Answer :
To determine which rational number does not lie between [tex]\frac{3}{5}[/tex] and [tex]\frac{2}{3}[/tex], we need to convert all the fractions to the same denominator for easy comparison.
Convert [tex]\frac{3}{5}[/tex] and [tex]\frac{2}{3}[/tex] to have a common denominator:
The least common multiple of 5 and 3 is 15.
Convert [tex]\frac{3}{5}[/tex] to [tex]\frac{9}{15}[/tex] by multiplying both the numerator and denominator by 3.
Convert [tex]\frac{2}{3}[/tex] to [tex]\frac{10}{15}[/tex] by multiplying both the numerator and denominator by 5.
Therefore, [tex]\frac{3}{5} = \frac{9}{15}[/tex] and [tex]\frac{2}{3} = \frac{10}{15}[/tex].
Convert the given options to have the common denominator of 15 (for easy comparison):
[tex]\frac{50}{75}[/tex] simplifies to [tex]\frac{2}{3}[/tex] by dividing both numerator and denominator by 25. As we already know, [tex]\frac{2}{3} = \frac{10}{15}[/tex].
[tex]\frac{46}{75}[/tex]: To simplify, let's multiply numerator and denominator by 2 to get [tex]\frac{92}{150}[/tex]. This is approximately [tex]\frac{46}{75} \approx \frac{9.2}{15}[/tex].
[tex]\frac{47}{75}[/tex]: Similarly multiply numerator and denominator by 2, giving us [tex]\frac{94}{150}[/tex], which approximately becomes [tex]\frac{47}{75} \approx \frac{9.4}{15}[/tex].
[tex]\frac{49}{75}[/tex]: Again, multiply numerator and denominator by 2, giving us [tex]\frac{98}{150}[/tex], which approximately becomes [tex]\frac{49}{75} \approx \frac{9.8}{15}[/tex].
Determine which fractions lie between [tex]\frac{9}{15}[/tex] and [tex]\frac{10}{15}[/tex]:
[tex]\frac{92}{150} \approx \frac{9.2}{15}[/tex] is between [tex]\frac{9}{15}[/tex] and [tex]\frac{10}{15}[/tex].
[tex]\frac{94}{150} \approx \frac{9.4}{15}[/tex] is between [tex]\frac{9}{15}[/tex] and [tex]\frac{10}{15}[/tex].
[tex]\frac{98}{150} \approx \frac{9.8}{15}[/tex] is between [tex]\frac{9}{15}[/tex] and [tex]\frac{10}{15}[/tex].
Since [tex]\frac{50}{75}[/tex] is exactly equivalent to [tex]\frac{10}{15}[/tex], which lies comfortably at the end of our range, we now see all but one reside properly between the provided limits.
None of the tested options lie outside this range, but [tex]\frac{50}{75} = \frac{2}{3}[/tex] really doesn't sit between but instead sits directly upon one range limit (the highest).
Conclusion: The rational number not lying technically between [tex]\frac{3}{5}[/tex] and [tex]\frac{2}{3}[/tex] is: 'a. \frac{50}{75}'.
