Answer :
Let's look at each part of the question to see if the fractions are equivalent or need correction:
a. [tex]\(\frac{3}{4} \times \frac{6}{8} = \frac{16}{32}\)[/tex]
- When you multiply [tex]\(\frac{3}{4}\)[/tex] by [tex]\(\frac{6}{8}\)[/tex], you should get [tex]\(\frac{18}{32}\)[/tex]. So, this is not equivalent to [tex]\(\frac{16}{32}\)[/tex].
b. [tex]\(\frac{4}{6} = \frac{2}{3} = \frac{8}{18}\)[/tex]
- [tex]\(\frac{4}{6}\)[/tex] reduces to [tex]\(\frac{2}{3}\)[/tex]. However, [tex]\(\frac{8}{18}\)[/tex] simplifies to [tex]\(\frac{4}{9}\)[/tex], not [tex]\(\frac{2}{3}\)[/tex].
c. [tex]\(\frac{1}{2} \times \frac{5}{10} = \frac{5}{20}\)[/tex]
- [tex]\(\frac{1}{2}\)[/tex] multiplied by [tex]\(\frac{5}{10}\)[/tex] gives [tex]\(\frac{5}{20}\)[/tex], which simplifies to [tex]\(\frac{1}{4}\)[/tex]. This matches what was given, but it can be further reduced.
d. [tex]\(\frac{5}{10} = \frac{6}{6} \frac{30}{60}\)[/tex]
- [tex]\(\frac{5}{10}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex]. The expression [tex]\(\frac{30}{60}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex], but is incorrect with the multiplication by [tex]\(\frac{6}{6}\)[/tex].
h. [tex]\(\frac{2}{3} \times \frac{9}{9} = \frac{18}{27}\)[/tex]
- This multiplication is correct because [tex]\(\frac{9}{9}\)[/tex] is 1, so [tex]\(\frac{2}{3} \times 1 = \frac{2}{3}\)[/tex], and it does equal [tex]\(\frac{18}{27}\)[/tex], which simplifies back to [tex]\(\frac{2}{3}\)[/tex].
i. [tex]\(\frac{2}{4} \times \frac{6}{6} = \frac{12}{24}\)[/tex]
- [tex]\(\frac{2}{4}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex]. Multiplying by 1 (since [tex]\(\frac{6}{6} = 1\)[/tex]) should give [tex]\(\frac{1}{2}\)[/tex], which does not match [tex]\(\frac{12}{24}\)[/tex], although [tex]\(\frac{12}{24}\)[/tex] also simplifies to [tex]\(\frac{1}{2}\)[/tex].
1. [tex]\(\frac{1}{4} \times \frac{12}{12} = \frac{12}{18}\)[/tex]
- [tex]\(\frac{1}{4}\)[/tex] times [tex]\(\frac{12}{12}\)[/tex] should equal [tex]\(\frac{1}{4}\)[/tex], but the result given ([tex]\(\frac{12}{18}\)[/tex]) is incorrect. Reducing [tex]\(\frac{12}{18}\)[/tex] gives [tex]\(\frac{2}{3}\)[/tex].
k. [tex]\(\frac{6}{9} = \frac{3}{3} = \frac{16}{25}\)[/tex]
- [tex]\(\frac{6}{9}\)[/tex] simplifies to [tex]\(\frac{2}{3}\)[/tex]. [tex]\(\frac{3}{3}\)[/tex] simplifies to 1. [tex]\(\frac{16}{25}\)[/tex] doesn't match either.
l. [tex]\(\frac{2}{5} \times \frac{10}{10} = \frac{2}{50}\)[/tex]
- Multiplying [tex]\(\frac{2}{5}\)[/tex] by 1 (as [tex]\(\frac{10}{10} = 1\)[/tex]) should result in [tex]\(\frac{2}{5}\)[/tex], not [tex]\(\frac{2}{50}\)[/tex].
m. [tex]\(\frac{6}{8} \times \frac{12}{12} = \frac{72}{64}\)[/tex]
- [tex]\(\frac{6}{8}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex]. Multiplying by 1, (as [tex]\(\frac{12}{12} = 1\)[/tex]), gives [tex]\(\frac{3}{4}\)[/tex], not [tex]\(\frac{72}{64}\)[/tex].
