Let us study some of the common methods of identifying out the largest or smallest of a given set of fractions which are useful for Mental Maths.
Type – 1
When the numerators are same and the denominators are different, the fraction with the largest denominator is the smallest.
Have a look at the following example.
Example: Which of the following fractions is the smallest?
(3/5) , (3/7) , (3/13), (3/8)
Here, 13 is the largest denominator, so, (3/13) is the smallest fraction. 5 is the smallest denominator, hence (3/5) is the largest fraction.
Here logic is very simple,
Situation: (i) Assume that you are 5 Children in your family. Your Dad brought an Apple and mom cut it into 5 pieces and distributed among all the children including you. (1/5)
Situation (ii) : Assume that you are 8 Children in your family. Your Dad brought an Apple and mom cut it into 8 pieces and distributed among all the children including you. (1/8)
Type - 2
When the numerators are different and the denominators are same, the fraction with the largest numerator is the largest. Have a look at the following example.
Example: Which of the following fractions is the smallest?
(7/5) , (9/5), (4/5), (11/5)
As 4 is the smallest numerator, the fraction 4/5 is the smallest.
As 11 is the largest numerator, the fraction 11/5 is the largest.
Here too logic is very simple,
Situation 1 : Assume that you are 4 Children in your family. Your Dad brought 8 Apples and mom distributed them among all the children including you. (8/4)
Situation 2 : Assume that you are 4 Children in your family. Your Dad brought 12 Apples and mom distributed them among all the children including you. (12/4)
Type - 3
The fraction with the largest numerator and the smallest denominator is the largest.
Example: Which of the following fractions is the largest?
(19/16), (24/11), (17/13), (21/14), (23/15)
Solution : As 24 is the largest numerator and 11 is the smallest denominator, 24/11 is the largest fraction.
Type - 4
When the numerators of two fractions are unequal, we try and equate them by suitably cancelling factors or by suitably multiplying the numerators. Thereafter we compare the denominators as in TYPE 1. Have a look at the following examples.
Example: Which of the following fractions is the largest?
(64/328), (28/152), (36/176), (49/196)
Solution : 64/328 = 32/164 = 16/82 = 8/41 this is approximately equal to 1/5
Note : In these type of problems, approximate values will be enough. No need to get EXACT values.
25/152 = 14/76 = 7/38 this is approximately equal to 1/5.5
36/176 = 18/88 = 9/44 this is approximately equal to 1/5
49/196 = 7/28 = ¼
As all the numerators are 1 and the least denominator is 4, the fraction 49/196 is the largest
Example: Which of the following fractions is the largest?
(71/181), (214/519), (429/1141)
Solution : (71/181) = (71 X 6) / (181 X 6) = 426/1086
(214/519) = (214 X 2) / (519 X 2) = 428/1038
The numerators are now all ALMOST equal (426, 428 and 429). The smallest denominator is 1038.
So, the largest fraction must be 428/1038 that is 214/519 :)
Type - 5
For a fraction Less than 1 :
If the difference between the numerator and the denominator is same then the fraction with the larger values of numerator and denominator will be the largest. Have a look at the following example.
Example: Which of the following fractions is the largest?
(31/37), (23/29), (17/23), (35/41), (13/19)
Solutions: difference between the numerator and the denominator of each fraction is 6.... So the fraction with the larger numerals i.e., 35/41 is the greatest and the fraction with smaller numerals i.e., 13/19 is the smallest.
Type - 6
For a fraction Greater than 1
If the difference between the numerator and denominator is same, then the fraction with the smaller values will be the largest.
Example: Which of the following fraction is largest ?
(31/27), (43/39), (57/53), (27/23), (29/25)
Solution : As the difference between the numerator and the denominator is same, the fraction with the smaller values i.e., 27/23 is the largest.
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