What are fractions ?
How many pizzas do you see? Clearly, it is one whole pizza. Now, what if we cut the pizza into two equal parts and Chris eats one of the parts.
In this situation, we say that Chris ate half of the pizza. We know what is half of something, or how to cut something into two equal halves, but does half have a mathematical significance?
This can be understood by studying fractions. When we divide a whole of any entity, quantity, or number, we get fractions. For example, if we divide 1 pizza into two equal parts, then this can be mathematically written as, 1 ÷ 2.
We can see that this type of division is new to us. This is mainly because the divisor is greater than the dividend, and this results in an expression called a fraction.
Now, fractions also have a unique way of representation. Fractions such as 1 ÷ 2 are written as \(\frac{1}{2}\), where the divisor is written below a horizontal line and the dividend is written above the horizontal line.
We also have special names for numbers written above and below the line. The number above the horizontal line is called the numerator and the number written below this line is called the denominator. A small tip to remember this is “Down is the Denominator”.
In the above circle, one fourths of it is shaded green. This part, one fourth, represents one part from a whole circle that is cut into four equal pieces. As discussed earlier, while we write fractions, we use a horizontal line or a bar.
The number written above this bar is called the numerator and it signifies the number of equal parts being counted. While the number written below the bar is called the denominator, which signifies the total number of equal parts of the whole.
We can also understand fractions by breaking geometrical figures into equal parts. Let us understand this by drawing a rectangle and making equal parts of it.
The Following Rectangle Represents the Whole
Now, let’s try to divide this rectangle into several equal parts, such as:
However, if the parts are not equal, then, we cannot consider that as equal parts of the whole. Here are some ways you should not cut a pizza, especially if you want to divide it equally amongst your siblings.
In the partition of the circle shown above, we cannot say that the yellow part is one-third of a circle. This is because of unequal partitions in the circle.
Some more fractions
Divide a pizza equally amongst 6 members of an archery team.
The answer would be sixths. This is because there are six friends and each friend should get an equal piece. This can also be shown as,
1 ÷ 6 or \(\frac{1}{6}\)
Let us take another example of dividing a pizza.
You cut the pizza into 8 equal parts and take one of them.
To write this as fraction you first draw a horizontal line with places above and below to write numbers,
Write how many equal parts you have taken above the line or in the upper box,
Write the total number of parts below the line,
So, you obtained the fraction as, \(\frac{1}{8}\) or this is the part that you took out of eight parts.
Let us take another example,
There are six chairs. 5 of them are occupied. What is the fraction of people sitting on these chairs?
First, draw a horizontal line with spaces above and below,
Write the number of seats occupied above the line,
Write the total number of seats below the line,
So we can see that \(\frac{5}{6}\) is the fraction here showing us that 5 people are sitting on 5 of the 6 chairs.
Unit fraction
A fraction is used to represent the number of equal parts taken out of a whole. Similarly, a unit fraction is a fraction that represents one equal part of the whole. We can understand this by looking at the following figure.
Writing fractions of a whole
Writing fractions of the whole is quite similar to the unit fraction. To write fractions of the whole one can count the number of equally shaded regions, and write that as the numerator.
For example: If one makes three equal parts of a circle and shades one part, then the fraction is a unit fraction which is represented as, \(\frac{1}{3}\).
Similarly, if two parts are shaded out of the three equal parts of the circle, then the fraction is a non-unit fraction and is \(\frac{2}{3}\).
If all the three parts are shaded then this can be represented as a fraction which is \(\frac{3}{3}\) . However, shading all the equal parts gives us the entire circle and we can say that it represents 1 circle.
From what we just saw, we get to know that all the fractions where the numerators and denominators have the same values represent unity, or in other words 1.
For example \(\frac{4}{4}\) \(\frac{9}{9}\) \(\frac{13}{13}\) and so on, are fractions that have a value of 1.
We also discussed that the fraction bar represents division. This would verify that \(\frac{3}{3}\) is actually 3 ÷ 3, which is equal to 1.
Fractions on a number line
Fractions on a number line less than one:
Every number on the number line represents a fixed distance from 0. The distance from 0 to 1 is taken as one whole. As the number line represents all types of numbers, we can also represent fractions of the number line. This is possible because we can divide the number line into as many equal parts as required.
Let us understand this with an example. If we want to plot \(\frac{3}{4}\) on the number line then we will follow the following steps.
Step 1:
Draw a number line and make equal divisions between 0 and 1 as shown in the diagram. The number of equal divisions or parts are always taken from the denominator. As the denominator given is 4 , hence we will draw four equal divisions.
Step 2:
Now we need to mark the equal distances starting from 0. This is the method to do this.
As the first point is 0 mark it as 0. Now this can also be written as \(\frac{0}{4}\) = 0
Now, the first part is \(\frac{1}{4}\) as we have made four equal divisions of the distance from 0 to 1.
The second part involves two fractions of \(\frac{1}{4}\) , or \(\frac{2}{4}\) .
The third part involves three fractions of \(\frac{1}{4}\) or \(\frac{3}{4}\) .
The fourth part involves four fractions of \(\frac{1}{4}\), which is \(\frac{4}{4}\) or simply 1.