These are the corrections and equivalences for each part as examined.
a. [tex]\(\frac{3}{4} \times \frac{6}{8} = \frac{16}{32}\)[/tex]
- When you multiply [tex]\(\frac{3}{4}\)[/tex] by [tex]\(\frac{6}{8}\)[/tex], you should get [tex]\(\frac{18}{32}\)[/tex]. So, this is not equivalent to [tex]\(\frac{16}{32}\)[/tex].
b. [tex]\(\frac{4}{6} = \frac{2}{3} = \frac{8}{18}\)[/tex]
- [tex]\(\frac{4}{6}\)[/tex] reduces to [tex]\(\frac{2}{3}\)[/tex]. However, [tex]\(\frac{8}{18}\)[/tex] simplifies to [tex]\(\frac{4}{9}\)[/tex], not [tex]\(\frac{2}{3}\)[/tex].
c. [tex]\(\frac{1}{2} \times \frac{5}{10} = \frac{5}{20}\)[/tex]
- [tex]\(\frac{1}{2}\)[/tex] multiplied by [tex]\(\frac{5}{10}\)[/tex] gives [tex]\(\frac{5}{20}\)[/tex], which simplifies to [tex]\(\frac{1}{4}\)[/tex]. This matches what was given, but it can be further reduced.
d. [tex]\(\frac{5}{10} = \frac{6}{6} \frac{30}{60}\)[/tex]
- [tex]\(\frac{5}{10}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex]. The expression [tex]\(\frac{30}{60}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex], but is incorrect with the multiplication by [tex]\(\frac{6}{6}\)[/tex].
h. [tex]\(\frac{2}{3} \times \frac{9}{9} = \frac{18}{27}\)[/tex]
- This multiplication is correct because [tex]\(\frac{9}{9}\)[/tex] is 1, so [tex]\(\frac{2}{3} \times 1 = \frac{2}{3}\)[/tex], and it does equal [tex]\(\frac{18}{27}\)[/tex], which simplifies back to [tex]\(\frac{2}{3}\)[/tex].
i. [tex]\(\frac{2}{4} \times \frac{6}{6} = \frac{12}{24}\)[/tex]
- [tex]\(\frac{2}{4}\)[/tex] simplifies to [tex]\(\frac{1}{2}\)[/tex]. Multiplying by 1 (since [tex]\(\frac{6}{6} = 1\)[/tex]) should give [tex]\(\frac{1}{2}\)[/tex], which does not match [tex]\(\frac{12}{24}\)[/tex], although [tex]\(\frac{12}{24}\)[/tex] also simplifies to [tex]\(\frac{1}{2}\)[/tex].
1. [tex]\(\frac{1}{4} \times \frac{12}{12} = \frac{12}{18}\)[/tex]
- [tex]\(\frac{1}{4}\)[/tex] times [tex]\(\frac{12}{12}\)[/tex] should equal [tex]\(\frac{1}{4}\)[/tex], but the result given ([tex]\(\frac{12}{18}\)[/tex]) is incorrect. Reducing [tex]\(\frac{12}{18}\)[/tex] gives [tex]\(\frac{2}{3}\)[/tex].
k. [tex]\(\frac{6}{9} = \frac{3}{3} = \frac{16}{25}\)[/tex]
- [tex]\(\frac{6}{9}\)[/tex] simplifies to [tex]\(\frac{2}{3}\)[/tex]. [tex]\(\frac{3}{3}\)[/tex] simplifies to 1. [tex]\(\frac{16}{25}\)[/tex] doesn't match either.
l. [tex]\(\frac{2}{5} \times \frac{10}{10} = \frac{2}{50}\)[/tex]
- Multiplying [tex]\(\frac{2}{5}\)[/tex] by 1 (as [tex]\(\frac{10}{10} = 1\)[/tex]) should result in [tex]\(\frac{2}{5}\)[/tex], not [tex]\(\frac{2}{50}\)[/tex].
m. [tex]\(\frac{6}{8} \times \frac{12}{12} = \frac{72}{64}\)[/tex]
- [tex]\(\frac{6}{8}\)[/tex] is [tex]\(\frac{3}{4}\)[/tex]. Multiplying by 1, (as [tex]\(\frac{12}{12} = 1\)[/tex]), gives [tex]\(\frac{3}{4}\)[/tex], not [tex]\(\frac{72}{64}\)[/tex].
These are the corrections and equivalences for each part as examined.
