A pool can be filled through four different pipes. If the first and second pipes are operating, it takes 2 hours to fill the pool. With the second and third pipes it takes 3 hours, and with the third and fourth it takes 4 hours. How long does it take to fill the pool if the first and fourth pipes are operating?
Let $a, b, c, d$ denote the portion of the pool the pipes fill per hour (with $a$ corresponding to the first pipe). Because the first and second pipes fill $\frac{1}{2}$ of the pool per hour, we have that

$$
a+b=\frac{1}{2} \text {. }
$$

Analogously, we derive

$$
b+c=\frac{1}{3}
$$

and

$$
c+d=\frac{1}{4} .
$$

But now

$$
a+d=(a+b)+(c+d)-(b+c)=\frac{1}{2}+\frac{1}{4}-\frac{1}{3}=\frac{5}{12}
$$

so together the first and fourth pipes fill $\frac{5}{12}$ of the pool per hour. Hence they fill the entire pull in $\frac{12}{5}$ hours (which may also be written as 2.4 hours, or 2 hours and 24 minutes).