Pool Filling Rate with Two Hoses and an Open Drain
You have an empty pool in your backyard. Hose A fills it in 0.1 hours on its own. Hose B fills it in 0.2 hours on its own. There is also a drain that empties a full pool in 0.15 hours.
If you run both hoses simultaneously with the drain open the whole time, how long does it take to fill the pool from empty?
Hints
- Convert times to rates: if a hose fills the pool in $T$ hours, its rate is /T$ pools per hour. Rates from independent sources add directly.
- The net fill rate is total inflow minus outflow: $r_A + r_B - r_{\text{drain}}$.
- Once you have the net rate $r_{\text{net}}$, the fill time is simply
/r_{\text{net}}$.Worked Solution
How to Think About It: Whenever you see a "how long until..." problem involving rates working together, the key move is to switch from times to rates. Times do not add -- rates do. Hose A fills the pool in 0.1 hours, which means it fills
/0.1 = 10$ pools per hour. Once you have rates, you just add the inflows and subtract the outflows to get the net rate, then invert to get the fill time.Quick Estimate: Hose A is twice as fast as Hose B, so Hose A dominates. Without the drain, the two hoses together fill about
0 + 5 = 15$ pools per hour, meaning the fill time is/15 \approx 0.067$ hours. The drain removes about 6.7 pools per hour, so the net rate is roughly5 - 6.7 = 8.3$ pools per hour, giving a fill time around/8.3 \approx 0.12$ hours. The exact answer should be near 0.12.Formal Solution:
Convert each device to a rate (pools per hour):
$r_A = \frac{1}{0.1} = 10, \quad r_B = \frac{1}{0.2} = 5, \quad r_{\text{drain}} = \frac{1}{0.15} = \frac{20}{3}$
Net filling rate with both hoses and drain active:
$r_{\text{net}} = r_A + r_B - r_{\text{drain}} = 10 + 5 - \frac{20}{3} = 15 - \frac{20}{3} = \frac{45 - 20}{3} = \frac{25}{3} \text{ pools/hour}$
Time to fill one pool:
$t = \frac{1}{r_{\text{net}}} = \frac{3}{25} = 0.12 \text{ hours}$
Answer: The pool fills in $\dfrac{3}{25} = 0.12$ hours.
Intuition
The rate-reciprocal trick is the cleanest tool for any problem involving multiple independent processes working toward a shared goal. The reason it works is that rate is the derivative of progress with respect to time, and derivatives of independent processes add linearly. Times, by contrast, are reciprocals of rates, and reciprocals do not add.
This pattern shows up in quant contexts whenever you have parallel channels: multiple market makers quoting on the same asset (their combined liquidity adds), or multiple risk factors contributing to portfolio variance (though covariance complicates the sum). The key discipline is always to ask: what is the natural unit for this quantity to add? For filling problems it is pools-per-hour. For option Greeks it is the individual Greek of each leg. For portfolio volatility it is variance (not standard deviation). Forcing a problem into the additive representation is often the entire insight.