Three taps A,b and C can till a cistern in 10 min. , 12 min, and 15 min respectively.They are all turned on at once , but after 1 1/2 min. B and C are turned off.How many minutes longer will A take then to fill the cistern ?

Options

A. 6 1/4 min.
B. 71/4 min.
C. 6 3/4 min.
D. 8 3/4 min.

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the filling rates of taps A, B, and C.
- Tap A fills the cistern in 10 minutes, so its rate is \( \frac{1}{10} \) of the cistern per minute.
- Tap B fills the cistern in 12 minutes, so its rate is \( \frac{1}{12} \) of the cistern per minute.
- Tap C fills the cistern in 15 minutes, so its rate is \( \frac{1}{15} \) of the cistern per minute.

### Step 2: Calculate the combined rate of all three taps.
The combined rate when all taps are open is:
\[ \text{Rate of A + Rate of B + Rate of C} = \frac{1}{10} + \frac{1}{12} + \frac{1}{15} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 10, 12, and 15 is 60.

Converting each rate:
- \( \frac{1}{10} = \frac{6}{60} \)
- \( \frac{1}{12} = \frac{5}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)

Now, adding these rates:
\[ \frac{6}{60} + \frac{5}{60} + \frac{4}{60} = \frac{15}{60} = \frac{1}{4} \]
Thus, the combined rate of A, B, and C is \( \frac{1}{4} \) of the cistern per minute.

### Step 3: Calculate how much of the cistern is filled in 1.5 minutes.
In 1.5 minutes (or \( \frac{3}{2} \) minutes), the amount filled by all three taps is:
\[ \text{Amount filled} = \text{Rate} \times \text{Time} = \frac{1}{4} \times \frac{3}{2} = \frac{3}{8} \]

### Step 4: Determine the remaining capacity of the cistern.
The total capacity of the cistern is 1 (full cistern). Therefore, the remaining capacity after 1.5 minutes is:
\[ 1 - \frac{3}{8} = \frac{5}{8} \]

### Step 5: Calculate how long tap A will take to fill the remaining capacity.
Now, only tap A is open, and it fills at a rate of \( \frac{1}{10} \) of the cistern per minute. To find the time \( t \) needed to fill the remaining \( \frac{5}{8} \) of the cistern, we set up the equation:
\[ \frac{1}{10} \times t = \frac{5}{8} \]
Solving for \( t \):
\[ t = \frac{5}{8} \div \frac{1}{10} = \frac{5}{8} \times 10 = \frac{50}{8} = 6.25 \text{ minutes} \]

### Step 6: Calculate the total time taken by tap A.
The total time taken by tap A to fill the cistern is the initial 1.5 minutes plus the additional time calculated:
\[ \text{Total time} = 1.5 + 6.25 = 7.75 \text{ minutes} \]

### Step 7: Determine how many minutes longer tap A takes compared to the time it would take alone.
If tap A were to fill the cistern alone, it would take 10 minutes. Therefore, the difference in time is:
\[ 10 - 7.75 = 2.25 \text{ minutes} \]

### Final Answer:
Tap A takes 2.25 minutes longer to fill the cistern.

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