[Arithmetic 6] Converting many fractions to the same denominator

❄ CONVERTING MANY FRACTIONS TO THE SAME DENOMINATOR

To convert many fractions to the same denominator, we do as follows:
Step 1: Reduce the given fractions, convert their negative denomonators to positive denominators.
Step 2: Find a common denominator (usually LCM)
Step 3: Multiply numerator and denominator of each fraction by corresponding sub-factor (find sub-factor by dividing the common denominator by each denominator)

Example: Convert the following fractions to the same denominator:

$\dfrac{-4}{{7}}$, $\dfrac{16}{{18}}$ and $\dfrac{20}{{-42}}$

Solution:

$\dfrac{{16}}{{18}} = \dfrac{{8}}{{9}}$, $\dfrac{{20}}{{-42}} = \dfrac{{-10}}{{21}}$

In order to find a common denominator, we often use the following ways:

Way 1: LCM(7, 9, 21) = 63 (To find LCM, review here)

Way 2: In the three denominators 7, 9, 21, we get the greatest denominator that is 21. We'll multiply 21 by 1, 2, 3, etc until receiving the answer which is divisible by the remaining numbers, that receiving answer is a common denominator. In this case, 21 $\times$ 3 = 63 is divisible by 7, 9, 21. Thus, 21 is common denominator which we have to find.

CD = 63

The 1st sub-factor 63 : 7 = 9, the 2nd sub-factor 63 : 9 = 7, the 3rd sub-factor 63 : 21 = 3

$\dfrac{{ - 4}}{7} = \dfrac{{ - 4.9}}{{7.9}} = \dfrac{{ - 36}}{{63}}$

$\dfrac{8}{9} = \dfrac{{8.7}}{{9.7}} = \dfrac{{56}}{{63}}$

$\dfrac{{ - 10}}{{21}} = \dfrac{{ - 10.3}}{{21.3}} = \dfrac{{ - 30}}{{63}}$