correct answer is: b) \large\frac{4}{225}

Explanation

To find the greatest fraction, we need to HCF the given fraction \large\frac{4}{15},\frac{8}{45},\frac{16}{75}
The formula we use to get the HCF of two or more fractions is:
\large\frac{The\ HCF\ of\ Numerator}{The\ LCM\ of\ Denominator}

Here, given numerators are 4,8 and 16.
\therefore The HCF of 4,8 and 16 is =4
And given denominators are 15,45 and 75.
\therefore The LCM of 15,45 and 75 is =5\times3\times5\times3=225
So that, the HCF of the fraction \large\frac{4}{15},\frac{8}{45},\frac{16}{75} is =\large\frac{4}{225}
\therefore\large\frac{4}{225} is the greatest fraction that is exactly divisible by the giving integers \large\frac{4}{15},\frac{8}{45}, and \large\frac{16}{75} as quotients.
Ans: \large\frac{4}{225} is the greatest fraction.

Another method to articulate this particular math query is available:

1. Finding the maximum fraction quotient for evenly dividing by 4/15, 8/45 and 16/75 into whole numbers.
2. What is the greatest fraction divisible by 4/15, 8/45 and 16/75 to ensure integer results?
3. Determining the largest common denominator for 4/15, 8/45 and 16/75 to ensure integer outcomes.