Problem:
A water tank has three inlets, each of which can independently fill the tank in 20, 40, and 30 minutes respectively. We need to determine how much time it takes to fill the tank when all three inlets work together.

Solution:

To find the time taken to fill the tank when all three inlets work together, we can use the concept of rates. Let's calculate the rates at which each inlet fills the tank.

Rate of filling:
- Inlet 1: 1/20 tank per minute
- Inlet 2: 1/40 tank per minute
- Inlet 3: 1/30 tank per minute

Combining the rates:
When multiple inlets work together, their rates of filling are additive. So, the combined rate at which all three inlets fill the tank is the sum of their individual rates.

Rate of filling when all three inlets work together = 1/20 + 1/40 + 1/30

Finding the common denominator:
To add the fractions, we need to find a common denominator. In this case, the least common multiple (LCM) of 20, 40, and 30 is 120.

Converting the fractions:
To have a common denominator of 120, we need to multiply each fraction by a suitable form of 1.

1/20 = (1/20) * (6/6) = 6/120
1/40 = (1/40) * (3/3) = 3/120
1/30 = (1/30) * (4/4) = 4/120

Now, let's add the fractions:

1/20 + 1/40 + 1/30 = 6/120 + 3/120 + 4/120 = 13/120

Therefore, the combined rate of filling when all three inlets work together is 13/120 tank per minute.

Calculating the time taken:
To find the time taken to fill the tank, we can use the formula:

Time = Amount / Rate

Since we want to find the time, and the amount is 1 tank, we can substitute these values into the formula:

Time = 1 / (13/120)

To divide by a fraction, we can multiply by its reciprocal:

Time = 1 * (120/13) = 120/13

Therefore, it takes approximately 9.23 minutes to fill the tank when all three inlets work together.

Conclusion:
When all three inlets work together, it takes approximately 9.23 minutes to fill the tank.