Sure! Let's use Ellie's method to compare the given fractions by converting the mixed numbers into improper fractions and then comparing them.
### Task 1
#### Example 1: [tex]\(3 \frac{1}{5}\)[/tex] and [tex]\(2 \frac{2}{5}\)[/tex]
1. First, convert the mixed numbers to improper fractions:
- For [tex]\(3 \frac{1}{5}\)[/tex], we multiply the whole number 3 by the denominator 5 and then add the numerator 1:
[tex]\[
3 \cdot 5 + 1 = 15 + 1 = 16
\][/tex]
So, [tex]\(3 \frac{1}{5}\)[/tex] is equivalent to [tex]\(\frac{16}{5}\)[/tex].
- For [tex]\(2 \frac{2}{5}\)[/tex], we multiply the whole number 2 by the denominator 5 and then add the numerator 2:
[tex]\[
2 \cdot 5 + 2 = 10 + 2 = 12
\][/tex]
So, [tex]\(2 \frac{2}{5}\)[/tex] is equivalent to [tex]\(\frac{12}{5}\)[/tex].
2. Compare the improper fractions [tex]\(\frac{16}{5}\)[/tex] and [tex]\(\frac{12}{5}\)[/tex]:
- Since both fractions have the same denominator (5), we can directly compare the numerators:
[tex]\[
16 > 12
\][/tex]
Therefore, [tex]\(3 \frac{1}{5}\)[/tex] is greater than [tex]\(2 \frac{2}{5}\)[/tex]:
[tex]\[
3 \frac{1}{5} \; \boxed{>} \; 2 \frac{2}{5}
\][/tex]
#### Example 3: [tex]\(6 \frac{6}{12}\)[/tex] and [tex]\(4 \frac{4}{12}\)[/tex]
1. First, convert the mixed numbers to improper fractions:
- For [tex]\(6 \frac{6}{12}\)[/tex], we multiply the whole number 6 by the denominator 12 and then add the numerator 6:
[tex]\[
6 \cdot 12 + 6 = 72 + 6 = 78
\][/tex]
So, [tex]\(6 \frac{6}{12}\)[/tex] is equivalent to [tex]\(\frac{78}{12}\)[/tex].
- For [tex]\(4 \frac{4}{12}\)[/tex], we multiply the whole number 4 by the denominator 12 and then add the numerator 4:
[tex]\[
4 \cdot 12 + 4 = 48 + 4 = 52
\][/tex]
So, [tex]\(4 \frac{4}{12}\)[/tex] is equivalent to [tex]\(\frac{52}{12}\)[/tex].
2. Compare the improper fractions [tex]\(\frac{78}{12}\)[/tex] and [tex]\(\frac{52}{12}\)[/tex]:
- Since both fractions have the same denominator (12), we can directly compare the numerators:
[tex]\[
78 > 52
\][/tex]
Therefore, [tex]\(6 \frac{6}{12}\)[/tex] is greater than [tex]\(4 \frac{4}{12}\)[/tex]:
[tex]\[
6 \frac{6}{12} \; \boxed{>} \; 4 \frac{4}{12}
\][/tex]
### Summary
1. [tex]\(3 \frac{1}{5} \; \boxed{>} \; 2 \frac{2}{5}\)[/tex]
3. [tex]\(6 \frac{6}{12} \; \boxed{>} \; 4 \frac{4}{12}\)[/tex]
By following these steps, we have successfully compared the mixed numbers and filled in the boxes with the correct comparison symbols.
