In the previous section we completed the discussion on Equivalent fractions and the methods to find them. In this section we will discuss about the 'simplest form' of a fraction.
Consider the fraction 12/18. If we divide both numerator and denominator of this fraction by a same number, we will get a new fraction. And this new fraction will be an equivalent fraction. That means., the new fraction will be having the same value as the original fraction 12/18. We can divide 12 and 18 by 2. So let us do it:
We get 6/9. This new numerator 6 and the new denominator 9 are divisible by 3.
So we get: 2/3. It is not possible to do further division because, there is no such number with which we can divide both 2 and 3 (except '1'). So 2/3 is the simplest form possible.
We can write: 12/18 = 6/9 = 2/3
Let us analyse the steps that we did above:
• First we divided both 12 and 18 by the same number 2. We were able to do the divisions because '2' is a common factor of both 12 and 18. The resulting fraction that we got is 6/9
• Next we divided 6 and 9 by the same number 3. We were able to do the divisions because 3 is a common factor of both 6 and 9. The resulting fraction that we got is 2/3
• We had to stop with 2/3. Because 2 and 3 have no common factors except '1'
A fraction is said to be in the simplest form if its numerator and denominator have no common factor except 1. The simplest form is some times referred to as lowest form.
In the above steps, first we divided by 2, and then by 3. These steps can be avoided, if we divide both 12 and 18 by ‘6’. This is shown below:
We get the same final answer: 2/3. Note that ‘6’ is the Highest common factor (HCF) of 12 and 18. So, the easiest method to obtain the simplest form of a given fraction is to divide both it’s numerator and denominator by their HCF. Now we will see some solved examples based on the above discussion:
Solved example 5.12
Write in the simplest form: (i) 50/125 (ii) 21/70 (iii) 36/90
Solution:
(i) HCF of 50 and 125 is 25 (ii) HCF of 21 and 70 is 7 (iii) HCF of 36 and 90 is 18.
With this information we can do the calculations as shown below:
Another method is to use continuous division on the given fraction itself. It is demonstrated for the last problem as follows:
In the above fig. identical colors at the top and bottom indicates results obtained in a same step. For example 12 and 30 were obtained in the same step. 4 and 10 are obtained in the next step. The fraction progressively gets smaller and smaller, and we get the final answer as 2/5. This is because 2 and 5 do not have any common factor.
Solved example 5.13
Is the fraction 39/79 in it's 'simplest form' ?
Solution:
The numbers 39 and 79 do not have any common factors. So there is not even a single number by which we can divide both numerator and denominator of this fraction. Thus we can say: 39/79 is in it's 'simplest form'
Solved example 5.14
Fig.5.19 below shows three rectangles. All three are of the same size. Each is divided into a number of equal parts. Out of these equal parts, a few are shaded with green color.
(i) Write the fraction represented by each rectangle
(ii) Are there any equivalent fractions?
Solution:
(i) Fraction represented by each rectangle:
(a) The rectangle is divided into 40 equal parts. Out of them, 8 are shaded. So the fraction represented by the rectangle is 8/40
(b) The rectangle is divided into 40 equal parts. Out of them, 15 are shaded. So the fraction represented by the rectangle is 15/40
(c) The rectangle is divided into 8 equal parts. Out of them, 3 are shaded. So the fraction represented by the rectangle is 3/8
(ii) To find if there are any equivalent fractions, we will first write each into the 'simplest form'.
(a) 8/40 = 1/5 (dividing numerator and denominator by 8)
(b) 15/40 = 3/8 (dividing numerator and denominator by 5)
(c) 3/8 can not be further reduced.
So we find that (b) and (c) are equivalent fractions. If we combine the 15 green portions in (b), that combined area will be equal to the combined area of the 3 green portions in (c)
Solved example 5.15
Fig.5.20 below shows three circles. All three are of the same size. Each is divided into a number of equal parts. Out of these equal parts, a few are shaded with yellow color.
(i) Write the fraction represented by each circle
(ii) Are there any equivalent fractions?
Solution:
(i) Fraction represented by each circle:
(a) The circle is divided into 6 equal parts. Out of them, 1 is shaded. So the fraction represented by the circle is 1/6
(b) The circle is divided into 4 equal parts. Out of them, 1 is shaded. So the fraction represented by the circle is 1/4
(c) The circle is divided into 12 equal parts. Out of them, 3 are shaded. So the fraction represented by the rectangle is 3/12
(ii) To find if there are any equivalent fractions, we will first write each into the 'simplest form'.
(a) 1/6 can not be further reduced.
(b) 1/4 can not be further reduced.
