Least Common Denominator in Fractions - Algebra 2

To simplify this problem we need to find the least common denominator between the two fractions. To do this we look at 5 and at 8. The least common number between these two is 40.

In order to rewrite each fraction in terms of a denominator of 40 we need to muliple as follows:

we are able to mulitply by 8/8 and 5/5 because those fractions are really just 1 written in a different format.

Now using order of opperations we get the following

Now we have a common denominator and can do our addition to get the simplfied number:

In order to be able to find , we must first find the least common denominator. In this case, it is :

The equation can now be written as:

Solving for , we get:

The least common denominator is the lowest common multiple of the denominators.

Multiple of 27: 27, 54, 81, 108, 135, 162, 189, 216, 243, 270

Multiple of 9: 9, 18, 27, 36, 45, 54, 63, 72, 81, 90

The first step of finding the LCD of a set of fractions is to make sure each of the fractions are simplified. and are already simplified. However, can be reduced to . This makes the problem much easier because we now only have two different denominators to work with. From here, we simply multiply each denominator by increasing integers until we get a common denominator. It is important to always increase the lower of the two denominators. For instance, we have 4 and 3 as denominators in this problem. Since 3 is lower, we will multiply it by 2, getting 6. Now we have 4 and 6. 4 is lower, so we multiply it by 2 to get 8. Now we have 8 and 6. 6 is lower, so we multiply the original denominator of 3 by 3, resulting in denominators of 8 and 9. Following this trend, we get: 12 and 9, then 12 and 12. Therefore, 12 will be the least common denominator.

While simply multiplying all of the denominators will get you a common denominator between the fractions, it does not always give you the LCD.

The first step is to find the least common denominator. In this case, it is .

Then, you convert each fraction by multiplying the first fraction by and the second fraction by .

Once you have both fractions with a common denominator, you can add the numerators.

.

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case and are both primes and don't share any factors other than .

We can multiply them to get as the final answer.

Another approach is to list out all the factors of each number and see which factor is in both sets first.

Notice appears in both sets before any other number therefore, this is the least common denominator.

When finding the least common denominator, the quickest way is to multiply the numbers out.

In this case and share a factor other than which is . We can divide those numbers by to get and leftover.

Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

Another approach is to list out the factors of both number and find the factor that appears in both sets first.

We can see that appears in both sets before any other number thus, this is our answer.

When finding the least common denominator, the quickest way is to multiply the numbers out. In this case and share a factor other than which is . We can divide those numbers by to get and leftover. Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is .

Another approach is to list out the factors of each number. The factor that appears first in both set is the least common denominator.

We see that appears first in both sets and thus, is the least common denominator.

When finding the least common denominator, the quickest way is to multiply the expression out. In this case and don't share any factors other than . We can multiply this to get as the final answer.

Remember when foiling, you multiply the numbers/variables that first appear in each binomial, followed by multiplying the outer most numbers/variables, then multiplying the inner most numbers/variables and finally multiplying the last numbers/variables.

When finding the least common denominator, the quickest way is to multiply the expression out. In this case and share a factor other than which is . If you don't see that. just break down the quadratic equation to simple factors. Remember, we need to find two terms that are factors of the c term that add up to the b term.

The quadratic becomes . By factoring out , we get and . Just multiply the leftovers and the factored expression to get .

When finding the least common denominator, the quickest way is to multiply the numbers out. In the case of finding least common denominators among three or more numbers, it's critical there are no common factors between two of the denominators and of course all 3. This will ensure the answer will always be the least common denominator.

Say we just multiplied the numbers out. It's basically or . That number seems big but lets cut this in half and check divides evenly into , , and . Lets check . doesn't divide evenly into so is the answer.

So this goes back to the statement: "In the case of finding least common denominators among three or more numbers, it's critical there are no common factors between two of the denominators and of course all 3." If I factored a , I can reduce the and but not the . That is ok. Now the leftover values are , , and . They only share a factor of . So let's multiply the leftover values and the factored value to get

The least common denomiator (or least common multiple...same concept) is the least expression that both denominators can go into. I like to work step by step with each term. Let's start with the numerical coefficients.

The least common multiple of 12 and 60 is 60. You can figure this out by writing out multiples of 12 and 60 and seeing the first one they have in common.

Now let's move on to the a's.

There's an and an Therefore, their LCM is .

Do the same with the b's and the c's; and , respectively.

Now put those all together to get .

Since fractions cannot be subtracted, we will need to find the least common denominator. In order to find the least common denominator, we will need to multiply the and .

There is no number below this value where the factors of nine and sixteen will intersect.

The question only asked for the least common denominator, so we do not need to evaluate this.

The least common denominator is .

When there are two terms involved in a denominator, I like to look at each separately and combine at the end. For 2 and 7, the least common denominator is what thier product is, which is 14. Then, for and , is the LCD for that pair since it's the least common multiple of the two. Now, put your answers together to get: .

The least common denominator is found by finding what the least common multiple is.

The multiples of are

The multiples of are

The multiples of are

The lowest number that occurs in the three numbers listed is , which means you could add or subtract the fractions by converting them all to fractions with a denominator of .

To simplify the sum of the two fractions, we must find the common denominator.

Simplifying the denominator of the first fraction, we get

because the denominator is a difference of two squares, which follows the form

Now, we can rewrite the sum as

It is far easier to see the common denominator now:

To solve this equation, we must find the least common denominator to add the fractions.

Keep in mind that the denominator of the second term is the difference of squares, which can be rewritten as

This is the least common denominator.

Now, we multiply both sides of the equationby the LCD on top and bottom (this is essentially 1):

After canceling terms, we get

Now, we solve for x:

Let's first get all the known terms on one side of the equation:

We need to find a common denominator for all the fractions. One easy way to do this is to multiply each fraction by the denominator on the other two:

We can now add the numerators together:

And simplify:

The least common denominator would be all of the denominators multiplied together:

To find the Least Common Denominator, we multiply the denominator of both terms together: