Certainly! Let's check each pair of fractions to see if they are equivalent.
### Step-by-Step Solution:
#### a) [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex]
To determine if these fractions are equivalent, we can cross-multiply and compare the products.
For [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[
4 \times 30 = 120 \\
5 \times 24 = 120
\][/tex]
Since [tex]\(120 = 120\)[/tex], [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex] are equivalent.
#### b) [tex]\(\frac{7}{11}\)[/tex] and [tex]\(\frac{21}{32}\)[/tex]
For [tex]\(\frac{7}{11}\)[/tex] and [tex]\(\frac{21}{32}\)[/tex]:
[tex]\[
7 \times 32 = 224 \\
11 \times 21 = 231
\][/tex]
Since [tex]\(224 \neq 231\)[/tex], [tex]\(\frac{7}{11}\)[/tex] and [tex]\(\frac{21}{32}\)[/tex] are not equivalent.
#### c) [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]
For [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex]:
[tex]\[
12 \times 5 = 60 \\
35 \times 3 = 105
\][/tex]
Since [tex]\(60 \neq 105\)[/tex], [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] are not equivalent.
#### d) [tex]\(\frac{45}{81}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex]
For [tex]\(\frac{45}{81}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex]:
[tex]\[
45 \times 7 = 315 \\
81 \times 5 = 405
\][/tex]
Since [tex]\(315 \neq 405\)[/tex], [tex]\(\frac{45}{81}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex] are not equivalent.
### Summary:
- [tex]\(\frac{4}{5}\)[/tex] and [tex]\(\frac{24}{30}\)[/tex] are equivalent (True)
- [tex]\(\frac{7}{11}\)[/tex] and [tex]\(\frac{21}{32}\)[/tex] are not equivalent (False)
- [tex]\(\frac{12}{35}\)[/tex] and [tex]\(\frac{3}{5}\)[/tex] are not equivalent (False)
- [tex]\(\frac{45}{81}\)[/tex] and [tex]\(\frac{5}{7}\)[/tex] are not equivalent (False)
Hence, the results are [tex]\([True, False, False, False]\)[/tex].
