Work and Time Calculation Strategy
This problem requires calculating the total time to complete a task considering different work efficiencies and a condition where one person leaves before the task finishes.
The approach involves determining individual work rates, calculating total work, and then analyzing the work done during different phases.
Calculating Individual Work Rates
Let's define the work rate of person A. We'll assume A's work rate is 1 unit per day.
- A's work rate: $R_A = 1$ unit/day.
- B is 50% more efficient than A. This means B's rate is A's rate plus 50% of A's rate: $R_B = R_A + (0.50 \times R_A) = 1.5 \times R_A = 1.5$ units/day.
- C is 3 times as fast as A: $R_C = 3 \times R_A = 3$ units/day.
Determining Total Work and Combined Rates
A completes the work in 45 days. The total amount of work ($W$) can be calculated as:
$W = R_A \times 45 \text{ days} = 1 \text{ unit/day} \times 45 \text{ days} = 45 \text{ units}$
When A, B, and C work together, their combined work rate ($R_{ABC}$) is:
$R_{ABC} = R_A + R_B + R_C = 1 + 1.5 + 3 = 5.5 \text{ units/day}$
When only A and B work together, their combined work rate ($R_{AB}$) is:
$R_{AB} = R_A + R_B = 1 + 1.5 = 2.5 \text{ units/day}$
Analyzing Work Done in Phases
C left 7 days before the work was completed. This implies that A and B worked together for the final 7 days.
Calculate the work done by A and B during these last 7 days ($W_{last7}$):
$W_{last7} = R_{AB} \times 7 \text{ days} = 2.5 \text{ units/day} \times 7 \text{ days} = 17.5 \text{ units}$
The work done by all three (A, B, and C) before C left ($W_{ABC}$) is the total work minus the work done in the last 7 days:
$W_{ABC} = W - W_{last7} = 45 \text{ units} - 17.5 \text{ units} = 27.5 \text{ units}$
Calculating Time for Completion
The number of days A, B, and C worked together ($Days_{ABC}$) is found by dividing the work they did together by their combined rate:
$Days_{ABC} = \frac{W_{ABC}}{R_{ABC}} = \frac{27.5 \text{ units}}{5.5 \text{ units/day}} = 5 \text{ days}$
The total time taken to complete the entire work is the sum of the days A, B, and C worked together and the final 7 days when only A and B worked.
Total Days = Days A, B, C worked + Days A, B worked alone
Total Days = 5 days + 7 days = 12 days
Therefore, the entire work was completed in 12 days.
Similar Questions
- Raju is thrice as good as a workman as Vinod and together they can finish a task in 21 days. In how many days can Vinod alone complete the work?
- A can do $\frac{1}{5}^{\text{th}}$ of some work in 12 days, B can do $20\%$ of the same work in 10 days, C can do $\frac{1}{6}^{\text{th}}$ of the work in 8 days and D can do $\frac{1}{5}^{\text{th}}$ of the work in 12 days. Who will complete the work first if all four started to work at the same time?
- If Raju is thrice as good workman as Ravi and takes 20 days less than him to complete a piece of work, then find the time taken by Ravi to complete the work.
- Raj is 60% more efficient than Sundar and Sundar can finish a certain task in 16 days. How many days will Raj alone take to complete the same task?
- Kamal can complete a work in 14 days. Vimal is 40% more efficient than Kamal. The number of days Vimal will take to complete the same piece of work is:
- John is only 40% as efficient as Jesse. If John alone can do a piece of work in 35 days, how many days will it take the duo to complete the work, if they work together?
- 21 women can complete a task in 40 days. 30 men can complete the task in 21 days. What is the ratio of the number of days a man will take to complete the task alone to the number of days a woman will need to do that task?
Manoj is twice as efficient a fisherman as Anuraj, and together they complete a piece of work in 22 days. In how many days can Anuraj alone complete the same work?
- Five women can do a work in thirty six days. If the ratio between the capacity of a man and a woman is 3:1, then find how many days it will take 5 men to complete the same work?
- If 15 persons can do a job in 30 days, then 30 persons with twice the efficiency can do the same job in how many days?
Important Questions from Work Efficiency
A and B together can complete a certain work in 20 days whereas B and C together can complete it in 24 days. If A is twice as good a workman as C, then in what time will B alone do 40% of the same work?
14 men can complete a work in 15 days. If 21 men are employed, then in how many days will they complete the same work?
A can do a certain work in 15 days, while B can do the same work in 21 days. If they work together, then in how many days will the same work be completed?
To do a certain work, A and B work on alternate days with B beginning the work on the first day. A alone can complete the same work in 24 days. If the work gets completed in \(11 \frac{1}{3}\) days, then B alone can complete \(\rm \frac{7}{9}^{th}\) part of the original work in:
Two men and 7 women can complete a work in 28 days whereas 6 men and 16 women can do the same work in 11 days. In how many days can 7 men complete the same work?
