# Algebra II : Least Common Denominator in Fractions

## Example Questions

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### Example Question #1 : Least Common Denominator In Fractions

Simplify

Possible Answers:

Correct answer:

Explanation:

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:

### Example Question #2 : Least Common Denominator In Fractions

Solve the following equation to find

Possible Answers:

Correct answer:

Explanation:

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:

### Example Question #3 : Fractions

Find the least common denominator for the following fractions:

Possible Answers:

Correct answer:

Explanation:

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

### Example Question #6 : Linear Equations With Fractions

What is the least common denominator between the following fractions: .

Possible Answers:

Correct answer:

Explanation:

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.

### Example Question #4 : Fractions

What's the least common denominator between  and ?

Possible Answers:

Correct answer:

Explanation:

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.

### Example Question #8 : Linear Equations With Fractions

What's the least common denominator of  and ?

Possible Answers:

Correct answer:

Explanation:

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.

### Example Question #5 : Fractions

What's the least common denominator of  and ?

Possible Answers:

Correct answer:

Explanation:

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.

### Example Question #3 : Least Common Denominator In Fractions

What's the least common denominator of  and ?

Possible Answers:

Correct answer:

Explanation:

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.

### Example Question #4 : Least Common Denominator In Fractions

What's the least common denominator of  and ?

Possible Answers:

Correct answer:

Explanation:

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 .

### Example Question #5 : Least Common Denominator In Fractions

What's the least common denominator among , and ?

Possible Answers:

Correct answer:

Explanation:

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

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