An in-depth guide on comparing work done by individuals or groups relative to time for CBSE Class 11 Applied Mathematics, covering unitary methods, work rates, and efficiency problems.
Work done is directly proportional to the number of people working and the time taken. If a person takes 'n' days to complete a task, their work rate is 1/n of the task per day. When comparing two entities, the ratio of work done is inversely proportional to the ratio of time taken, assuming the effort remains constant.
• Calculate A's daily work rate: 1/10.
• Calculate B's daily work rate: 1/15.
• Sum the rates: 1/10 + 1/15 = (3+2)/30 = 5/30 = 1/6.
• The time taken is the reciprocal of the combined rate: 6 days.
Answer: They will finish the task together in 6 days.
When dealing with groups of people, we use the principle that the product of workers (M), days (D), and hours (H) per unit of work (W) is constant, provided the efficiency remains the same. This allows us to equate scenarios using the M1D1/W1 = M2D2/W2 formula.
• Identify variables: M1=12, D1=20, D2=15.
• Set up the equation: 12 * 20 = M2 * 15.
• Solve for M2: M2 = (12 * 20) / 15.
• Calculate: M2 = 240 / 15 = 16.
Answer: 16 men are required to complete the project in 15 days.