The correct answer is $\boxed{\frac{1}{3}}$.
BOD, or biochemical oxygen demand, is a measure of the amount of oxygen that is used by microorganisms to break down organic matter in water. The ultimate BOD is the theoretical maximum amount of oxygen that will be used by microorganisms to break down organic matter, while the 5-day BOD is the amount of oxygen that is used in 5 days.
The ratio of 5-day BOD to ultimate BOD is typically about $\frac{1}{3}$. This means that about one-third of the organic matter in water will be broken down by microorganisms in 5 days. The remaining two-thirds of the organic matter will be broken down more slowly over time.
The ratio of 5-day BOD to ultimate BOD can vary depending on the type of organic matter present in the water, the temperature of the water, and the presence of other factors that can affect the activity of microorganisms.
Option A is incorrect because the ratio of 5-day BOD to ultimate BOD is typically about $\frac{1}{3}$, not $\frac{1}{2}$.
Option B is incorrect because the ratio of 5-day BOD to ultimate BOD is typically about $\frac{1}{3}$, not $\frac{2}{3}$.
Option C is incorrect because the ratio of 5-day BOD to ultimate BOD is typically about $\frac{1}{3}$, not $\frac{3}{4}$.
Option D is incorrect because the ratio of 5-day BOD to ultimate BOD is typically about $\frac{1}{3}$, not $1$.