(a) (i) Apply addition modulo to positive integers 17 and 13 for modulo 30. (ii) Find subtraction modulo 8 for numbers 11 and 3. OR (b) Three pipes A, B and C can together fill a tank in 8 hours. After working at it together for 2 hours, B is closed and A and C fill the remaining part in 9 hours. Determine the time in which pipe B alone can fill the tank.
Answer: Pipe B alone can fill the tank in 24 hours.
Step-by-step solution
Step 1: Define rates and total work
Let the time taken by pipes A, B, and C to fill the tank individually be , , and hours, respectively. Their individual rates are , , and per hour. When working together, their combined rate is the sum of their individual rates, which is equal to since they fill the tank in 8 hours.
Step 2: Calculate work done in the first 2 hours
The three pipes work together for 2 hours. We can calculate the amount of work done in these 2 hours by multiplying their combined rate by the time they worked.
Step 3: Calculate remaining work
The total work to fill the tank is considered as 1 unit. After the first 2 hours, we subtract the work done from the total work to find the remaining portion of the tank that needs to be filled.
Step 4: Formulate equation for remaining work
After pipe B is closed, pipes A and C work together to fill the remaining of the tank in 9 hours. We can set up an equation where their combined rate multiplied by 9 hours equals the remaining work.
Step 5: Solve for the combined rate of A and C
From the previous step, we can solve for the combined rate of pipes A and C by dividing the remaining work by the time they took to complete it.
Step 6: Determine the rate of pipe B
We know the combined rate of A, B, and C, and the combined rate of A and C. By subtracting the latter from the former, we can find the individual rate of pipe B.
Step 7: Calculate the time taken by pipe B alone
After finding the rate of pipe B, we can determine the time it takes for pipe B alone to fill the tank by taking the reciprocal of its rate. The least common multiple of 8 and 12 is 24.