There are four pipes connected to a tank - A, B, C and D. A & D are inlet pipes and B & C are outlet pipes. When all four pipes are opened together, then the tank will be filled in 40 minutes. When B & D are opened together, then the tank will be filled in 60 minutes. If D is twice as efficient than C and A is 25% more efficient than C, then find in how much time the tank will be filled when A & C are opened together?

Options

A. A)120 minutes
B. B)100 minutes
C. C)90 minutes
D. D)70 minutes

Answer

To solve the problem step by step, we will first define the efficiencies of the pipes and then use the information given to find the time taken to fill the tank when pipes A and C are opened together.

### Step 1: Define the efficiencies of the pipes
Let the efficiency of pipe C be \( x \) units (liters per minute).
Then, the efficiency of pipe D, which is twice as efficient as C, will be \( 2x \).
The efficiency of pipe A, which is 25% more efficient than C, will be \( x + 0.25x = 1.25x \).
The efficiency of pipe B is unknown and will be denoted as \( y \).

### Step 2: Calculate the total efficiency when all pipes are opened
When all four pipes (A, B, C, and D) are opened together, the tank is filled in 40 minutes.
The total capacity of the tank can be taken as 120 liters (as derived from the LCM of 40 and 60).
Thus, the combined efficiency of A, B, C, and D is:
\[ \frac{120 \text{ liters}}{40 \text{ minutes}} = 3 \text{ units} \]
So, we have:
\[ 1.25x + y + x + 2x = 3 \]
This simplifies to:
\[ 4.25x + y = 3 \quad \text{(Equation 1)} \]

### Step 3: Calculate the total efficiency when B and D are opened
When pipes B and D are opened together, the tank is filled in 60 minutes.
Thus, the combined efficiency of B and D is:
\[ \frac{120 \text{ liters}}{60 \text{ minutes}} = 2 \text{ units} \]
So, we have:
\[ y + 2x = 2 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
We now have two equations:
1. \( 4.25x + y = 3 \)
2. \( y + 2x = 2 \)

From Equation 2, we can express \( y \) in terms of \( x \):
\[ y = 2 - 2x \]

Substituting this expression for \( y \) into Equation 1:
\[ 4.25x + (2 - 2x) = 3 \]
This simplifies to:
\[ 4.25x - 2x + 2 = 3 \]
\[ 2.25x + 2 = 3 \]
\[ 2.25x = 1 \]
\[ x = \frac{1}{2.25} = \frac{4}{9} \]

Now substituting \( x \) back into the equation for \( y \):
\[ y = 2 - 2\left(\frac{4}{9}\right) = 2 - \frac{8}{9} = \frac{10}{9} \]

### Step 5: Find the efficiency of A and C
Now we can find the efficiencies of A and C:
- Efficiency of C: \( x = \frac{4}{9} \)
- Efficiency of A: \( 1.25x = 1.25 \times \frac{4}{9} = \frac{5}{9} \)

### Step 6: Calculate the combined efficiency of A and C
The combined efficiency of A and C is:
\[ \text{Efficiency of A} + \text{Efficiency of C} = \frac{5}{9} + \frac{4}{9} = \frac{9}{9} = 1 \text{ unit} \]

### Step 7: Calculate the time taken to fill the tank with A and C
The time taken to fill the tank when A and C are opened together can be calculated as:
\[ \text{Time} = \frac{\text{Work}}{\text{Efficiency}} = \frac{120 \text{ liters}}{1 \text{ unit}} = 120 \text{ minutes} \]

### Final Answer
The tank will be filled in **120 minutes** when pipes A and C are opened together.