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Comparison of Fractions

Comparison of fractions: Suppose, some fractions are to be arranged in ascending or descending order of magnitude.

Fractions can be compared in many ways. Find below 3 differnet ways of doing it.
If denominators are same, like 56/98 and 57/98, then just compare the numerators,
So the rule is:
a     b    --- > ---  if  a > b     n     n
Since 56 < 57, 56/98 < 57/98. They are in the same order as their numerators.
Second, suppose the numerators are the same, as in your problem, 5/9 and 5/6. Then it works the opposite way: The bigger the denominator, the smaller the fraction. So the rule is:     n     n    --- > ---  if  a < b     a     b
Since 9 > 6, 5/9 < 5/6. They are in the reverse order of their denominators.

Thirdly, in general cases, you simply convert the fractions to the first case, by giving them a common denominator. You do not really have to worry about finding the least common denominator, though sometimes that will save a lot of work. Let us compare 5/9 and 4/7. Since we do not see any common factors immediately (and in fact there are not any), we can just multiply the denominators to get a common denominator, 63. To convert 5/9 to 63rds, we multiply by 7; to convert 4/7 to 63rds, we multiply by 9:
5     4    --- ? ---     9     7
5*7   4*9    --- ? ---    9*7   7*9
35 < 36, so
5     4    --- < ---     9     7
You may not calculate the value of denominator because it will be same in both cases.

5*7=35          4*9=36; since this is bigger, 4/7 is bigger          \        /           5     4          --- ? ---           9     7
If there is a common multiple in denominators, for instance, which is bigger, 5/9 or 44/81? I see that 81 is a multiple of 9, so I do not have to go to the trouble of multiplying 5 by 81 and 44 by 9; I just multiply 5 by 9 and compare to 44:     5    44    --- ? --     9    81
5*9   44    --- ? --    9*9   81
45 > 44 so
5    44    --- > --     9    81
Finally, convert each one of the given fractions in the decimal form. Now, arrange them in ascending order

as per requirements.

Recurring Decimal: - If, in a decimal fraction, a figure or a set of figures is repeated continuously, then such a number is called a recurring decimal. In a recurring decimal, if a single figure is repeated, then it is expressed by putting a dot on it.  If a set of figures is repeated, it is expressed by putting a bar on the set.

Pure Recurring Decimal: A decimal fraction, in which all the figures after the decimal point are repeated, is called a pure recurring decimal e.g.  2/3 = 0.666…=0.6.

Converting a Pure Recurring Decimal into Vulgar Fraction:-Rule: - Write the repeated figures only once in the numerator and take as many nines in the denominator as is the number of repeating figures.

Decimal: - a decimal fraction in which some figures do not repeat and some of them are repeated is called a mixed recurring decimal, e.g. 0.173333…= 0.173.

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