A tank is filled in 4 hours by three pipes A, B and C. The pipe C is twice as fast as B and pipe B is thrice as fast as A. How much time will pipe A alone take to fill the tank?

Options

A. 25 hours
B. 40 hours
C. 30 hours
D. 32 hours

Answer

To solve the problem, we need to determine how much time pipe A alone will take to fill the tank. Let's break down the solution step by step.

### Step 1: Define the efficiencies of pipes A, B, and C
Let the efficiency of pipe A be \( x \) (the part of the tank filled by A in one hour).
Since pipe B is thrice as fast as A, the efficiency of pipe B will be \( 3x \).
Since pipe C is twice as fast as B, the efficiency of pipe C will be \( 2 \times 3x = 6x \).

### Step 2: Write the total efficiency equation
The total efficiency of pipes A, B, and C working together is:
\[ \text{Efficiency of A} + \text{Efficiency of B} + \text{Efficiency of C} = x + 3x + 6x = 10x \]

### Step 3: Relate total efficiency to the time taken to fill the tank
We know that together they fill the tank in 4 hours. Therefore, the total work done (which we can consider as 1 tank) can be expressed as:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Time} \]
Substituting the known values:
\[ 1 = 10x \times 4 \]

### Step 4: Solve for x
From the equation above, we can solve for \( x \):
\[ 1 = 40x \implies x = \frac{1}{40} \]
This means pipe A fills \(\frac{1}{40}\) of the tank in one hour.

### Step 5: Calculate the time taken by pipe A to fill the tank alone
To find out how long it will take for pipe A to fill the entire tank alone, we take the reciprocal of its efficiency:
\[ \text{Time taken by A} = \frac{1}{\text{Efficiency of A}} = \frac{1}{\frac{1}{40}} = 40 \text{ hours} \]

Thus, pipe A alone will take **40 hours** to fill the tank.

### Final Answer
The time taken by pipe A alone to fill the tank is **40 hours**.

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