (not multiply) between the integer and fractional parts.
For example: \(2\frac{3}{5}=2+\frac{3}{5}\)
It doesn’t need to be memorized, just get the point. Think about what in practice means, for example, the entry: “there are \(2\)\(\frac{3}{5}\) sacks of flour left in the warehouse”? That there are two full bags in the warehouse and another one full of \(\frac{3}{5}\).
Where is the place for multiplication? Obviously that’s two plus \(\frac{3}{5}\) sacks of flour! It is very important to understand this point, because a huge number of errors are made here when calculating with mixed fractions (see below).
Improper mixed fraction conversion
To convert a mixed fraction to an improper one, you need to multiply the integer part by the denominator of the fractional and add the numerator to the result — you get the numerator of the improper fraction. The denominator does not change. That is,
\(a\)\(\frac{m}{n}\)\(=\)\(\frac{a n + m}{n}\).
For example, when transforming \(2\)\(\frac{3}{5}\) we get \(\frac{2 5 + 3}{5}=\frac{13}{5}\).
Why is the calculation done this way? It’s all about the plus, standing between the integer and the fractional part (see above). In fact, the complete transformation looks like this:
But to paint everything in such detail for too long, and there is no need, it is easier to immediately get an answer using the formula above.
Improper fraction conversion
To convert an improper fraction to a mixed one, you need to select the whole part in it.
To achieve this, we ask ourselves the question — how many times does the denominator «fit» entirely in the numerator?
For example , let’s represent \(\frac{13}{5}\) as a mixed fraction. How many times does a five «fit» in a thirteen? Twice. The third time already «will not fit». This means that the integer part will be equal to two, and the fractional part will be equal to the remainder, that is, \(\frac{3}{5}\). We make out: \(\frac{13}{5}\)\(=\)\(\frac{10 + 3}{5}\)\(=\)\(\frac{10}{5}\) \(+\)\(\frac{3}{5}\)\(=2+\)\(\frac{3}{5}\)\(=2\)\(\frac{3}{ 5}\). Here are some more examples with the correct conversion:
\(\frac{37}{11}\)\(=\)\(\frac{33 + 4}{11}\)\(=\)\(\frac{33}{11}\)\( +\)\(\frac{4}{11}\)\(=3+\)\(\frac{4}{11}\)\(=3\)\(\frac{4}{11} \)
\(\frac{26}{3}\)\(=\)\(\frac{24 + 2}{3}\)\(=\)\(\frac{24}{3}\)\( +\)\(\frac{2}{3}\)\(=8+\)\(\frac{2}{3}\)\(=8\)\(\frac{2}{3} \)
And here is an example of incorrect selection of the integer part:
\(\frac{7}{2}\)\(=\)\(\frac{4 + 3}{2}\)\(=\)\(\frac{4}{2}\)\( +\)\(\frac{3}{2}\)\(=2+\)\(\frac{3}{2}\)\(=2\)\(\frac{3}{2} \)
Where is the mistake? That the fractional part must be a proper fraction.
But here it is not so — it means that the whole part is not completely highlighted. Indeed, after all, a deuce in a seven is completely placed three times, not two. So the following would be correct:
\(\frac{7}{2}\)\(=\)\(\frac{6 + 1}{2}\)\(=\)\(\frac{6}{2}\)\( +\)\(\frac{1}{2}\)\(=3+\)\(\frac{1}{2}\)\(=3\)\(\frac{1}{2} \)
Converting a mixed fraction to a decimal
To convert a mixed fraction to a decimal, you need to divide the numerator by the denominator in the fractional part, and then add the result to the integer part.
For example: \(2\)\(\frac{3}{5}\)\(=2+\)\(\frac{3}{5}\)\(=2+0.6=2 ,6\)
\(7\)\(\frac{11}{25}\)\(=7+\)\(\frac{11}{25}\)\(=7+0.44=7.44\)
Hence the conclusion:
A mixed fraction is an ordinary number, with the integer part representing what will appear before the decimal point, and the fractional part after.
Most common mistakes when working with mixed shot
The main reason for most mistakes is forgetting the point described above — between the integer and the fractional part is «plus», not «multiply» .
