Solving the Problem

The problem informs us that the land will be subdivided into two lots: either $80$ or $100$ feet of lake frontage. The problem then informs us that $\frac{1}{9}$ of the lots are to have $80$ feet of lake frontage. From this piece of information, we can make a very important deduction:

Very important deduction: If $\frac{1}{9}$ of the lots have $80$ feet of lake frontage, and there are only two possible lots, then $1 - \frac{1}{9} = \frac{8}{9}$ of the lots must have $100$ feet of lake frontage.

This is a good start, but we still don't know how many lots have $80$ feet of lake frontage and how many have $100$ feet. If we say that the total number of lots is $a$, then we can set up the following equation:

$$x = \frac{1}{9}a \times 80 + \frac{8}{9}a \times 100$$

The next piece of information from the problem allows us to crack the puzzle and solve for the total number of lots, $a$. It says the "remaining 40 lots are to have $100$ feet of lake frontage each." Ahh, so $\frac{8}{9} \times total = 40$. Solving for the total, $a$, we calculate $a$ to be $45$ lots.

$$x = \frac{1}{9}45 \times 80 + \frac{8}{9}45 \times 100$$

$$x = (5)(80) + (40)(100)$$

$$x = 400 + 4000$$

Thus, the correct answer is D, ($4$,$400$).