To find the least common denominator, list out the multiples of both denominators until you find the smallest multiple that is shared by both.

4: 4, 8, 12, 16, 20, 24

5: 5, 10, 15, 20, 25

Because 20 is the first shared multiple of 4 and 5, it must be the least common denominator for these two fractions.

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. \(\displaystyle \frac{3}{4}\) and \(\displaystyle \frac{1}{3}\) are already simplified. However, \(\displaystyle \frac{2}{8}\) can be reduced to \(\displaystyle \frac{1}{4}\). 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.

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

In this case \(\displaystyle 2\) and \(\displaystyle 3\) are both primes and don't share any factors other than \(\displaystyle 1\).

We can multiply them to get \(\displaystyle 6\) 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.

\(\displaystyle 2: 2,4,6,8,10,12,...\)

\(\displaystyle 3:3,6,9,12,...\)

Notice \(\displaystyle 6\) 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 \(\displaystyle 2\) and \(\displaystyle 4\) share a factor other than \(\displaystyle 1\) which is \(\displaystyle 2\). We can divide those numbers by \(\displaystyle 2\) to get \(\displaystyle 1\) and \(\displaystyle 2\) leftover.

Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is \(\displaystyle 4\).

\(\displaystyle 2\cdot 4\rightarrow2(1\cdot 2)=4\)

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

\(\displaystyle 2:2,4,6,8,10,...\)

\(\displaystyle 4:4,8,12,16,...\)

We can see that \(\displaystyle 4\) 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 \(\displaystyle 4\) and \(\displaystyle 6\) share a factor other than \(\displaystyle 1\) which is \(\displaystyle 2\). We can divide those numbers by \(\displaystyle 2\) to get \(\displaystyle 2\) and \(\displaystyle 3\) leftover. Now, they don't share a common factor so basically multiply them out with the shared factor. Answer is \(\displaystyle 12\).

\(\displaystyle 4\cdot6\rightarrow2(2\cdot3)=12\)

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

\(\displaystyle 4:4,8,12,16,20,...\)

\(\displaystyle 6:6,12,18,24,...\)

We see that \(\displaystyle 12\) 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 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 \(\displaystyle 4\cdot5\cdot6\) or \(\displaystyle 120\). That number seems big but lets cut this in half and check \(\displaystyle 60.\) \(\displaystyle 60\) divides evenly into \(\displaystyle 4\), \(\displaystyle 5\), and \(\displaystyle 6\). Lets check \(\displaystyle 30\). \(\displaystyle 30\) doesn't divide evenly into \(\displaystyle 4\) so \(\displaystyle 60\) 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 \(\displaystyle 2\), I can reduce the \(\displaystyle 4\) and \(\displaystyle 6\) but not the \(\displaystyle 5\). That is ok. Now the leftover values are \(\displaystyle 2\), \(\displaystyle 3\), and \(\displaystyle 5\). They only share a factor of \(\displaystyle 1\). So let's multiply the leftover values and the factored value to get \(\displaystyle 60.\)

\(\displaystyle 4\cdot5\cdot6\rightarrow2(2\cdot3)\cdot5=60\)

Start by adding 10 to both sides.

\(\displaystyle \frac{5}{9}x=64\)

Multiply both side by 9 to get rid of the fraction.

\(\displaystyle 5x=576\)

Divide by 5

\(\displaystyle x=\frac{576}{5}\)

Since all the answer choices have mixed fractions, you will also need to reduce down to a mixed fraction

\(\displaystyle \frac{576}{5}=100\frac{76}{5}=115\frac{1}{5}\)

Add both sides by 9 to isolate the x on one side.

\(\displaystyle \frac{9}{5}x=18\)

Multiply both sides by 5.

\(\displaystyle 9x=90\)

Divide boths ides b 9.

\(\displaystyle x=10\)

First, add 10 to both sides so the term with "z" is isolated on one side.

\(\displaystyle \frac{2}{3}z=22\)

To get rid of the fraction, multiply both sides by 3.

\(\displaystyle 2z=66\)

Divide by 2.

\(\displaystyle z=33\)