A, B, and C worked together for 5 days to build a wall and then B left the work. A and C together finished the remaining work in 5 days. In how many days can A and C working together finish building the whole wall if B alone can do it in 25 days?
12 1/2
Analyzing the Work Problem
This problem is a classic example of a work and time question. We are given information about the combined work of three individuals (A, B, and C) for a certain period, followed by the work of two of them (A and C) to complete the remaining task. We are also given the time one individual (B) takes to complete the entire task alone. Our goal is to find the time A and C together would take to complete the entire task.
Understanding Work and Time Concepts
In work and time problems, the total work is usually considered as 1 unit. The work rate (or efficiency) of a person is the amount of work they can do in one unit of time (usually a day). If a person can complete a task in \(d\) days, their work rate per day is \(1/d\).
- Work Rate: Work done per unit of time.
- Total Work: Usually represented as 1 unit for completing a whole task.
- Relationship: Total Work = Work Rate \(\times\) Time Taken.
Step-by-Step Solution
Step 1: Determine the work rate of B
We are told that B alone can build the wall in 25 days.
This means B's work rate per day is the reciprocal of the time taken:
\[ \text{Work rate of B} = \frac{1}{\text{Time taken by B}} = \frac{1}{25} \text{ of the wall per day} \]Step 2: Analyze the work done in the first 5 days
A, B, and C worked together for the first 5 days. Let their individual work rates be \(r_A\), \(r_B\), and \(r_C\) respectively.
Their combined work rate is \(r_A + r_B + r_C\).
The work done by A, B, and C in the first 5 days is:
\[ \text{Work done in first 5 days} = (\text{Combined rate of A, B, C}) \times \text{Time} \] \[ \text{Work done in first 5 days} = (r_A + r_B + r_C) \times 5 \]Step 3: Analyze the work done in the next 5 days
After 5 days, B left. The remaining work was finished by A and C together in another 5 days. Let the remaining work be \(W_{\text{remaining}}\).
The combined work rate of A and C is \(r_A + r_C\).
The work done by A and C in these 5 days is:
\[ \text{Work done by A and C in 5 days} = (\text{Combined rate of A, C}) \times \text{Time} \] \[ \text{Work done by A and C in 5 days} = (r_A + r_C) \times 5 \]This work done by A and C is exactly the remaining work after the first 5 days.
Step 4: Relate the work done to the total work
The total work to build the wall is 1 unit.
The work done in the first 5 days by A, B, C plus the work done in the next 5 days by A, C equals the total work:
\[ (\text{Work done by A, B, C in first 5 days}) + (\text{Work done by A, C in next 5 days}) = \text{Total Work} \] \[ 5 \times (r_A + r_B + r_C) + 5 \times (r_A + r_C) = 1 \]Step 5: Substitute the known work rate of B
We know \(r_B = \frac{1}{25}\). Substitute this into the equation from Step 4:
\[ 5 \times \left(r_A + \frac{1}{25} + r_C\right) + 5 \times (r_A + r_C) = 1 \]Expand and simplify the equation:
\[ 5r_A + 5 \times \frac{1}{25} + 5r_C + 5r_A + 5r_C = 1 \] \[ 5r_A + \frac{5}{25} + 5r_C + 5r_A + 5r_C = 1 \] \[ 5r_A + \frac{1}{5} + 5r_C + 5r_A + 5r_C = 1 \] \[ (5r_A + 5r_A) + (5r_C + 5r_C) + \frac{1}{5} = 1 \] \[ 10r_A + 10r_C + \frac{1}{5} = 1 \]Step 6: Solve for the combined work rate of A and C
We want to find the combined work rate of A and C, which is \(r_A + r_C\). Rearrange the equation from Step 5 to isolate terms with \(r_A\) and \(r_C\):
\[ 10r_A + 10r_C = 1 - \frac{1}{5} \] \[ 10(r_A + r_C) = \frac{5}{5} - \frac{1}{5} \] \[ 10(r_A + r_C) = \frac{4}{5} \]Now, divide both sides by 10 to find the combined rate \(r_A + r_C\):
\[ r_A + r_C = \frac{4/5}{10} = \frac{4}{5 \times 10} = \frac{4}{50} \]Simplify the fraction:
\[ r_A + r_C = \frac{2}{25} \text{ of the wall per day} \]This is the combined work rate of A and C.
Step 7: Calculate the time A and C together take to finish the whole wall
The time taken by A and C together to complete the entire wall is the reciprocal of their combined work rate:
\[ \text{Time taken by A and C} = \frac{\text{Total Work}}{\text{Combined rate of A and C}} = \frac{1}{r_A + r_C} \] \[ \text{Time taken by A and C} = \frac{1}{2/25} = 1 \times \frac{25}{2} = \frac{25}{2} \text{ days} \]Convert the improper fraction to a mixed number:
\[ \frac{25}{2} \text{ days} = 12 \frac{1}{2} \text{ days} \]This means A and C working together can finish the whole wall in \(12 \frac{1}{2}\) days.
| Individual/Group | Work Rate (per day) | Time Taken for Whole Wall |
|---|---|---|
| B | \(1/25\) | 25 days (given) |
| A and C together | \(2/25\) | \(1 / (2/25) = 25/2 = 12.5\) days (calculated) |
| A, B, and C together (rate) | \(r_A + r_B + r_C\) | - |
Conclusion
By calculating the work rate of B, setting up an equation based on the work done in the two phases of the project, and solving for the combined work rate of A and C, we found that A and C together can build the entire wall in \(12 \frac{1}{2}\) days.
Revision Table: Work and Time Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Work Rate | Amount of work done per unit of time (e.g., per day). | If work takes \(T\) days, Rate = \(1/T\) per day. |
| Total Work | The complete task, usually considered as 1 unit. | Total Work = Rate \(\times\) Time |
| Combined Rate | Sum of individual rates when people work together. | If rates are \(r_1, r_2, \dots, r_n\), Combined Rate = \(r_1 + r_2 + \dots + r_n\). |
| Time for Combined Work | Time taken by a group to complete the total work. | Time = Total Work / Combined Rate |
Additional Information: Work and Time Problems
Work and time problems often involve understanding how individuals or groups contribute to a task at different rates. Key strategies include:
- Converting time taken into work rate (rate = 1/time).
- Converting work rate into time taken (time = 1/rate).
- Adding work rates when people work together.
- Subtracting work rates when one person's rate is relative to another's (e.g., A is twice as efficient as B).
- Calculating the fraction of work done in a given time.
- Determining the remaining work and the time taken to complete it.
These problems can sometimes involve negative work (e.g., a leak emptying a tank while a pipe fills it), but the principle of adding/subtracting rates applies.
Important Questions from Time and Work
A man, a woman, and a boy can do a work in 6, 9, and 18 days respectively. How many boys must assist one man and one woman to do the work in 1 day?
Three men can complete a work in 6 days and five women can complete the same work in 18 days. In how many days can 4 men and 10 women together complete the same work?
Mother, daughter, and son can do a work in 3, 4, and 5 days respectively. How many days will they take to complete the work, if they work together?
Rajan does 7/11 of a work in 21 days. How many more days will he take to complete the work?
By looking in a mirror, it appears that it is 6:30 in the clock. What is the real time?
