Pipes A and B can fill a tank in 18 minutes and 30 minutes, respectively. Pipe C, attached to the tank, can drain off 125, litres of water per minute. If all the pipes are opened together, the tank is filled in 45 minutes. The capacity of the tank, in litres, is :

Options

A. 1500
B. 1375
C. 1875
D. 2000

Answer

To solve the problem step by step, we will first determine the filling rates of pipes A and B, then calculate the combined effect of all three pipes (A, B, and C) when they are opened together.

### Step 1: Determine the filling rates of pipes A and B.
- Pipe A can fill the tank in 18 minutes.
- Therefore, the rate of Pipe A = \( \frac{1 \text{ tank}}{18 \text{ minutes}} = \frac{1}{18} \text{ tanks per minute} \).

- Pipe B can fill the tank in 30 minutes.
- Therefore, the rate of Pipe B = \( \frac{1 \text{ tank}}{30 \text{ minutes}} = \frac{1}{30} \text{ tanks per minute} \).

### Step 2: Calculate the combined filling rate of pipes A and B.
- Combined rate of A and B = Rate of A + Rate of B
- \( = \frac{1}{18} + \frac{1}{30} \)

To add these fractions, we need a common denominator. The least common multiple (LCM) of 18 and 30 is 90.

- Convert the rates:
- \( \frac{1}{18} = \frac{5}{90} \)
- \( \frac{1}{30} = \frac{3}{90} \)

- Now add them:
- \( \frac{5}{90} + \frac{3}{90} = \frac{8}{90} = \frac{4}{45} \text{ tanks per minute} \)

### Step 3: Determine the draining rate of pipe C.
- Pipe C drains 125 liters per minute.
- To find the draining rate in terms of tanks, we need to know the capacity of the tank. Let’s denote the capacity of the tank as \( V \) liters.

- The draining rate of pipe C in terms of tanks per minute = \( \frac{125}{V} \text{ tanks per minute} \).

### Step 4: Set up the equation for the combined effect of all pipes.
- When all pipes are opened together, they fill the tank in 45 minutes.
- Therefore, the combined rate of A, B, and C is:
- \( \frac{V}{45} \text{ tanks per minute} \)

### Step 5: Write the equation.
- The equation combining all three pipes is:
- \( \frac{4}{45} - \frac{125}{V} = \frac{V}{45} \)

### Step 6: Solve for V.
- Multiply through by 45V to eliminate the denominators:
- \( 4V - 125 \cdot 45 = V^2 \)

- Rearranging gives:
- \( V^2 - 4V + 5625 = 0 \)

### Step 7: Use the quadratic formula to find V.
- The quadratic formula is given by:
- \( V = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
- Here, \( a = 1, b = -4, c = 5625 \).

- Calculate the discriminant:
- \( b^2 - 4ac = (-4)^2 - 4 \cdot 1 \cdot 5625 = 16 - 22500 = -22484 \)

Since the discriminant is negative, we made an error in our calculations. Let's check the equation again.

### Correct Step 6:
- The correct equation should be:
- \( \frac{4}{45} - \frac{125}{V} = \frac{1}{45} \)

### Step 7: Solve for V again.
- Rearranging gives:
- \( \frac{4}{45} - \frac{1}{45} = \frac{125}{V} \)
- \( \frac{3}{45} = \frac{125}{V} \)
- \( V = \frac{125 \cdot 45}{3} = 1875 \text{ liters} \)

### Final Answer:
The capacity of the tank is **1875 liters**.

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