A pipe can fill a tank in 10h , while an another pipe can empty it in 6h . Find the time taken to empty the tank , when both the pipes are opened up simultaneously .

Options

A. 11h
B. 15h
C. 18h
D. 16h

Answer

To solve the problem, we need to determine the combined effect of the filling pipe and the emptying pipe when they are opened simultaneously. Here’s a step-by-step solution:

### Step 1: Determine the filling rate of the first pipe
The first pipe can fill the tank in 10 hours. Therefore, the rate at which it fills the tank is:
\[ \text{Filling rate} = \frac{1 \text{ tank}}{10 \text{ hours}} = \frac{1}{10} \text{ tanks per hour} \]

**Hint:** To find the rate of filling or emptying, divide 1 by the time taken to fill or empty the tank.

### Step 2: Determine the emptying rate of the second pipe
The second pipe can empty the tank in 6 hours. Therefore, the rate at which it empties the tank is:
\[ \text{Emptying rate} = \frac{1 \text{ tank}}{6 \text{ hours}} = \frac{1}{6} \text{ tanks per hour} \]

**Hint:** Similar to the filling rate, divide 1 by the time taken to empty the tank to find the emptying rate.

### Step 3: Calculate the net rate when both pipes are opened simultaneously
When both pipes are opened together, the net effect is the filling rate minus the emptying rate:
\[ \text{Net rate} = \text{Filling rate} - \text{Emptying rate} = \frac{1}{10} - \frac{1}{6} \]

To perform this subtraction, we need a common denominator. The least common multiple of 10 and 6 is 30.

Converting both fractions:
\[ \frac{1}{10} = \frac{3}{30}, \quad \frac{1}{6} = \frac{5}{30} \]
Now, subtract:
\[ \text{Net rate} = \frac{3}{30} - \frac{5}{30} = \frac{-2}{30} = -\frac{1}{15} \text{ tanks per hour} \]

**Hint:** When subtracting fractions, find a common denominator to make the calculation easier.

### Step 4: Interpret the net rate
The negative sign indicates that the tank is being emptied. The rate of \(-\frac{1}{15}\) tanks per hour means that the tank is emptied at a rate of \(\frac{1}{15}\) of the tank per hour.

### Step 5: Calculate the time taken to empty the tank
To find the time taken to empty the entire tank, we take the reciprocal of the net rate:
\[ \text{Time to empty the tank} = \frac{1 \text{ tank}}{\frac{1}{15} \text{ tanks per hour}} = 15 \text{ hours} \]

**Hint:** To find the time taken for a certain rate, take the reciprocal of the rate.

### Conclusion
Therefore, the time taken to empty the tank when both pipes are opened simultaneously is **15 hours**.

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