In the previous section we discussed the basics of equivalent fractions. We have see the following examples:
• 1/5 = 3/15 = 2/10
• 3/5 = 9/15 = 6/10
• 1/2 = 2/4 = 3/6 = 4/8 = 5/10
• 3/4 = 9/12
Numerous examples like those given above can be written for equivalent fractions. We will not need to draw figures to prove their 'equivalence'. In this section we will discuss the various features of equivalent fractions. While discussing it, we will learn methods to determine any equivalent fraction of a given fraction.
Consider the first example given above:
1/5 = 3/15 = 2/10
Do you see any thing peculiar? Yes.
The numerator 3 in the second fraction can be obtained by multiplying the first numerator 1 by 3
The denominator 15 in the second fraction can be obtained by multiplying the first denominator 5 by 3
That is., If we multiply both numerator and denominator of 1/5 by the same number 3, we will get the numerator and denominator of 3/15
Similarly:
If we multiply both numerator and denominator of 1/5 by the same number 2, we will get the numerator and denominator of 2/10.
Mathematically this can be written as:
In this way, the rest of the examples given above can also be written. We will find that in each equivalent fraction, the numerator and denominator of the 'original fraction' is multiplied by the same number. This is shown below:
So we get an easy method for calculating the equivalent fraction of any given fraction:
• Just multiply the numerator and the denominator by the same number.
The resulting fraction will be an equivalent fraction.
Let us see some solved examples:
Solved example 5.5
Find 3 equivalent fractions for each of the following:
(i) 2/3 (ii) 4/5 (iii) 2/9 (iv) 5/7
Solution: We can use any numbers. Let us use 2, 3 and 4 to multiply both the numerator and the denominator of the above fractions. The results along with the calculation steps are given below:
So now we have an easy method to obtain any equivalent fraction. From that method, a second method can be derived:
We know that division is just the opposite of multiplication. So, if we divide both the numerator and the denominator by the same number, we should get an equivalent fraction. This can be proved with the help of the following fig.5.18
• Fig.(a) shows the cake divided into 6 pieces.
• Fig.(b) shows the cake divided into 3 pieces.
• Fig.(c) shows the first 4 pieces out of the 6 in (a). So it is 4/6.
• Fig (d) shows the first 2 pieces out of the 3 in (b). So it is 2/3.
• (c) and (d) are same in size. So we see that 4/6 is equal to 2/3.
If we divide the numerator and denominator of 4/6 by the same number 2, we will get 2/3. This can be written mathematically as:
Thus we see that division can also be used to determine equivalent fractions.
Solved example 5.6
Find the equivalent fraction of 18/42 such that, the numerator of the equivalent fraction is 3
Solution:
• The numerator of the given fraction is 18.
• The numerator of the new equivalent fraction should be 3.
• 3 is less than 18. So we have to divide the original numerator by a number to obtain the new numerator.
• 18 ÷ ? = 3. The answer is 6.
• We have to divide the denominator also by 6. The result and the calculation steps are given below:
So the required equivalent fraction is 3/7
Solved example 5.7
Find the equivalent fraction of 4/9 such that, the denominator of the equivalent fraction is 36
Solution:
• The denominator of the given fraction is 9.
• The denominator of the new equivalent fraction should be 36.
• 36 is greater than 9. So we have to multiply the original denominator by a number to obtain the new denominator.
• 9 × ? = 36. The answer is 4.
• We have to multiply the numerator also by 4. The result and the calculation steps are given below:
So the required equivalent fraction is 16/36
Thus we have learned the method for determining an equivalent fraction of any given fraction. In the next section we will see another method.
• 1/5 = 3/15 = 2/10
• 3/5 = 9/15 = 6/10
• 1/2 = 2/4 = 3/6 = 4/8 = 5/10
• 3/4 = 9/12
Numerous examples like those given above can be written for equivalent fractions. We will not need to draw figures to prove their 'equivalence'. In this section we will discuss the various features of equivalent fractions. While discussing it, we will learn methods to determine any equivalent fraction of a given fraction.
Consider the first example given above:
1/5 = 3/15 = 2/10
Do you see any thing peculiar? Yes.
The numerator 3 in the second fraction can be obtained by multiplying the first numerator 1 by 3
The denominator 15 in the second fraction can be obtained by multiplying the first denominator 5 by 3
That is., If we multiply both numerator and denominator of 1/5 by the same number 3, we will get the numerator and denominator of 3/15
Similarly:
If we multiply both numerator and denominator of 1/5 by the same number 2, we will get the numerator and denominator of 2/10.
Mathematically this can be written as:
So we get an easy method for calculating the equivalent fraction of any given fraction:
• Just multiply the numerator and the denominator by the same number.
The resulting fraction will be an equivalent fraction.
Let us see some solved examples:
Solved example 5.5
Find 3 equivalent fractions for each of the following:
(i) 2/3 (ii) 4/5 (iii) 2/9 (iv) 5/7
Solution: We can use any numbers. Let us use 2, 3 and 4 to multiply both the numerator and the denominator of the above fractions. The results along with the calculation steps are given below:
So now we have an easy method to obtain any equivalent fraction. From that method, a second method can be derived:
We know that division is just the opposite of multiplication. So, if we divide both the numerator and the denominator by the same number, we should get an equivalent fraction. This can be proved with the help of the following fig.5.18
 |
| Fig.5.18 |
• Fig.(a) shows the cake divided into 6 pieces.
• Fig.(b) shows the cake divided into 3 pieces.
• Fig.(c) shows the first 4 pieces out of the 6 in (a). So it is 4/6.
• Fig (d) shows the first 2 pieces out of the 3 in (b). So it is 2/3.
• (c) and (d) are same in size. So we see that 4/6 is equal to 2/3.
If we divide the numerator and denominator of 4/6 by the same number 2, we will get 2/3. This can be written mathematically as:
Thus we see that division can also be used to determine equivalent fractions.
Solved example 5.6
Find the equivalent fraction of 18/42 such that, the numerator of the equivalent fraction is 3
Solution:
• The numerator of the given fraction is 18.
• The numerator of the new equivalent fraction should be 3.
• 3 is less than 18. So we have to divide the original numerator by a number to obtain the new numerator.
• 18 ÷ ? = 3. The answer is 6.
• We have to divide the denominator also by 6. The result and the calculation steps are given below:
So the required equivalent fraction is 3/7
Solved example 5.7
Find the equivalent fraction of 4/9 such that, the denominator of the equivalent fraction is 36
Solution:
• The denominator of the given fraction is 9.
• The denominator of the new equivalent fraction should be 36.
• 36 is greater than 9. So we have to multiply the original denominator by a number to obtain the new denominator.
• 9 × ? = 36. The answer is 4.
• We have to multiply the numerator also by 4. The result and the calculation steps are given below:
Thus we have learned the method for determining an equivalent fraction of any given fraction. In the next section we will see another method.
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