There are two inlets A and B connected to a tank . A and B can fill the tank in 16h and 10 h ,respectively. If both the pipes are opened alternately for 1 h, starting from A, then how much time will the tank take to be filled?

Options

A. `13(1)/(4) h`
B. `11(6)/(8) h`
C. `12(2)/(5)`
D. `12 (1)/(4) h`

Answer

To solve the problem step by step, we will first determine the rates at which pipes A and B fill the tank, then calculate how much of the tank is filled when both pipes are opened alternately for 1 hour each.

### Step 1: Determine the filling rates of pipes A and B
- Pipe A can fill the tank in 16 hours.
- Pipe B can fill the tank in 10 hours.

To find the filling rates:
- Rate of pipe A = 1 tank / 16 hours = 1/16 tank per hour
- Rate of pipe B = 1 tank / 10 hours = 1/10 tank per hour

### Step 2: Find a common capacity for the tank
To make calculations easier, we can assume the capacity of the tank to be the least common multiple (LCM) of 16 and 10.
- LCM of 16 and 10 = 80 units

### Step 3: Calculate the amount filled by each pipe in one hour
- Amount filled by pipe A in 1 hour = 80 units / 16 hours = 5 units
- Amount filled by pipe B in 1 hour = 80 units / 10 hours = 8 units

### Step 4: Calculate the total amount filled in 2 hours (1 hour by A and 1 hour by B)
- In the first hour, pipe A fills 5 units.
- In the second hour, pipe B fills 8 units.
- Total amount filled in 2 hours = 5 units + 8 units = 13 units

### Step 5: Determine how many complete cycles of 2 hours fit into the total capacity of the tank
- Total capacity of the tank = 80 units
- Amount filled in 2 hours = 13 units
- Number of complete cycles in 80 units = 80 units / 13 units = 6 complete cycles (which fills 78 units)

### Step 6: Calculate the total time for the complete cycles
- Each cycle takes 2 hours, so for 6 cycles:
- Total time for 6 cycles = 6 cycles * 2 hours/cycle = 12 hours

### Step 7: Calculate the remaining amount to fill after 78 units
- Amount left to fill = 80 units - 78 units = 2 units

### Step 8: Determine how much more time is needed to fill the remaining 2 units
- In the next hour, pipe A will fill 5 units. However, we only need 2 units.
- Time taken by pipe A to fill 2 units = (2 units / 5 units/hour) = 2/5 hours

### Step 9: Calculate the total time taken to fill the tank
- Total time = Time for complete cycles + Time to fill remaining units
- Total time = 12 hours + 2/5 hours = 12 2/5 hours

### Final Answer:
The tank will take **12 2/5 hours** to be filled.

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