Introduction: In the competition exams, questions related to the comparison of fractions are given in four different ways.

(i) Which one fraction is the smallest in the given fractions and which is the largest ?

(ii) Which fraction groups in the given group are in ascending order and which fraction group is in descending order (Descending order)?

Note:

** (i) Ascending order –** The smallest difference in this sequence is written first and largest in last.

Like ^{1}/_{16 } ^{1}/_{8 }^{1}/_{4 } ^{ 1}/_{2 }

** (ii) Descending order** – The largest fraction in this sequence is at the first and the smallest of the most is in the last.

Such as ^{ 7}/_{8 } ^{5}/_{8 }^{3}/_{8 }^{1}/_{8}

Example 1. Which of the following fractions is the smallest and the largest?

(i) ^{ 13}/_{17} (ii) ^{ 19}/_{23} (iii) ^{37}/_{41} (iv) ^{ 29}/_{33} (v) ^{ 59}/_{63}

Explanation: Here ^{13}/_{17} is the smallest and ^{ 59}/_{63} is the largest.

Example 2 . Which set of fractions is in ascending order in the fraction series below and which is in the Descending Order?

(i) ^{ 5}/_{7 } ^{3}/_{5} ^{9}/_{12} (ii) ^{ 3}/_{5} ^{5}/_{7 } ^{ 9} /_{12} (iii) ^{ 5}/_{7} ^{9}/_{12} ^{3}/_{5 }(iv) ^{ 9}/_{12} ^{3}/_{5 }^{5}/_{7 }(v) ^{ 9}/_{12} ^{5}/_{7 } ^{ 3}/_{5}

Explanation: here ^{ 3}/_{5} ^{5}/_{7 } ^{9}/_{12} i.e. the fraction categories given in the option (2) is in ascending order. Whereas the fraction category given in the Option (5) is in descending order.

Noble Suggestion: In Banking, S.S.C., R.R.B. and other competitive exams, questions are often asked on the basis of fractions. Most of the questions based on fraction are those which can only be solved by mental action. There are some questions that require a little calculation too. Solving such questions by short-cut methods is advisable.

** How to Identify the Different Types of Fractions ?**

Points to be Noted: Before solving all fraction related questions, it is necessary to first consider this point as by which method can be solved as quickly as possible. i.e. – Difference Rule, Cross multiplication rule, LCM Rule or Division Rule. Questions should be resolved only after this decision.

If the numerator and the denominator of the given fractions are the same, then they should be solved by “by the difference rule”, “if the difference is uneven,” then solve it by the cross multiplication rule. If the LCM of numerator and denominator is being small “by the LCM rule and the difference between being unequal and also the LCM is large then by “by Division” rule.

** TYPE-1 Difference Rule **

TRICK – If the difference of numerator and denominator is same, then the fraction made by the small numbers become smallest and made by the large numbers is the largest fraction.

Example – 1. What is the highest value of a fraction in the given fractions.

(i) ^{6}/_{7} (ii) ^{4}/_{5} (iii) ^{8}/_{9} (iv) ^{ 3}/_{4} (v) ^{ 7}/_{8}

Explanation: The difference between numerator and denominator of each fraction is ‘1’ and the fraction made by largest numbers is ^{ 8}/_{9}. Therefore, the value of ^{ 8}/_{9} is highest in them.

Example – 2. Which set of fractions is in Descending Order in the groups given below?

(i) ^{ 3}/_{7} ^{7}/_{11} ^{13}/_{17} ^{5}/_{9} (ii) ^{3}/_{7} ^{13}/_{17} ^{7}/_{11} ^{5}/_{9} (iii) ^{5}/_{9} ^{13}/_{17} ^{7}/_{11} ^{3}/_{7}

(iv) ^{13}/_{17} ^{7}/_{11} ^{5}/_{9} ^{3}/_{7 }(v) None of these

Explanation: The difference between numerator and denominator of each of the given fractions is 4 and the fractions made with the smallest and largest numbers are ^{3}/_{7} and ^{ 13}/_{17 } respectively. Therefore, the smallest of them is ^{ 3}/_{7} and the largest difference is ^{ 13}/_{17} . Therefore, in the option (4) the fractions are placed in descending order.

** TYPE-2 Cross Multiplication Rule **

**TYPE-3 L. C. M. Rule **

TRICK –

(i) If the LCM of numerators is smaller than the LCM of Denominators, then by multiplying the numerator and denominator of each fraction by the appropriate number. the numerators of each fraction is made equal to the LCM of those fractions. Now the fraction whose denominator is smallest becomes the smallest fraction and the fraction whose denominator is largest becomes the largest fraction.

(ii) If the LCM of denominators is smaller than the LCM of Numerators, then by multiplying the numerator and denominator of each fraction by the appropriate number. the denominators of each fraction is made equal to the LCM of those fractions. Now the fraction whose numerator is smallest becomes the smallest fraction and the fraction whose numerator is largest becomes the largest fraction.

** TYPE-4 Division Rule**

TRICK – There are some fraction series that are easy to compare with division method. Fractions of this type of series are divided by dividing numerator by denominator of each fraction and then by comparing the results we will get the result.

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