Answer :
To solve the problem, we need to determine which given expressions have the same value as [tex]\(\frac{3}{5} \div \frac{6}{8}\)[/tex].
### Step-by-Step Solution
1. Simplify the Original Expression:
[tex]\[ \frac{3}{5} \div \frac{6}{8} = \frac{3}{5} \times \frac{8}{6} \][/tex]
Simplifying it by multiplying the fractions:
[tex]\[ \frac{3}{5} \times \frac{8}{6} = \frac{3 \times 8}{5 \times 6} = \frac{24}{30} \][/tex]
Now, simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[ \frac{24}{30} = \frac{4}{5} \text{ (by dividing both the numerator and the denominator by 6)} \][/tex]
2. Evaluate Each Choice:
(A) [tex]\(\frac{3}{5} \div \frac{8}{6}\)[/tex]:
[tex]\[ \frac{3}{5} \div \frac{8}{6} = \frac{3}{5} \times \frac{6}{8} = \frac{3 \times 6}{5 \times 8} = \frac{18}{40} = \frac{9}{20} \][/tex]
[tex]\(\frac{9}{20}\)[/tex] is not equal to [tex]\(\frac{4}{5}\)[/tex], so this is not correct.
(B) [tex]\(\frac{3}{5} \times \frac{8}{6}\)[/tex]:
[tex]\[ \frac{3}{5} \times \frac{8}{6} = \frac{3 \times 8}{5 \times 6} = \frac{24}{30} \][/tex]
[tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex]. This matches the original simplified expression. Therefore, (B) is correct.
(C) [tex]\(\frac{5}{3} \times \frac{6}{8}\)[/tex]:
[tex]\[ \frac{5}{3} \times \frac{6}{8} = \frac{5 \times 6}{3 \times 8} = \frac{30}{24} = \frac{5}{4} \][/tex]
[tex]\(\frac{5}{4}\)[/tex] is not equal to [tex]\(\frac{4}{5}\)[/tex], so this is not correct.
(D) [tex]\(\frac{24}{40} \div \frac{30}{40}\)[/tex]:
[tex]\[ \frac{24}{40} \div \frac{30}{40} = \frac{24}{40} \times \frac{40}{30} = \frac{24 \times 40}{40 \times 30} = \frac{24}{30} \][/tex]
[tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex]. This matches the original simplified expression. Therefore, (D) is correct.
(E) [tex]\(\frac{24}{40} \times \frac{40}{30}\)[/tex]:
[tex]\[ \frac{24}{40} \times \frac{40}{30} = \frac{24 \times 40}{40 \times 30} = \frac{24}{30} \][/tex]
[tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex]. This matches the original simplified expression. Therefore, (E) is correct.
### Conclusion
The expressions that have the same value as [tex]\(\frac{3}{5} \div \frac{6}{8}\)[/tex] are:
[tex]\[ \boxed{B, D, E} \][/tex]
### Step-by-Step Solution
1. Simplify the Original Expression:
[tex]\[ \frac{3}{5} \div \frac{6}{8} = \frac{3}{5} \times \frac{8}{6} \][/tex]
Simplifying it by multiplying the fractions:
[tex]\[ \frac{3}{5} \times \frac{8}{6} = \frac{3 \times 8}{5 \times 6} = \frac{24}{30} \][/tex]
Now, simplify [tex]\(\frac{24}{30}\)[/tex]:
[tex]\[ \frac{24}{30} = \frac{4}{5} \text{ (by dividing both the numerator and the denominator by 6)} \][/tex]
2. Evaluate Each Choice:
(A) [tex]\(\frac{3}{5} \div \frac{8}{6}\)[/tex]:
[tex]\[ \frac{3}{5} \div \frac{8}{6} = \frac{3}{5} \times \frac{6}{8} = \frac{3 \times 6}{5 \times 8} = \frac{18}{40} = \frac{9}{20} \][/tex]
[tex]\(\frac{9}{20}\)[/tex] is not equal to [tex]\(\frac{4}{5}\)[/tex], so this is not correct.
(B) [tex]\(\frac{3}{5} \times \frac{8}{6}\)[/tex]:
[tex]\[ \frac{3}{5} \times \frac{8}{6} = \frac{3 \times 8}{5 \times 6} = \frac{24}{30} \][/tex]
[tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex]. This matches the original simplified expression. Therefore, (B) is correct.
(C) [tex]\(\frac{5}{3} \times \frac{6}{8}\)[/tex]:
[tex]\[ \frac{5}{3} \times \frac{6}{8} = \frac{5 \times 6}{3 \times 8} = \frac{30}{24} = \frac{5}{4} \][/tex]
[tex]\(\frac{5}{4}\)[/tex] is not equal to [tex]\(\frac{4}{5}\)[/tex], so this is not correct.
(D) [tex]\(\frac{24}{40} \div \frac{30}{40}\)[/tex]:
[tex]\[ \frac{24}{40} \div \frac{30}{40} = \frac{24}{40} \times \frac{40}{30} = \frac{24 \times 40}{40 \times 30} = \frac{24}{30} \][/tex]
[tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex]. This matches the original simplified expression. Therefore, (D) is correct.
(E) [tex]\(\frac{24}{40} \times \frac{40}{30}\)[/tex]:
[tex]\[ \frac{24}{40} \times \frac{40}{30} = \frac{24 \times 40}{40 \times 30} = \frac{24}{30} \][/tex]
[tex]\(\frac{24}{30} = \frac{4}{5}\)[/tex]. This matches the original simplified expression. Therefore, (E) is correct.
### Conclusion
The expressions that have the same value as [tex]\(\frac{3}{5} \div \frac{6}{8}\)[/tex] are:
[tex]\[ \boxed{B, D, E} \][/tex]
