Thursday, July 14, 2016

Chapter 5.14 - Comparing Unequal fractions

In the previous section we learned about  properties of equivalent fractions. In this section we will see the comparison of unequal fractions.

Consider the following points:
• We are given some fractions
• All those given fractions have the same denominators. The numerators are different
■ Then, the fraction with the smallest numerator will be the smallest fraction
■ The fraction with the largest numerator will be the largest fraction
• Example: Given some fractions: 611311811411 ,911
• The smallest is 311, and the largest is 911


We have discussed the reason behind the above result here. Now we consider another such property:
Consider the following points:
• We are given some fractions
• All those given fractions have the same numerators. The denominators are different
■ Then, the fraction with the smallest denominator will be the largest fraction
■ The fraction with the largest denominator will be the smallest fraction
• Example: Given some fractions: 917912914921910
• The smallest is 921, and the largest is 910

We have discussed the reason behind the above result here.


Let us see some applications of the above two properties:
■ Compare 34 and 311
Ans: 311 < 3(∵ when two fractions have the same numerator, that fraction with the largest denominator is the smallest)
■ Compare 75 and 79
Ans: 79 < 7(∵ when two fractions have the same numerator, that fraction with the largest denominator is the smallest)
■ Compare 312 and 512
Ans: 312 512 (∵ when two fractions have the same denominator, that fraction with the smallest numerator is the smallest)
■ Compare 47 and 27
Ans: 27 < 47 (∵ when two fractions have the same denominator, that fraction with the smallest numerator is the smallest)

In some situations, the above two properties can be used together. Let us see an example:
■ Compare 37 and 45
In this case, neither the numerators nor the denominators are equal. So how do we proceed?
1. We have to compare 37 and 45
2. Set up a new fraction. 
    ♦ It must have the same numerator as the first fraction 37
    ♦ It must have the same denominator as the second fraction 45 
3. So the new fraction is: 35
4. Compare the new fraction with the first fraction. We get: 37 < 35. (∵ when two fractions have the same numerator, that fraction with the largest denominator is the smallest)
5. Compare the new fraction with the second fraction. We get 35 < 45. (∵ when two fractions have the same denominator, that fraction with the smallest numerator is the smallest) 
6. So we have two results: 37 < 35 and 35 < 45
7. we can make an important conclusion from the above:
37 is less than 35. But this 35 is less than 45. So obviously, 37 will be less than 45
8. So the result of the comparison between 37 and 45 is: 37 < 45

■ The above steps can be summarised as follows:
• We have a fraction 37 in hand
• We are given a second fraction 45 for comparison
• If the second fraction has an increased numerator, and a decreased denominator, then that second fraction will be greater than the first fraction

Another example:
■ Compare 56 and 49
In this case, neither the numerators nor the denominators are equal. So how do we proceed?
1. We have to compare 56 and 49
2. Set up a new fraction. 
    ♦ It must have the same numerator as the first fraction 56
    ♦ It must have the same denominator as the second fraction 49 
3. So the new fraction is: 59
4. Compare the new fraction with the first fraction. We get: 59 < 56. (∵ when two fractions have the same numerator, that fraction with the largest denominator is the smallest)
5. Compare the new fraction with the second fraction. We get 49 < 59. (∵ when two fractions have the same denominator, that fraction with the smallest numerator is the smallest) 
6. So we have two results: 59 < 56 and 49 < 59
7. we can make an important conclusion from the above:
5is greater than 59. But this 59 is greater than 49. So obviously, 56 will be greater than 49
8. So the result of the comparison between 56 and 49 is: 49 < 56

■ The above steps can be summarised as follows:
• We have a fraction 56 in hand
• We are given a second fraction 49 for comparison
• If the second fraction has a decreased numerator, and an increased denominator, then that second fraction will be less than the first fraction.

A general form of the results of the above two examples can be written as:

It is easy to remember the results shown above if we know this:
■ When numerator increases, the fraction increases and vice versa (denominator remains constant)
■ When denominator increases, the fraction decreases and vice versa (numerator remains constant)

We will now see some solved examples:
Find the larger of the two fractions in each of the following pairs:
 13/17, 14/15
Solution: 
First fraction: 13/17
second fraction: 14/15. Numerator increased, denominator decreased
So Second fraction 14/15 is larger
 13/17, 11/18
solution:
First fraction: 13/17
second fraction: 11/18. Numerator decreased, denominator increased
So First fraction 13/17 is larger
 14/15, 11/18
solution:
First fraction: 14/15
second fraction: 11/18. Numerator decreased, denominator increased
So First fraction 14/15 is larger

We will now see an advanced case:
■ Compare 12 and 23
The methods that we have seen so far cannot be applied here. This is because:
• Neither numerators nor denominators are the same
• Numerator of the first fraction increased. The denominator of the second fraction also increased.

In such cases, we have to look for other methods. The most common method is to convert each of them into fractions with the same denominator. We have seen this method earlier here.
• Multiply both numerator and denominator of 12 by '3' (the other denominator). This will give 36
• Multiply both numerator and denominator of 23 by '2' (the other denominator). This will give 46
• Now we compare 36 and 46. Obviously, 46 is larger. That is., 36 < 46
• This is same as 123

Another example:

From the above two examples, it is clear that, once we convert each of the fractions into equivalent fractions of same denominators, we can ignore those 'same denominators'. All we have to do is to compare the 'new numerators'. We can write the steps algebraically as follows:


In the next section we will see a more advanced case.

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