Time And Work, Pipes And Cistern

Explanation
A and B can complete the work in 15 and 10 days respectively. In 2 days, they would have completed a fraction of the work, which can be calculated as (2/15 + 2/10). To find the remaining work, we subtract this fraction from 1 (1 - 2/15 - 2/10). Simplifying this gives us 11/30. Since A can complete 1/15th of the work in a day, it would take him 11/30 * 15 = 5.5 days to complete the remaining work. Therefore, the total time taken to complete the work is 2 + 5.5 = 7.5 days, which is approximately 8 days. However, since the options do not include 7.5, the closest option is 12 days.

Explanation
Since A takes twice as much time as B to finish the work, and A, B, and C can finish the work together in 2 days, it means that B's contribution to the work is half of A's contribution. Therefore, B can finish the work alone in half the time it takes A, which is 6 days.

Explanation
Since B can complete the work in 12 days independently, A can complete the same work in half the time, which is 6 days. Therefore, if A and B work together, they can complete the work in less time than either of them individually. The time taken to complete the work together can be found by taking the reciprocal of the sum of the reciprocals of their individual times. In this case, the reciprocal of 6 is 1/6 and the reciprocal of 12 is 1/12. Adding these reciprocals gives 1/6 + 1/12 = 1/4. Taking the reciprocal of 1/4 gives 4, which means A and B can complete the work together in 4 days.

Explanation
The given information states that twenty women can complete a work in sixteen days, while sixteen men can complete the same work in fifteen days. This implies that the capacity of a woman is greater than that of a man, as fewer women can complete the work in the same amount of time. Therefore, the ratio between the capacity of a man and a woman is 4:3.

Explanation
A can do the work alone in 20 days, while B and C can do the work together in (30+60) = 90 days. Since A is assisted by B and C on every third day, it means that A will work alone for 2 days and then with B and C on the third day. This pattern will continue. So, in every 3 days, A will do 2/20 = 1/10th of the work alone, and with B and C, they will do 1/90th of the work. Therefore, in 30 days (10 sets of 3 days), A will do 10*(1/10) + 10*(1/90) = 1 + 1/9 = 10/9th of the work. To complete the remaining 1/9th of the work, A will take 1 more day. Hence, A can do the work in 30+1 = 31 days.

Explanation
A is thrice as good as workman as B means that A is three times more efficient than B. This implies that A can complete 3 units of work in the same time it takes B to complete 1 unit of work.

Let's assume that B takes x days to complete the job alone. Therefore, A takes x/3 days to complete the job alone.

According to the given information, A is able to finish the job in 60 days less than B. This means that A takes x - 60 days to complete the job alone.

When they work together, their combined efficiency is 1/x + 1/(x-60) units of work per day.

To find the time taken when they work together, we need to find the reciprocal of their combined efficiency, which is (x(x-60))/(2x-60).

We need to find the value of x that makes the equation equal to 1/20 (since they can do the job together in 20 days).

By solving the equation (x(x-60))/(2x-60) = 1/20, we find that x = 75.

Therefore, when they work together, they can complete the job in 75 days.

Since the answer options are not in terms of 75, we need to calculate the actual time.

When x = 75, the time taken is (75(75-60))/(2(75)-60) = 22.5 days.

Therefore, the correct answer is 22.5 days.

Explanation
The given question is based on the concept of the inverse variation of work and time. It states that the number of people required to complete a piece of work is inversely proportional to the time taken to complete the work. Therefore, if the number of people increases, the time taken will decrease and vice versa.

From the given information, we can form the equation: (6 men + 8 boys) * 10 days = (26 men + 48 boys) * 2 days.

Simplifying this equation, we find that (3 men + 4 boys) = (13 men + 24 boys).

Now, we need to find the time taken by 15 men and 20 boys.

Using the same logic, we can form the equation: (3 men + 4 boys) * 10 days = (15 men + 20 boys) * x days.

Simplifying this equation, we find that x = 4 days.

Therefore, the time taken by 15 men and 20 boys in doing the same type of work will be 4 days.

Explanation
Machine P can print one lakh books in 8 hours, so it can print 12,500 books per hour. Machine Q can print the same number of books in 10 hours, so it can print 10,000 books per hour. Machine R can print the same number of books in 12 hours, so it can print 8,333 books per hour.

Machine P is closed at 11 A.M., which means it has worked for 2 hours and printed 25,000 books. The remaining two machines, Q and R, need to print the remaining 75,000 books. Together, they can print 18,333 books per hour.

To print 75,000 books, it will take approximately 4.1 hours. Adding this to the 2 hours that machine P worked, the work will be finished at approximately 1:00 P.M.

Explanation
A completes 80% of the work in 20 days, which means he can complete the entire work in 25 days. Then, A and B together finish the remaining 20% of the work in 3 days. This implies that they can complete the entire work in 3/20 * 100 = 15 days. Since A takes 25 days to complete the work alone, B alone would take 15 - 25 = -10 days to complete the work, which is not possible. Therefore, B alone would take longer than A, so the correct answer is 37.5 days.

