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How to Cross Multiply Fractions











Introduction:

             The real numbers are those which are represented on the number line. We can easily identify the greater and smaller whole numbers or integers once we see them. But in the case of fractions, it is difficult to identify the greater fraction by observation. For example  among the fractions $\frac{3}{7}$ and $\frac{4}{9}$, which is greater? It is hard to identify the greater fraction. In  this section let us  discuss with the method of identifying greater fraction.

Comparing Like Fractions:

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What are like fractions: Fractions with same denominator are called like fractions.

Example : $\frac{3}{8}$, $ \frac{5}{8}$, $\frac{7}{8}$.

These fractions have the same denominator 8.
Comparing like fractions:
The fraction which has the greater numerator is a greater fraction.
Let us compare the pictorial representation of these fractions.

fraction23
It is clear from the above figure that $\frac{7}{8}> \frac{5}{8}> \frac{3}{8}$

Example : Arrange the following in descending order.
$\frac{5}{11},  \frac{9}{11}, \frac{2}{11}$


Solution:           We have the fractions, $\frac{5}{11},  \frac{9}{11}, \frac{2}{11}$
The numerators are , 5 , 9 and 2
since 9 > 5 > 2,
the descending order of the fractions is  $\frac{9}{11}>\frac{5}{11}> \frac{2}{11}$

Comparing unlike fractions (LCM Method) :

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When we have two fractions $\frac{3}{7}$ and $\frac{4}{9}$, we can make the denominators same by finding the Least Common Multiple of the denominator.
To find the LCM of the denominators, 7 and 9
Since 7 and 9 are co-prime numbers the Least Common Multiple of 7 and 9 is its product.
Therefore LCM = 7 x 9 = 63
Let us convert the given fractions into like fractions with the denominator as 63.
To do this we must multiply the numerator and the denominator by the appropriate number.
Hence we have,

                                  $\frac{3}{7}$ and $\frac{4}{9}$


                              =>  $\frac{3\times 9}{7\times 9} $ ,  $\frac{4\times 7}{9\times 7}$
 
                              =>  $\frac{27}{63}$ , $\frac{28}{63}$ [ like fractions with denominator as 63 ]

                             => $\frac{28}{63}$ > $\frac{27}{63}$

                             => $\frac{4}{9}$  > $\frac{3}{7}$  [ second fraction(4/9) is greater than the first fraction (3/7)]


Comparing unlike fractions (Cross multiplication method):

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When we have two fractions $\frac{a}{b} and   \frac{c}{d}$,
we follow the following procedure to identify greater or smaller fraction.
Step 1. Multiply the numerator of the first fraction with the denominator of the second fraction.
Step 2: Multiply the denominator of the first fraction with the numerator of the second fraction.
Step 3: Compare the products obtained. IF the first product is greater then $\frac{a}{b} >   \frac{c}{d}$ .
            If the second product is greater then, $\frac{a}{b}$  <   $\frac{c}{d}$,

Example: $\frac{4}{9},\frac{5}{8}$


Solution: We have $\frac{4}{9},\frac{5}{8}  $
fraction24
Step 1:   4 x 8 = 40
Step 2:   9 x 5 = 45
Step 3:  45 > 40 [ second product is greater than the first product ]
Step 4:  $ \frac{5}{8}$ > $\frac{4}{9} $

Practice Questions:

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Use the appropriate sign , =, > or < for the following  pair of fractions.

1. $\frac{4}{7}, \frac{2}{7}$

2. $\frac{2}{5},\frac{3}{7}$ [ use LCM method ]

3. $\frac{5}{11},\frac{7}{15}$

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