In the previous sections we completed the discussion on positive and negative integers. In this section we will discuss about fractions. Consider the following situation:
A cake is to be divided equally between 8 students. So we cut the cake into 8 equal pieces as shown in the fig.5.1 below:
Each student gets one piece each. So the share got by each student is ‘one out of eight’. Mathematically we write this as 1/8. So each student gets 1/8 of the whole cake. Suppose one student was absent. And his share was given to another 'fat student'. Then this fat student would get two pieces. The share got by him is then ‘two out of 8’. Mathematically this is written as 2/8. We can say the fat student got 2/8 of the whole cake.
Consider another situation: A sack of rice is available in a store. It is to be divided equally among 12 families. The weighing apparatus is not available at that time. How would they divide it? They would first divide the rice into 12 equal heaps. This is shown in the fig.5.2 below:
Each family would get one part. So the share got by each family is ‘one out of 12’. Mathematically it is written as 1/12. That is., each family get 1/12 of the total rice available. This is shown in the fig.5.3 below:
Suppose all the families are not present there at the time of division. Only 7 families were present. Each of them will take one part and go home. Thus 7 parts out of 12 are removed. We say 7/12 of the whole rice is removed. 5 parts are remaining. We say 5/12 of the whole rice is remaining. This is shown in the fig.5.4 below:
Quantities like 1/8, 2/8, 7/12, 5/12 etc., are called fractions. They help us to express 'small quantities' or 'parts of large quantities'. Take for example 5/12. In it, 12 is the number of equal parts into which the whole is divided. 5 is the number of equal parts (out of the 12) taken into consideration. It is read as 'five twelfths'. That is., five times one twelfths. Similarly, 2/8 is read as 'two eighths'. That is., two times one eighths. Note that in the first case above, if each student was given a full cake, then the need for division would not arise. consequently, we will not need to use any fractions. But cakes are expensive, and so we cannot give each student an undivided cake. Similarly, in the second case, if each family is given a full sack each, we will not need to use fractions. So it is when dealing with small quantities, that we feel the need for fractions.
We will now see some examples.
Solved example 5.1
Write the fraction representing the portions shaded in red color in each case in the fig.5.5 below:
Solution:
(a) The circle is divided into two equal parts. The red portion is one among the two. So the red portion is 1/2 of the whole circle.
(b) The circle is divided into three equal parts. The red portion is one among the three. So the red portion is 1/3 of the whole circle.
(c) The circle is divided into four equal parts. The red portions are two among the four. So the red portions is 2/4 of the whole circle.
(d) The circle is divided into five equal parts. The red portions are two among the five. So the red portions is 2/5 of the whole circle.
Solved example 5.2
Write the fraction representing the portions shaded in red color in the fig.5.6 below:
Solution:
The rectangle is divided into eight equal parts. The red portions are three among the eight. So the red portions is 3/8 of the whole rectangle.
A cake is to be divided equally between 8 students. So we cut the cake into 8 equal pieces as shown in the fig.5.1 below:
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Fig.5.1 Division of cake into 8 equal parts
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Consider another situation: A sack of rice is available in a store. It is to be divided equally among 12 families. The weighing apparatus is not available at that time. How would they divide it? They would first divide the rice into 12 equal heaps. This is shown in the fig.5.2 below:
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Fig.5.2 A sack of rice divided into 12 equal heaps or parts
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Fig.5.3
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| Fig.5.4 |
We will now see some examples.
Solved example 5.1
Write the fraction representing the portions shaded in red color in each case in the fig.5.5 below:
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| Fig.5.5 |
(a) The circle is divided into two equal parts. The red portion is one among the two. So the red portion is 1/2 of the whole circle.
(b) The circle is divided into three equal parts. The red portion is one among the three. So the red portion is 1/3 of the whole circle.
(c) The circle is divided into four equal parts. The red portions are two among the four. So the red portions is 2/4 of the whole circle.
(d) The circle is divided into five equal parts. The red portions are two among the five. So the red portions is 2/5 of the whole circle.
Solved example 5.2
Write the fraction representing the portions shaded in red color in the fig.5.6 below:
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| Fig.5.6 |
The rectangle is divided into eight equal parts. The red portions are three among the eight. So the red portions is 3/8 of the whole rectangle.
Let us now see the various features of fractions:
• We can see that a fraction has two numbers: An upper number and a lower number. The upper number is called the numerator, and the lower number is called the denominator.
Example: In the fraction 3/4, 3 is the numerator, and 4 is the denominator.
• The denominator is the number of parts into which the large object is divided into. For example,
♦ The cake was divided into 8 parts. So the denominator is 8
♦ The rice was divided into 12 parts. So the denominator is 12.
• One important point to note:
We have seen that a large object is divided into small parts. Each of these parts should be of the same size. In other words, all the smaller parts should be equal.
• The numerator is the number of parts which are taken out. For example we take 1 part of the cake for each student. Here 1 is the numerator. In the second example, 5 parts are remaining at the store. So the portion remaining is 5/12. Here 5 is the numerator.
• The numerator is always less than the denominator. This is because the denominator is the total number of divisions. The numerator is the number of divisions taken out from the total divisions. So numerator will always be less than denominator. Such fractions are called Proper fractions. In later sections we will discuss special occasions where the numerator is larger than the denominator.
In the next section we will see another situation where fractions become essential for calculations.
In the next section we will see another situation where fractions become essential for calculations.
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