6. (10 points) There are three water pools of the same capacity, named A, B, and C. A type-A water pump fills pool A alone, a type-B water pump fills pool B alone, and a type-A and a type-B water pump together fill pool C. It is known that it takes 4 hours more to fill pool B than to fill pool C, and it takes 5 hours more to fill pool A than to fill pool B. Then, to fill two-thirds of pool C, it will take $\qquad$ hours.
【Analysis】First, according to the problem, let the time required to fill the Yi water pool be $x$ hours, then the time required to fill the Jia water pool is $x+5$ hours, and the time required to fill the Bing water pool is $x-4$ hours. Then, based on work efficiency $=$ work volume $\div$ work time, we use 1 divided by $x+5$, $x$ to find out what fraction of the work one A-type water pump and one B-type water pump can complete per hour; then, according to: the work efficiency of one A-type water pump + the work efficiency of one B-type water pump $=1 \div$ the time required to fill the Bing water pool, we set up an equation, solve for the value of $x$, and then subtract 4 from $x$ to find out how long it takes to fill the Bing water pool. Then, multiply it by $\frac{2}{3}$ to find out how many hours it takes to fill two-thirds of the Bing water pool.

【Solution】Solution: Let the time required to fill the Yi water pool be $x$ hours, then the time required to fill the Jia water pool is $x+5$ hours, and the time required to fill the Bing water pool is $x-4$ hours,
$$
\begin{aligned}
\frac{1}{\mathrm{x}+5}+\frac{1}{\mathrm{x}} & =\frac{1}{\mathrm{x}-4} \\
\left(\frac{1}{\mathrm{x}+5}+\frac{1}{\mathrm{x}}\right) x(x-4)(x+5) & =\frac{1}{\mathrm{x}-4} \cdot x(x-4)(x+5) \\
x(x-4)+(x-4)(x+5) & =x(x+5) \\
2 x^{2}-3 x-20 & =x^{2}+5 x \\
x^{2}-8 x-20 & =0
\end{aligned}
$$

Solving, we get $x=10$ or $x=-2$ (discard)
$$
\begin{aligned}
& (10-4) \times \frac{2}{3} \\
= & 6 \times \frac{2}{3} \\
= & 4 \text { (hours) }
\end{aligned}
$$

Answer: It takes 4 hours to fill two-thirds of the Bing water pool.
The answer is: 4 .