Introduction:
In this question, we are given four fractions: 4/5, 5/6, 5/3, and 5/2. We need to determine which fraction is the greatest among them.

Method:
To compare fractions, we can use a common denominator. The common denominator for all these fractions is 30. We will convert each fraction to an equivalent fraction with a denominator of 30 and then compare their numerators.

Conversion to a common denominator:
To convert 4/5 to an equivalent fraction with a denominator of 30, we can multiply both the numerator and denominator by 6: (4/5) x (6/6) = 24/30.

To convert 5/6 to an equivalent fraction with a denominator of 30, we can multiply both the numerator and denominator by 5: (5/6) x (5/5) = 25/30.

To convert 5/3 to an equivalent fraction with a denominator of 30, we can multiply both the numerator and denominator by 10: (5/3) x (10/10) = 50/30.

To convert 5/2 to an equivalent fraction with a denominator of 30, we can multiply both the numerator and denominator by 15: (5/2) x (15/15) = 75/30.

Comparison:
Now, we can compare the numerators of the converted fractions.

24/30, 25/30, 50/30, 75/30

Among these numerators, 75 is the greatest. Therefore, the fraction 5/2 (or 75/30) is the greatest among the given fractions.

Conclusion:
The fraction 5/2 (or 75/30) is the greatest among the given fractions: 4/5, 5/6, 5/3, and 5/2.