A cistern which could be filled in 9 hours takes one hour more to be filled owing to a leak in its bottom. If the cistern is full, in what time will the leak empty it?

Options

A. 19 hours
B. 1 hour
C. 90 hours
D. `(10)/9` hours

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the filling and leaking rates
The cistern can be filled in 9 hours. This means that in one hour, the filling rate is:
\[ \text{Filling rate} = \frac{1}{9} \text{ of the cistern per hour} \]

### Step 2: Account for the leak
Due to a leak, the cistern takes 10 hours to fill instead of 9 hours. This means that the effective filling rate when the leak is present is:
\[ \text{Effective filling rate} = \frac{1}{10} \text{ of the cistern per hour} \]

### Step 3: Determine the rate of the leak
The difference between the filling rate and the effective filling rate gives us the rate at which the leak empties the cistern. We can express this as:
\[ \text{Rate of leak} = \text{Filling rate} - \text{Effective filling rate} \]
Substituting the values we have:
\[ \text{Rate of leak} = \frac{1}{9} - \frac{1}{10} \]

### Step 4: Find a common denominator
To subtract these fractions, we need a common denominator. The least common multiple of 9 and 10 is 90. Therefore, we rewrite the fractions:
\[ \frac{1}{9} = \frac{10}{90} \quad \text{and} \quad \frac{1}{10} = \frac{9}{90} \]
Now we can subtract:
\[ \text{Rate of leak} = \frac{10}{90} - \frac{9}{90} = \frac{1}{90} \]

### Step 5: Calculate the time taken by the leak to empty the cistern
The rate of the leak tells us how much of the cistern it can empty in one hour. Since the leak empties \(\frac{1}{90}\) of the cistern in one hour, it will take:
\[ \text{Time to empty the cistern} = 90 \text{ hours} \]

### Conclusion
Thus, the leak will empty the full cistern in **90 hours**.

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