(c) 3/12 = 1/4 (dividing numerator and denominator by 3)
So we find that (b) and (c) are equivalent fractions. If we combine the 3 yellow portions in (c), that combined area will be equal to the yellow portion in (b).
In the next section we will discuss about 'like fractions'.
Consider the fraction 12/18. If we divide both numerator and denominator of this fraction by a same number, we will get a new fraction. And this new fraction will be an equivalent fraction. That means., the new fraction will be having the same value as the original fraction 12/18. We can divide 12 and 18 by 2. So let us do it:
We can write: 12/18 = 6/9 = 2/3
Let us analyse the steps that we did above:
• First we divided both 12 and 18 by the same number 2. We were able to do the divisions because '2' is a common factor of both 12 and 18. The resulting fraction that we got is 6/9
• Next we divided 6 and 9 by the same number 3. We were able to do the divisions because 3 is a common factor of both 6 and 9. The resulting fraction that we got is 2/3
• We had to stop with 2/3. Because 2 and 3 have no common factors except '1'
A fraction is said to be in the simplest form if its numerator and denominator have no common factor except 1. The simplest form is some times referred to as lowest form.
In the above steps, first we divided by 2, and then by 3. These steps can be avoided, if we divide both 12 and 18 by ‘6’. This is shown below:
We get the same final answer: 2/3. Note that ‘6’ is the Highest common factor (HCF) of 12 and 18. So, the easiest method to obtain the simplest form of a given fraction is to divide both it’s numerator and denominator by their HCF. Now we will see some solved examples based on the above discussion:
Solved example 5.12
Write in the simplest form: (i) 50/125 (ii) 21/70 (iii) 36/90
Solution:
(i) HCF of 50 and 125 is 25 (ii) HCF of 21 and 70 is 7 (iii) HCF of 36 and 90 is 18.
With this information we can do the calculations as shown below:
Another method is to use continuous division on the given fraction itself. It is demonstrated for the last problem as follows:
In the above fig. identical colors at the top and bottom indicates results obtained in a same step. For example 12 and 30 were obtained in the same step. 4 and 10 are obtained in the next step. The fraction progressively gets smaller and smaller, and we get the final answer as 2/5. This is because 2 and 5 do not have any common factor.
Solved example 5.13
Is the fraction 39/79 in it's 'simplest form' ?
Solution:
The numbers 39 and 79 do not have any common factors. So there is not even a single number by which we can divide both numerator and denominator of this fraction. Thus we can say: 39/79 is in it's 'simplest form'
Solved example 5.14
Fig.5.19 below shows three rectangles. All three are of the same size. Each is divided into a number of equal parts. Out of these equal parts, a few are shaded with green color.
 |
| Fig.5.19 |
(ii) Are there any equivalent fractions?
Solution:
(i) Fraction represented by each rectangle:
(a) The rectangle is divided into 40 equal parts. Out of them, 8 are shaded. So the fraction represented by the rectangle is 8/40
(b) The rectangle is divided into 40 equal parts. Out of them, 15 are shaded. So the fraction represented by the rectangle is 15/40
(c) The rectangle is divided into 8 equal parts. Out of them, 3 are shaded. So the fraction represented by the rectangle is 3/8
(ii) To find if there are any equivalent fractions, we will first write each into the 'simplest form'.
(a) 8/40 = 1/5 (dividing numerator and denominator by 8)
(b) 15/40 = 3/8 (dividing numerator and denominator by 5)
(c) 3/8 can not be further reduced.
So we find that (b) and (c) are equivalent fractions. If we combine the 15 green portions in (b), that combined area will be equal to the combined area of the 3 green portions in (c)
Solved example 5.15
Fig.5.20 below shows three circles. All three are of the same size. Each is divided into a number of equal parts. Out of these equal parts, a few are shaded with yellow color.
 |
| Fig.5.20 |
(ii) Are there any equivalent fractions?
Solution:
(i) Fraction represented by each circle:
(a) The circle is divided into 6 equal parts. Out of them, 1 is shaded. So the fraction represented by the circle is 1/6
(b) The circle is divided into 4 equal parts. Out of them, 1 is shaded. So the fraction represented by the circle is 1/4
(c) The circle is divided into 12 equal parts. Out of them, 3 are shaded. So the fraction represented by the rectangle is 3/12
(ii) To find if there are any equivalent fractions, we will first write each into the 'simplest form'.
(a) 1/6 can not be further reduced.
(b) 1/4 can not be further reduced.
(c) 3/12 = 1/4 (dividing numerator and denominator by 3)
So we find that (b) and (c) are equivalent fractions. If we combine the 3 yellow portions in (c), that combined area will be equal to the yellow portion in (b).
In the next section we will discuss about 'like fractions'.
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