Step 1: Assume present ages in terms of x

Since the present ages are in the ratio $4: 5: 6$, let the present ages of the three persons be $4x$, $5x$, and $6x$ respectively.

$\text{Present Ages} = 4x, 5x, 6x$

Step 2: Express ages 8 years ago

Eight years ago, the ages of the three persons were $(4x - 8)$, $(5x - 8)$, and $(6x - 8)$.

$\text{Ages} = (4x - 8), (5x - 8), (6x - 8)$

Step 3: Set up the equation from the given sum

The sum of their ages 8 years ago was given as 51. Adding the expressions and equating to 51: $(4x - 8) + (5x - 8) + (6x - 8) = 51$.

$15x - 24 = 51$

Step 4: Solve for x

Adding 24 to both sides: $15x = 51 + 24 = 75$. Dividing by 15 gives $x = 5$.

$x = \frac{75}{15} = 5$

Step 5: Find the present age of the eldest person

The eldest person has a ratio of 6, so their present age is $6x = 6 \times 5 = 30$ years.

$\text{Eldest Age} = 6 \times 5 = 30 \text{ years}$