I just wanted to know is there any book or resource or perhaps online resource which could help me to do faster problem solving problems like below(so that i could do it mentally an perhaps a lot faster even if i write)
There are 12 pipes that are connected to a tank. Some of them are fill pipes and the others are drain pipes. Each of the fill pipes can fill the tank in 8 hours and each of the drain pipes can drain the tank completely in 6 hours. If all the fill pipes and drain pipes are kept open, an empty tank gets filled in 24 hours. How many of the 12 pipes are fill pipes?
Explanatory Answer Let there be 'n' fill pipes attached to the tank. Therefore, there will be 12 - n drain pipes attached to the tank Each fill pipe fills the tank in 8 hours. Therefore, each of the fill pipes will fill 1/8th of the tank in an hour. Hence, n fill pipes will fill n/8 of the tank in an hour. Each drain pipe will drain the tank in 6 hours. Therefore, each of the drain pipes will drain 1/6th of the tank in an hour. Hence, (12 - n) drain pipes will drain (12-n)/6 of the tank in an hour. When all these 12 pipes are kept open, it takes 24 hours for an empty tank to overflow. Therefore, in an hour 1/24th of the tank gets filled. Hence, (n/8)-((12-n)/6) = 1/24 i.e. or 7n - 48 = 1 => 7n = 49 or n = 7.
If A and B work together, they will complete a job in 7.5 days. However, if A works alone and completes half the job and then B takes over and completes the remaining half alone, they will be able to complete the job in 20 days. How long will B alone take to do the job if A is more efficient than B?
Explanatory Answer Let 'a' be the number of days in which A can do the job alone. Therefore, working alone, A will complete of the job in a day. Similarly, let 'b' be the number of days in which B can do the job alone. Hence, B will complete of the job in a day. Working together, A and B will complete of the job in a day. The problem states that working together, A and B will complete the job in 7.5 or 15/2 days. i.e they will complete 2/15th of the job in a day. Therefore, ...... (1) From the question, we know that if A completes half the job working alone and B takes over and completes the next half, they will take 20 days. As A can complete the job working alone in 'a' days, he will complete half the job, working alone, in days. Similarly, B will complete the remaining half of the job in days. Therefore, => a + b = 40 or a = 40 - b ...... (2) From (1) and (2) we get, => 600 = 2b(40 - b) => 600 = 80b - 2b2 => b2 - 40b + 300 = 0 => (b - 30)(b - 10) = 0 => b = 30 or b = 10. If b = 30, then a = 40 - 30 = 10 or If b = 10, then a = 40 - 10 = 30. As A is more efficient than B, he will take lesser time to do the job alone. Hence A will take only 10 days and B will take 30 days.