At a dairy​ plant, a milk tank can be filled in 4 hours​ (using the​ in-valve). Using the​ out-valve, the tank can be emptied in 8 hours. If both valves are open and milk is being pumped into the​ tank, how long will it take to fill the​ tank?

Let’s assume that the milk flows in at 10 kg per min. As it takes twice as long to empty it is flowing out at 5 kg per min.

If both valves are open, then the net (or total) increase = rate of inflow - rate of outflow, or 5 kg per min, so it would take 8 hrs to fill.

To find out how long it will take to fill the tank with both valves open, you need to use the concept of rates. The rate of filling with both valves open is the sum of the rates of the in-valve and the out-valve. The rate of the in-valve is 1 tank per 4 hours, and the rate of the out-valve is 1 tank per 8 hours. So, the combined rate of both valves open is ( \frac{1}{4} - \frac{1}{8} ) tanks per hour. Simplifying this expression, you get ( \frac{1}{4} - \frac{1}{8} = \frac{2}{8} - \frac{1}{8} = \frac{1}{8} ) tanks per hour. Therefore, it will take 8 hours to fill the tank with both valves open.