To solve the problem of finding the least common denominator (LCD) and expressing each fraction with that denominator, we will first outline the steps needed to achieve this.
### Step 1: Identify the Denominators
We'll start by identifying the denominators of the given fractions:
- The first fraction is [tex]\(\frac{1}{9}\)[/tex], so the denominator is [tex]\(9\)[/tex].
- The second fraction is [tex]\(\frac{2}{15}\)[/tex], so the denominator is [tex]\(15\)[/tex].
### Step 2: Find the Least Common Denominator (LCD)
To find the least common denominator, we'll determine the smallest number that is a multiple of both [tex]\(9\)[/tex] and [tex]\(15\)[/tex].
The multiples of [tex]\(9\)[/tex] are:
[tex]\[ 9, 18, 27, 36, 45, 54, 63, \ldots \][/tex]
The multiples of [tex]\(15\)[/tex] are:
[tex]\[ 15, 30, 45, 60, 75, 90, \ldots \][/tex]
The smallest common multiple of [tex]\(9\)[/tex] and [tex]\(15\)[/tex] is [tex]\(45\)[/tex]. Thus, the least common denominator (LCD) is [tex]\(45\)[/tex].
### Step 3: Convert Fractions to the LCD
Next, we'll express each fraction using the least common denominator of [tex]\(45\)[/tex].
#### For [tex]\(\frac{1}{9}\)[/tex]:
To convert [tex]\(\frac{1}{9}\)[/tex] to a fraction with a denominator of [tex]\(45\)[/tex]:
[tex]\[ \frac{1}{9} = \frac{1 \times 5}{9 \times 5} = \frac{5}{45} \][/tex]
#### For [tex]\(\frac{2}{15}\)[/tex]:
To convert [tex]\(\frac{2}{15}\)[/tex] to a fraction with a denominator of [tex]\(45\)[/tex]:
[tex]\[ \frac{2}{15} = \frac{2 \times 3}{15 \times 3} = \frac{6}{45} \][/tex]
### Step 4: Write the Final Fractions
Now that we've converted both fractions to have the common denominator, we can write them as:
- [tex]\(\frac{1}{9}\)[/tex] becomes [tex]\(\frac{5}{45}\)[/tex].
- [tex]\(\frac{2}{15}\)[/tex] becomes [tex]\(\frac{6}{45}\)[/tex].
### Summary
We found that the least common denominator for the fractions [tex]\(\frac{1}{9}\)[/tex] and [tex]\(\frac{2}{15}\)[/tex] is [tex]\(45\)[/tex]. Converted to this common denominator, the fractions are:
[tex]\[ \frac{1}{9} = \frac{5}{45} \][/tex]
[tex]\[ \frac{2}{15} = \frac{6}{45} \][/tex]
Thus, the final solution is:
[tex]\[ \frac{5}{45}, \frac{6}{45} \][/tex]