Explanation
When all three taps are opened together, they fill the cistern in 20 minutes. This means that in 1 minute, they fill 1/20th of the cistern's capacity. The first tap fills the cistern in 12 minutes, so in 1 minute it fills 1/12th of the cistern's capacity. Similarly, the second tap fills 1/15th of the cistern's capacity in 1 minute. Therefore, the combined rate of the first two taps is 1/12 + 1/15 = 9/60 + 4/60 = 13/60. Subtracting this rate from the overall rate of filling the cistern (1/20), we get the rate of the exhaust tap, which is 1/20 - 13/60 = 7/60. Inverting this rate, we find that the exhaust tap takes 60/7 minutes to empty the cistern, which is approximately 8.57 minutes or 9 minutes rounded to the nearest whole number. Therefore, the correct answer is 10 minutes.

Explanation
The first pipe can fill 1/10 of the tank per hour, the second pipe can fill 1/12 of the tank per hour, and the third pipe can empty 1/20 of the tank per hour. When all three pipes are opened together, the net amount of water being added to the tank per hour is (1/10 + 1/12 - 1/20). Simplifying this expression gives (6/60 + 5/60 - 3/60) = 8/60 = 1/7.5. Therefore, it will take 7.5 hours to fill the tank when all three pipes are opened together.

Explanation
To find the time at which the first pipe should be turned off, we need to determine the rate at which each pipe fills the cistern. Pipe P fills 1/24 of the cistern per minute, while pipe Q fills 1/32 of the cistern per minute. When both pipes are open, the combined rate of filling is (1/24) + (1/32) = 13/192 of the cistern per minute.

To find the time at which the cistern is just filled in 16 minutes, we multiply the combined rate by 16: (13/192) * 16 = 13/12 of the cistern.

Since pipe P fills the cistern at a rate of 1/24 per minute, it would have filled (1/24) * 12 = 1/2 of the cistern in 12 minutes. Therefore, the first pipe should be turned off after 12 minutes.

Explanation
Let's assume the capacity of the tank is x liters.
The leak empties the completely filled tank in 10 hours, so the leak rate is x/10 liters per hour.
When the tap is opened, it admits 4 liters of water per minute, which is 4 * 60 = 240 liters per hour.
So, the net inflow rate when the tap is opened is 240 - x/10 liters per hour.
Given that the leak takes 15 hours to empty the tank, we can set up the equation:
x / (240 - x/10) = 15
Solving this equation, we get x = 7200 liters.
Therefore, the tank holds 7200 liters of water.

Explanation
Pipe A can fill the tank in 4 hours, which means it fills 1/4 of the tank in 1 hour. Pipe B can fill the tank in 6 hours, which means it fills 1/6 of the tank in 1 hour. When they are opened alternately, in the first hour, pipe A fills 1/4 of the tank. In the second hour, pipe B fills 1/6 of the remaining tank, which is 3/4. In the third hour, pipe A fills 1/4 of the remaining tank, which is 1/2. In the fourth hour, pipe B fills 1/6 of the remaining tank, which is 1/3. Therefore, the tank shall be full in 4 (2/3) hours.

Explanation
If 20 workers can finish a work in 30 days, it means that the total work requires 600 worker-days (20 workers * 30 days). To complete the work in 35 days, the remaining workers need to work for 35 days. If 5 workers leave the job, the remaining workers need to complete the remaining work in 35 days. Since the total work requires 600 worker-days, the remaining workers need to work for 595 days (600 worker-days - 5 worker-days). Therefore, the 5 workers should leave the job after 15 days (595 worker-days / 35 days).

Explanation
At the beginning, there were 165 men. Each day, 10 men dropped out, so after 12 days, there would be 120 men remaining. Since the job got completed at the end of the 12th day, it means that all the men who were present at the beginning were able to complete the job. Therefore, the correct answer is 165.

Explanation
Pipe A can fill the tank in 10 hours, which means it fills 1/10th of the tank in 1 hour. Pipe B can empty the tank in 15 hours, which means it empties 1/15th of the tank in 1 hour. When both pipes are working together, the net fill rate is (1/10) - (1/15) = 1/30th of the tank per hour. Therefore, it will take 30 hours for both pipes to fill the tank together.

Explanation
In this question, we are given that 'A' can complete a piece of work in 12 days and 'A' and 'B' together can complete the same piece of work in 8 days. From this information, we can infer that in 1 day, 'A' completes 1/12th of the work and 'A' and 'B' together complete 1/8th of the work. To find out how many days 'B' alone can complete the work, we subtract the work done by 'A' in 1 day from the work done by 'A' and 'B' together in 1 day. So, 'B' alone completes 1/8th - 1/12th = 1/24th of the work in 1 day. Therefore, 'B' alone can complete the same piece of work in 24 days.

Explanation
If 56 men can complete a piece of work in 24 days, it means that the work requires a total of 56*24 = 1344 man-days to be completed. To find out how many days 42 men will take to complete the same piece of work, we divide the total man-days required by the number of men, which gives us 1344/42 = 32 days. Therefore, 42 men can complete the work in 32 days.

Explanation
Assume any value and check the answer. The total time taken will always be 6 days.