Understanding the Fish Tank Filling Problem

This problem asks us to determine the total time required to fill a fish tank when two taps, each with a different filling rate, are opened simultaneously. We need to find the combined time based on their individual filling times.

Calculating Individual Tap Rates

First, let's determine the rate at which each tap fills the tank.

• Fresh Water Tap Rate: This tap fills the entire tank in 40 minutes. So, in one minute, it fills $ \frac{1}{40} $ of the tank.
• Salt Water Tap Rate: This tap fills the same tank in 120 minutes. Therefore, in one minute, it fills $ \frac{1}{120} $ of the tank.

Calculating the Combined Filling Rate

When both taps are open, their rates add up. We need to sum the fractions representing their individual rates:

Combined Rate = Rate of Fresh Water Tap + Rate of Salt Water Tap

Combined Rate = $ \frac{1}{40} + \frac{1}{120} $

To add these fractions, we find a common denominator, which is 120:

$ \frac{1}{40} = \frac{1 \times 3}{40 \times 3} = \frac{3}{120} $

Now, add the fractions:

Combined Rate = $ \frac{3}{120} + \frac{1}{120} = \frac{3 + 1}{120} = \frac{4}{120} $

Simplify the combined rate:

Combined Rate = $ \frac{4}{120} = \frac{1}{30} $

This means that when both taps are open, they fill $ \frac{1}{30} $ of the tank every minute.

Calculating the Total Time to Fill the Tank

The time it takes to fill the tank together is the reciprocal of their combined rate. If they fill $ \frac{1}{30} $ of the tank per minute, the total time to fill 1 whole tank is:

Total Time = $ \frac{1}{\text{Combined Rate}} $

Total Time = $ \frac{1}{1/30} $

Total Time = 30 minutes

Conclusion

If both the fresh water tap and the salt water tap are open simultaneously, it will take 30 minutes to fill the fish tank.