4. Pjotr has a bathtub with a plug at the bottom that he can fill with two identical taps. The water flows from both taps at the same constant speed, and the bathtub drains (if the plug is not in) at a constant speed.

On Monday, Pjotr lets the bathtub fill to the brim by turning on one tap fully, after which he pulls the plug and waits until the bathtub is empty again. Only when the bathtub is empty does he turn the tap off. On Tuesday, he lets the bathtub fill to the brim by turning on both taps fully, after which he pulls the plug and waits until the bathtub is empty again. Only when the bathtub is empty does he turn the taps off. On both days, Pjotr was busy for exactly 45 minutes from turning on the tap/taps to turning them off again.

How many minutes does it take for a full bathtub to drain completely if both taps are off?
B4. 10 Given that water flows from a tap at a rate of $v$ liters per minute and the bathtub drains at a rate of $w$ liters per minute when the taps are off. Assume the bathtub can hold $b$ liters of water. We want to know how long it takes for a full bathtub to drain, which is $\frac{b}{w}$.

On Monday, the bathtub first fills at a rate of $v$ liters per minute, and then drains at a rate of $w-v$ liters per minute. Together, this takes $\frac{b}{v}+\frac{b}{w-v}$ minutes. On Tuesday, the bathtub first fills at a rate of $2v$ liters per minute, and then drains at a rate of $w-2v$ liters per minute. This takes the same amount of time as on Monday. Together, this gives the equation

$$
\frac{b}{v}+\frac{b}{w-v}=\frac{b}{2v}+\frac{b}{w-2v}
$$

Making the fractions have a common denominator, cross-multiplying, and simplifying yields $w=3v$. (You can easily verify this by substituting it into the equation above.) In other words, the bathtub drains three times as fast as it fills with one tap.

Finally, we use the given information that Pjotr spent 45 minutes on both days. Substituting $v=\frac{1}{3}w$ into the formula for Monday, we get

$$
45=\frac{b}{v}+\frac{b}{w-v}=\frac{b}{\frac{1}{3}w}+\frac{b}{\frac{2}{3}w}=\frac{9}{2} \cdot \frac{b}{w}.
$$

From this, it follows that $\frac{b}{w}=10$.