Example : Calculate \(2\)\(\frac{3}{5}\)\(:\)\(\frac{1}{5}\)
Wrong solution : \ (2\)\(\frac{3}{5}\)\(:\)\(\frac{1}{5}\)\(=2\)\(\frac{3}{5}\ )\(\)\(\frac{5}{1}\)\(=2\)\(\frac{3 5}{5 1}\)\(=2 3=6\)
Correct solution: \(2\)\(\frac{3}{5}\)\(:\)\(\frac{1}{5}\)\(=( 2+\)\(\frac{3}{5}\)\():\)\(\frac{1}{5}\)\(=\)\(\frac{2 5+3} {5}\)\(:\)\(\frac{1}{5}\)\(=\)\(\frac{13}{5}\)\(\)\(\frac{5 }{1}\)\(=\)\(\frac{13 5}{5 1}\)\(=13\)
Example: Compute \(3\)\(\frac{1}{5}\)\( 1\)\(\frac{1}{4}\)
Wrong solution : \(3\)\(\frac{1}{5}\)\( 1\)\(\frac{1}{4}\)\(=3\)\(\frac{1}{ 5}\)\( 1 )\(\frac{1}{4}\)\(=\)\(\frac{3}{5}\)\(\)\(\frac{ 1}{4}\)\(=\)\(\frac{3 1}{5 4}\)\(=\)\(\frac{3}{20}\)
Correct solution : \)\(\frac{1}{5}\)\() (1+\)\(\frac{1}{4}\)\()=\)\(\frac{3 5 + 1}{5}\)\(\)\(\frac{1 4 + 1}{4}\)\(=\)\(\frac{16}{5}\)\(\) \(\frac{5}{4}\)\(=\)\(\frac{16 5}{5 4}\)\(=4\)
From the fact that the integer and fractional parts are connected by a plus sign, one more conclusion follows:
If a mixed fraction is preceded by a minus sign, then it precedes both the integer part and the fractional part.
For example: \(-7\) \(\frac{5}{9}\)\(=-(7+\) \(\frac{5}{9}\)\()=-7-\ ) \(\frac{5}{9}\).
This is important to remember when subtracting mixed fractions.
Example . Compute \(4\)\(\frac{3}{5}\)\(-2\)\(\frac{1}{5}\).
Solution: \(4\)\(\frac{3}{5}\)\(-2\)\(\frac{1}{5}\)\(=(4+\) \(\frac{3}{5}\)\()-(2+\)\(\frac{1}{5}\)\()=4+\)\(\frac{3}{5 }\)\(-2-\)\(\frac{1}{5}\)\(=4-2+\)\(\frac{3}{5}\)\(-\)\( \frac{1}{5}\)\(=2+\)\(\frac{3-1}{5}\)\(=2+\)\(\frac{2}{5}\) \(=2\)\(\frac{2}{5}\).
In general, the subtraction (addition) of mixed fractions is conveniently carried out in two stages: first subtract (add) the whole parts separately, and then the fractional ones.
See also:
Fractions (cheat sheet)
Proper and improper fractions. Mixed fractions.
Definition
A common fraction is called proper if its numerator is less than its denominator.
For example: Fraction \(\
\frac{11}{23}
\) is correct because its numerator, which is 11, is less than its denominator, which is 23: 11 Definition
A fraction is called irregular if its numerator is greater than or equal to its denominator.
For example: Fractional \(\
\frac{23}{11}
\) is incorrect because 23> 11. The fraction \(\
\frac{3}{3}
\) incorrect is incorrect because the numerator of a fraction is equal to its denominator.
MIXED FROT
Definition
Numbers that contain an integer and a proper fraction are called mixed numbers.
An integer is called the integer part of the mixed number, and a regular fraction is called the fractional part of the mixed number.
For example: For a mixed fraction \(\
3 \frac{11}{23}=3+\frac{11}{23}
\) the number 3 is the integer part, \(\
\frac{11}{23}
\) fractional.
An improper fraction can be written as a mixed number by dividing the numerator by the denominator. The resulting partial coefficient will be the integer part of the mixed fraction, the remainder will be the numerator of the fractional part, and the denominator of the original irregular fraction will be the denominator of the fractional part.
EXAMPLE
Write an improper fraction \(\\frac{20}{3}
\) as mixed.
So we get that \(\
\frac{20}{3}=20 : 3=6(\text { remainder } 2)
\). And then the desired mixed fraction \(\
\frac{20}{3}=6 \frac{2}{3}
\)
\frac{20}{3}=6 \frac{2}{3}
\)
To write a mixed number as an improper fraction, multiply the integer part by the denominator of the fractional part, add the numerator of the fractional part to the resulting number and write this amount into the numerator, and leave the denominator of the fractional part unchanged.
EXAMPLE
8 \frac{2}{3}
\) is written as an improper fraction.
8 \frac{2}{3}=\frac{8 \cdot 3+2}{3}=\frac{26}{3}
\)
8 \frac{2}{3}=\frac{26}{3}
\)
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Differentiation formulas Ordinary fractions Concept of a fraction.