So, in this case we mark the number line as, \(\frac{1}{4}\) , \(\frac{2}{4}\), \(\frac{3}{4}\).
Step 3: Mark the fraction \(\frac{3}{4}\).
As three fractions of \(\frac{1}{4}\) is \(\frac{3}{4}\) we plot the required fraction on the number line after 3 equal divisions as shown in the diagram.
All the fractions we observed till now had numerators lesser than their denominators. Such fractions are called Proper Fractions.
Now, what if you have two pizzas and you eat one full pizza, and one half of the other pizza. You say that “I ate one and a half pizzas!”. Let us see how to understand this type of fractions.
Fractions on a number line greater than 1:
When the numerator is greater than the denominator, the fraction is greater than one or one whole. This type of fraction is also called improper fractions. For example, you have two pizzas and you divide each into 8 equal parts. Then, you take 3 equal parts from a pizza.
To represent the fraction of pizzas left, we draw a fraction bar. As you have taken 3 equal parts there is 1 whole pizza and 5 equal parts of another pizza left.
So, we write equal parts of pizza left as the numerator i.e. 13. And the total number of parts in each pizza as the denominator, which is 8.
Therefore the fraction of the pizza left is \(\frac{13}{8}\).
We can also plot fractions greater than 1 on a number line. Let us take an example.
Suppose we want to plot the fraction \(\frac{6}{4}\) on a number line then we follow the following steps.
We know that a fraction \(\frac{1}{4}\) is \(\frac{1}{4}\) , two of the fractions \(\frac{1}{4}\) is \(\frac{2}{4}\), and so on.
So, six of the fraction \(\frac{1}{4}\) will be \(\frac{6}{4}\).
Therefore, we make six equal divisions each equal to \(\frac{1}{4}\) on a number line to represent \(\frac{6}{4}\).
Step 1: Divide each whole into as many equal parts with a denominator of 4.
Step 2: Label each point on the number line.
Step 3: Plot \(\frac{6}{4}\).
Solved Examples on Fractions
Example 1: What fraction of the whole is shaded?
Solution:
To write a fraction for the diagram given, let us find the numerator and the denominator.
There are 6 equal parts of the whole, so 6 would be our denominator.
One of the equal parts is shaded hence 1 is our numerator.
Therefore, the fraction obtained is, \(\frac{1}{6}\).
Example 2: What fraction of the whole is shaded?
Solution:
To write a fraction for the above figure, let us find the numerator and the denominator.
There are 6 equal parts of the whole, so 6 would be our denominator.
Five of the equal parts are shaded, hence 5 is our numerator.
Therefore the fraction obtained is, \(\frac{5}{6}\).
Example 3: What fraction of the whole is shaded?
Solution:
To write a fraction for the above figure, let us find the numerator and the denominator.
There are 2 equal parts in the whole so 2 would be our denominator.
One of the equal parts is shaded hence 1 is our numerator.
Therefore, the fraction obtained is, \(\frac{1}{2}\).
Example 4: What fraction of the whole is shaded?
Solution:
To write a fraction for the above figure, let us find the numerator and denominator.
There are 8 equal parts in the whole, so 8 would be our denominator.
One of the equal parts is shaded so 1 is our numerator.
Therefore, the fraction obtained is, \(\frac{1}{8}\).
Example 5: Plot \(\frac{7}{4}\) on the number line.
Solution:
We know that the fraction \(\frac{1}{4}\) is \(\frac{1}{4}\), two of the fractions \(\frac{1}{4}\) is \(\frac{2}{4}\) and so on.
So, seven of the fractions \(\frac{1}{4}\) will be \(\frac{7}{4}\).
Therefore, we make seven equal divisions, each being \(\frac{1}{4}\) on a number line to represent \(\frac{7}{4}\).
Step 1: Divide each whole into as many equal parts as the denominator. The denominator is 4 so we make 4 parts between each whole number.
Step 2: Label each point on the number line.
Step 3: Plot \(\frac{7}{4}\).
There are three types of fractions, proper fractions, improper fractions and mixed fractions (also called mixed numbers).
If the numerator of a fraction is less than the denominator, then the fraction is called a proper fraction. For example \(\frac{1}{2}\) , \(\frac{2}{5}\) , \(\frac{7}{12}\) , and so on, are proper fractions that are always less than 1.
If the numerator is greater than the denominator, then the fraction is called an improper fraction. For example \(\frac{5}{4}\) , \(\frac{3}{2}\) , \(\frac{9}{5}\) , and so on. Improper fractions are always greater than 1.
If the fraction is a combination of a whole number and a fraction, then these fractions are called mixed fractions (also called mixed numbers). For example 1\(\frac{2}{3}\) , 3\(\frac{5}{6}\) , and so on.
If the denominator of any fractions are the same, then such fractions are called like fractions. For example \(\frac{4}{5}\) , \(\frac{3}{5}\), \(\frac{6}{5}\) , are examples of like fractions.
Similarly, if the denominator of fractions are not the same, then such fractions are called unlike fractions. For example \(\frac{1}{2}\) ,\(\frac{2}{3}\) , \(\frac{4}{7}\) ,are unlike fractions.