# Common denominator

🏆Practice least common denominator

A common denominator is a denominator that will be common and equal for all the fractions in the exercise. We will reach such a denominator by reducing or enlarging the fraction - an operation of multiplication or division.
We can arrive at several correct common denominators.

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

• The first case: one of the denominators appearing in the original exercise will be the common denominator.
In this case, we will notice that we only have to multiply one denominator by an integer to reach the same denominator as in the other fraction.
• The second case: find a number that both denominators in the exercise can reach by multiplication.
• The third case: find the common denominator by multiplying the denominators.

## Test yourself on least common denominator!

$$\frac{3}{5}+\frac{2}{15}=$$

## 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:

### The first case: one of the denominators that appears in the original exercise - will be the common denominator.

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.

#### Let's look at an example

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.

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### Now we will move on to the second case: finding a number that both denominators in the exercise can reach through multiplication.

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

#### Let's look at an example

$\frac{2}{4}+\frac{1}{6}$
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 $12$.

### The third case: finding the common denominator by multiplying the denominators

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.

#### Let's look at this in an exercise

$\frac{1}{13}+\frac{2}{5}$
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 $65$.

#### Another exercise

$\frac{2}{3}+\frac{1}{6}$
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:
$\frac{12}{18}+\frac{3}{18}$

## Examples and exercises with solutions of common denominator

### Exercise #1

$\frac{5}{10}-\frac{1}{6}=$

### Step-by-Step Solution

Let's try to find the lowest common multiple between 6 and 10

To find the lowest common multiple, we need to find a number that is divisible by both 6 and 10

In this case, the lowest common multiple is 30

Now let's multiply each number by an appropriate factor to reach the multiple of 30

We will multiply the first number by 3

We will multiply the second number by 5

$\frac{5\times3}{10\times3}-\frac{1\times5}{6\times5}=\frac{15}{30}-\frac{5}{30}$

Now let's subtract:

$\frac{15-5}{30}=\frac{10}{30}$

$\frac{10}{30}$

### Exercise #2

$\frac{4}{8}+\frac{4}{10}=$

### Step-by-Step Solution

Let's try to find the lowest common multiple between 8 and 10

To find the lowest common multiple, we need to find a number that is divisible by both 8 and 10

In this case, the lowest common multiple is 40

Now, let's multiply each number in the appropriate multiples to reach the number 40

We will multiply the first number by 5

We will multiply the second number by 4

$\frac{4\times5}{8\times5}+\frac{4\times4}{10\times4}=\frac{20}{40}+\frac{16}{40}$

Now let's calculate:

$\frac{20+16}{40}=\frac{36}{40}$

$\frac{36}{40}$

### Exercise #3

$\frac{2}{3}-\frac{1}{6}-\frac{6}{12}=$

### Step-by-Step Solution

Let's try to find the lowest common multiple of 3, 6 and 12

To find the lowest common multiple, we find a number that is divisible by 3, 6 and 12

In this case, the common multiple is 12

Now let's multiply each number in the appropriate multiple to reach the multiple of 12

We will multiply the first number by 4

We will multiply the second number by 2

We will multiply the third number by 1

$\frac{2\times4}{3\times4}-\frac{1\times2}{6\times2}-\frac{6\times1}{12\times1}=\frac{8}{12}-\frac{2}{12}-\frac{6}{12}$

Now let's subtract:

$\frac{8-2-6}{12}=\frac{6-6}{12}=\frac{0}{12}$

We will divide the numerator and the denominator by 0 and get:

$\frac{0}{12}=0$

$0$

### Exercise #4

$\frac{3}{5}+\frac{2}{15}=$

### Video Solution

$\frac{11}{15}$

### Exercise #5

$\frac{1}{5}+\frac{2}{15}=$

### Video Solution

$\frac{5}{15}$