What is the fundamental formula for work in relation to time and rate?

The fundamental formula for work is Work = Rate × Time. This indicates that the total work done can be calculated by multiplying the rate of work by the time taken.

If person A can complete a job in 8 hours and person B can complete the same job in 12 hours, how long will it take them to complete the job together?

First, determine their rates: A's rate = 1/8 jobs/hour, B's rate = 1/12 jobs/hour. Together, their combined rate = (1/8 + 1/12) = (3/24 + 2/24) = 5/24 jobs/hour. Thus, Time = 1 job / (5/24 jobs/hour) = 24/5 hours = 4.8 hours.

If A can finish a task in 9 days and B can finish it in 15 days, how much of the task can they complete together in 5 days?

A's work rate is 1/9 tasks/day and B's work rate is 1/15 tasks/day. Their combined rate = (1/9 + 1/15) = (5/45 + 3/45) = 8/45 tasks/day. In 5 days, they can complete (8/45 tasks/day) × 5 days = 40/45 = 8/9 of the task.

A can complete a work in 10 days, B can complete the same work in 15 days, and C can complete it in 30 days. How long will it take them to finish the work together?

Rates: A = 1/10, B = 1/15, C = 1/30. Combined rate = (1/10 + 1/15 + 1/30). Finding a common denominator (30), we get (3/30 + 2/30 + 1/30) = 6/30 = 1/5 jobs/day. Thus, Time = 1 / (1/5) = 5 days.

If worker A can complete a job in 20 hours and worker B can complete it in 30 hours, how long will it take them to complete the job if they work together?

A's rate = 1/20 jobs/hour, B's rate = 1/30 jobs/hour. Combined rate = (1/20 + 1/30) = (3/60 + 2/60) = 5/60 jobs/hour. Thus, Time = 1 / (5/60) = 12 hours.

A can finish a job in 14 days, while B can finish it in 21 days. If they work together for 6 days, how much of the job will be left?

A's rate = 1/14, B's rate = 1/21. Combined rate = (1/14 + 1/21) = (3/42 + 2/42) = 5/42 jobs/day. In 6 days, they will complete (5/42) × 6 = 30/42 = 5/7 of the job. Therefore, 2/7 of the job will be left.

A and B can complete a job in 15 days together. If A alone can do it in 25 days, how long will B take to finish the job alone?

Let B's time be x days. Combined rate = 1/15, A's rate = 1/25. Thus, 1/15 = 1/25 + 1/x. Solving for x gives x = 37.5 days.

If A can do 1/3 of a job in 6 hours, how long will it take A to complete the entire job?

If A does 1/3 of the job in 6 hours, then to complete the entire job, A would take 6 hours × 3 = 18 hours.

A can complete a work in 40 days, B can do it in 50 days. If they work together for 10 days, how much of the work remains?

A's rate = 1/40, B's rate = 1/50. Combined rate = (1/40 + 1/50) = (5/200 + 4/200) = 9/200 jobs/day. In 10 days, they complete (9/200) × 10 = 9/20. The remaining work is 1 - 9/20 = 11/20.

If A can finish a job in x days and B can finish it in y days, what is the formula for the time taken to complete the job when they work together?

The formula is Time = xy / (x + y) days, where x and y are the individual times taken by A and B, respectively.

A can do a job in 24 days, B can do it in 36 days. If they work together for 8 days, how much of the job is left?

A's rate = 1/24, B's rate = 1/36. Combined rate = (1/24 + 1/36) = (3/72 + 2/72) = 5/72 jobs/day. In 8 days, they complete (5/72) × 8 = 40/72 = 5/9 of the job. Therefore, 1 - 5/9 = 4/9 of the job remains.

A can do a piece of work in 16 days and B in 24 days. Working together, how long will they take to complete the work?

A's rate = 1/16, B's rate = 1/24. Combined rate = (1/16 + 1/24) = (3/48 + 2/48) = 5/48 jobs/day. Time = 1 / (5/48) = 48/5 = 9.6 days.

If A can paint a house in 12 hours and B can paint the same house in 18 hours, how long will it take them to paint the house together?

A's rate = 1/12, B's rate = 1/18. Combined rate = (1/12 + 1/18) = (3/36 + 2/36) = 5/36 jobs/hour. Time = 1 / (5/36) = 36/5 = 7.2 hours.

Two workers, A and B, can complete a task in 40 days and 60 days respectively. If they work together for 10 days, how much of the task is left?

A's rate = 1/40, B's rate = 1/60. Combined rate = (1/40 + 1/60) = (3/120 + 2/120) = 5/120 = 1/24 jobs/day. In 10 days, they complete (1/24) × 10 = 10/24 = 5/12 of the task. Therefore, the remaining work is 1 - 5/12 = 7/12.

If A can complete a job in 8 hours and B can complete the same job in 12 hours, how long will it take them to complete the job together?

First, determine the rates: A's rate = 1/8 jobs per hour, B's rate = 1/12 jobs per hour. Their combined rate = (1/8 + 1/12) = (3/24 + 2/24) = 5/24 jobs per hour. To find the time taken together, use Time = Work / Rate. Thus, Time = 1 job / (5/24 jobs/hour) = 24/5 hours = 4.8 hours.