What are Mixed Numbers?
Mixed numbers are numbers that have a whole number part and a fractional part. A mixed number represents an improper fraction — a number between two whole numbers. The general form of a fraction is \(\frac{a}{b}\). In an improper fraction, the numerator is greater than the denominator, that is, a > b. We use mixed fractions in place of improper fractions to easily identify the position of the fraction on the number line. Let’s observe the mixed fraction \(2\frac{3}{4}\).
In this case, 2 represents the whole number part and \(\frac{3}{4}\) represents the fractional part. This indicates that the mixed number lies between 2 and 3. Since the denominator of the fraction is 4, each whole is divided into four parts. The numerator of the fraction is 3; we only have three parts of the third whole.
How Do We Express Mixed Numbers?
Mixed numbers can also be expressed as improper fractions, which are in the form \(\frac{a}{b}\), where a > b. Consider the fraction \(1\frac{1}{2}\).
\(1\frac{1}{2}\) can be rewritten as \(\frac{2 {\times}1 + 1}{2}=\frac{2 + 1}{2}=\frac{3}{2}\).
In this case, \(\frac{3}{2}\) is an improper fraction. It means the numerator is greater than the denominator (a > b). A fraction whose numerator is greater than the denominator lies outside the region of 0 and 1.
And just like all fractions, a mixed number can also be expressed as \(a\div b\). So, \(1\frac{1}{2}\) can also be expressed as \(3\div 2\).
Representing Mixed Numbers
Mixed numbers can be expressed as quotients using two types of models: tape diagrams and area models. Let’s take a look at how these models can be used to represent mixed numbers.
Using tape diagram
Consider the following: \(5 \div 2\).
\(5 \div 2=\frac{5}{2}\)
We know that this number doesn’t lie between 0 and 1, as the numerator is greater than the denominator. We can use a tape diagram to find where this mixed number is located.
We have 5 wholes in this tape diagram. Each of these wholes is divided into two parts. So, each part is \(\frac{1}{2}\) of the whole. We get \(\frac{5}{2}\) by adding five halves.
Hence, 5 ÷ 2 = \(\frac{5}{2}\) = 2\(\frac{1}{2}\)
Using area model
\(5 \div 2\) can also be represented using an area model.
5 wholes are divided into two parts. One-half of the wholes is taken into a single group, and the other half is taken into another group.
Solved Examples
Example 1:
Use a model to represent \(4\div 3\).
Solution:
\(4\div 3\) is the same as \(\frac{4}{3}\).
We can represent this using a tape diagram by showing 4 wholes and dividing each of these wholes into 3 parts.
Here, 4 wholes are divided into 3 equal parts.
So, each part is \(\frac{1}{3}\) of the whole.
That means 4 ÷ 3 = \(\frac{4}{3}\)
= \(\frac{3+1}{3}=1\frac{1}{3}\)
Therefore, \(4\div 3=\frac{4}{3}\) or \(1\frac{1}{3}\)
Example 2:
Use an area model to represent \(5\div 4\).
Solution:
\(5\div 4\) can also be written as \(\frac{5} {4}\). This fraction can be represented using an area model by showing 5 wholes and dividing each of the wholes into 4 parts.
There are 5 wholes and they are divided into four parts. One part from each whole is grouped together.
So, each group gets \(\frac{1} {4}\) of each whole.
\(5\div 4=\frac{5} {4}=\frac{4+1} {4}\)
= \(1+\frac{1} {4}=1\frac{1} {4}\)
Therefore, \(5\div 4=\frac{5} {4}\) or \(1\frac{1} {4}\)
Example 3:
You are sharing 8 small pizzas with 4 of your friends. How many whole pizzas will each person get? Also, what fractional amount of a pizza will each person get?
Solution:
As you are sharing these pizzas with 4 of your friends, the pizza is divided among 5 people (including yourself).
To find how much pizza each person gets, we can divide 8 ÷ 5. Let’s express this using an area model.
\(8\div 5=\frac{8} {5}\)
\(\frac{8} {5}=\frac{5+3}{5}=\frac{5}{5}+\frac{3}{5}\)
\(=1+\frac{3}{5}=1\frac{3}{5}\)
Whole number part = 1
Fractional part = \(\frac{3} {5}\)
So, each person gets a whole pizza and \(\frac{3} {5}\) of another pizza.
Example 4:
If Leo runs 19 miles in 3 hours at a steady pace, is he running more than \(\frac{11}{2}\) miles in an hour?
Solution:
We know that Leo can run 19 miles in 3 hours. Since he is running at a steady pace, we can say that he runs \(\frac{19}{3}\) miles per hour.
Leo’s pace = \(\frac{19}{3}\) miles per hour
Now, we need to compare \(\frac{19}{3}\) with \(\frac{11}{2}\) to see if he runs more than \(\frac{11}{2}\) in an hour.
The easiest way to compare these improper fractions is to convert them into mixed numbers.
\(\frac{19}{3}=\frac{18+1}{3}=\frac{18}{3}+\frac{1}{3}\)
\(=6+\frac{1}{3}=6\frac{1}{3}\)
\(\frac{11}{2}=\frac{10+1}{2}=\frac{10}{2}+\frac{1}{2}\)
\(=5+\frac{1}{2}=5\frac{1}{2}\)
Leo can run \(6\frac{1}{3}\) miles in an hour. We know that \(6\frac{1}{3}\) miles is greater than \(5\frac{1}{2}\) miles.
Hence, Leo runs more than \(5\frac{1}{2}\) or \(\frac{11}{2}\) miles in an hour.
A mixed number is a combination of a whole number and a fraction. But just like proper fractions, mixed numbers lie between two whole numbers. That means mixed numbers are basically fractions. To be precise, mixed numbers are improper fractions.
The general form of a fraction is \(\frac{a}{b}\). If b > a or the denominator is greater than the numerator, it is a proper fraction. On the other hand, if a > b or the numerator is greater than the denominator, it is an improper fraction.
A fraction can be converted into a mixed fraction when the numerator is greater than the denominator (a > b in \(\frac{a}{b}\)). In other words, only improper fractions can be converted into mixed fractions.
A tape diagram helps us represent a single fraction. On the other hand, the area model helps us visualize real-life applications of fractions.