Common denominator

A common denominator is a topic that will accompany you for a long time from now until the end of your math studies, so you should know how to find it easily.
What is a common denominator?
It is a denominator that will be common and equal in all the fractions of the exercise. We will reach such a denominator by reducing or expanding the fraction - a multiplication or division operation.
We can arrive at several correct common denominators.

We will divide the search for the common denominator into 3 cases:

In this case, we will notice that we only have to multiply a denominator by an integer so that it reaches the same denominator as in the other fraction.

Find the common denominator in the exercise: $\frac{2}{3}+\frac{1}{6}$
Solution:
We will notice that we can multiply $3$ by $2$ to reach the denominator $6$.
and therefore the common denominator will be $6$.
We will obtain:
$\frac{4}{6}+\frac{1}{6}$

Pay attention: When we multiply the fraction to reach a common denominator, we must multiply both the numerator and the denominator so as not to change the value of the fraction.

Pay attention: sometimes, we might not reach that number and then we will move directly to the third case.

$\frac{2}{4}+\frac{1}{6}$
Find the common denominator in the exercise.

Solution:
If we look at the denominators, we can come to the fact that $12$ is a number whose denominators can reach it through a multiplication operation.
If we multiply $4$ by $3$ we get the result $12$
If we multiply $6$ by $2$ we get the result $12$
Remember that when we find a common denominator, we also perform the operation on the numerator and not just on the denominator.
We will obtain:
$\frac{6}{12}+\frac{2}{12}$
The common denominator is $12$.

Sometimes, we will not be able to reach a common denominator using the first and the second method, so we will resort to this method.
Keep in mind that multiplying denominators is always a safe way to find a common denominator and you can operate immediately (unless you want to find the least common denominator.

$\frac{1}{13}+\frac{2}{5}$
Find the common denominator.

Solution:
We multiply the denominator of the first fraction $13$ by the entire second fraction and the denominator of the second fraction $5$ We multiply by the entire first fraction.
Remember to perform the operation both in the numerator and the denominator.
Also, it is customary to mark the multiplication operation with a line above the fraction as follows:

$\frac{5}{65}+\frac{26}{65}$

The common denominator is $65$.

$\frac{2}{3}+\frac{1}{6}$
Find the common denominator in the exercise.

Solution:
We can solve this exercise in several ways. It is also suitable for the first case, as well as for the second and third.
Basically, we will arrive at$3$ different common denominators, all of which will have a correct answer.
Pay attention: You can use the method of multiplying the denominators in any exercise and, therefore, its use is always recommended.
But if you want to find the least common denominator that is possible to achieve (we will always go in order) first we will see if we can reach the common denominator by the first case, then the second and only then if we have not achieved it, we will move to the third.
Solution through the first case and finding the least common denominator:

We will notice that if we multiply $3$ by $2$ we will arrive at $6$.
We will obtain:

$\frac{4}{6}+\frac{1}{6}$
The common denominator is $6$.

Solution through the second case: finding a common number through a multiplication operation

If we multiply $6$ by $2$ we will arrive at $12$If we multiply $3$ by $4$ we will arrive at$12$.
We obtain:
$\frac{8}{12}+\frac{2}{12}$

Solution through the third case: multiplying the denominators

Multiply the denominator of the first fraction $3$ In the second fraction and in the denominator of the second fraction $6$ Multiply by the first fraction.
We obtain:
$\frac{12}{18}+\frac{3}{18}$