Certainly! Let's start by rewriting the fractions [tex]\(\frac{2}{15}\)[/tex] and [tex]\(\frac{3}{20}\)[/tex] so that they have a common denominator.
### Step 1: Find the Common Denominator
The first step is to find a common denominator, which is the least common multiple (LCM) of the two denominators (15 and 20):
- The prime factorization of 15 is [tex]\(3 \times 5\)[/tex].
- The prime factorization of 20 is [tex]\(2^2 \times 5\)[/tex].
The LCM of 15 and 20 is obtained by taking the highest power of each prime factor that appears in these factorizations:
- The highest power of 2 is [tex]\(2^2\)[/tex].
- The highest power of 3 is [tex]\(3\)[/tex].
- The highest power of 5 is [tex]\(5\)[/tex].
Therefore,
[tex]\[
\text{LCM}(15, 20) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60
\][/tex]
So, the common denominator is 60.
### Step 2: Rewriting and Converting Fractions
Next, we convert each fraction to have the common denominator of 60.
#### For [tex]\(\frac{2}{15}\)[/tex]:
- The common denominator is 60.
- To rewrite [tex]\(\frac{2}{15}\)[/tex] with a denominator of 60, we multiply both the numerator and the denominator by 4 (since [tex]\(60 \div 15 = 4\)[/tex]):
[tex]\[
\frac{2}{15} = \frac{2 \times 4}{15 \times 4} = \frac{8}{60}
\][/tex]
#### For [tex]\(\frac{3}{20}\)[/tex]:
- The common denominator is 60.
- To rewrite [tex]\(\frac{3}{20}\)[/tex] with a denominator of 60, we multiply both the numerator and the denominator by 3 (since [tex]\(60 \div 20 = 3\)[/tex]):
[tex]\[
\frac{3}{20} = \frac{3 \times 3}{20 \times 3} = \frac{9}{60}
\][/tex]
Now we have:
[tex]\[
\frac{2}{15} = \frac{8}{60} \quad \text{and} \quad \frac{3}{20} = \frac{9}{60}
\][/tex]
### Step 3: Compare the Fractions
To compare [tex]\(\frac{8}{60}\)[/tex] and [tex]\(\frac{9}{60}\)[/tex], we simply compare their numerators, as their denominators are now the same:
[tex]\[
8 < 9
\][/tex]
### Conclusion
Hence, we can say:
[tex]\[
\frac{2}{15} < \frac{3}{20}
\][/tex]
In summary:
[tex]\[
\frac{2}{15} = \frac{8}{60} \quad , \quad \frac{3}{20} = \frac{9}{60} \quad , \quad \frac{2}{15} < \frac{3}{20}
\][/tex]
So, the final order is:
[tex]\[
\boxed{\frac{2}{15} < \frac{3}{20}}
\][/tex]
