A and B can do a work together in 18 days. A is three times as efficient as B. In how many days can B alone complete the work?
72 days
Understanding the Problem: Time, Work, and Efficiency
This problem involves the concept of time and work, specifically focusing on the relationship between efficiency and the time taken to complete a task. We are given the combined time taken by two individuals, A and B, to complete a work, and a relationship between their efficiencies. We need to find the time taken by one individual, B, to complete the same work alone.
Key Concepts: Efficiency and Work
- Efficiency: Efficiency is defined as the amount of work done per unit of time. Higher efficiency means less time is required to complete a given amount of work.
- Work: The total amount of task to be completed. It is often considered as a fixed unit (e.g., 1 unit of work) or calculated based on efficiency and time (Work = Efficiency × Time).
Setting up the Problem
We are given two main pieces of information:
- A and B together can complete the work in 18 days.
- A is three times as efficient as B.
Step-by-Step Solution
Step 1: Define Efficiency Ratio
Let the efficiency of B be represented by $E_B$.
According to the problem, A is three times as efficient as B. So, the efficiency of A ($E_A$) is:
We can assume a unit of efficiency for B for simplicity. Let $E_B = 1$ unit of work per day.
Then, $E_A = 3 \times 1 = 3$ units of work per day.
Step 2: Calculate Combined Efficiency
When A and B work together, their efficiencies add up. The combined efficiency ($E_{A+B}$) is:
So, A and B together complete 4 units of work each day.
Step 3: Calculate Total Work
We know that Work = Efficiency × Time. A and B together complete the work in 18 days with a combined efficiency of 4 units/day.
The total work (W) is:
The total amount of work to be done is 72 units.
Step 4: Calculate Time Taken by B Alone
To find the time taken by B alone to complete the work, we use the formula:
The total work is 72 units, and B's efficiency ($E_B$) is 1 unit/day.
Therefore, B alone can complete the work in 72 days.
Verification
Let's check the result:
- B's efficiency = 1 unit/day. Time for B alone = 72 days. Total Work = 1 × 72 = 72 units.
- A's efficiency = 3 units/day. Time for A alone = Total Work / EA = 72 / 3 = 24 days.
- Combined Work Rate = 1/72 (by B) + 1/24 (by A) = 1/72 + 3/72 = 4/72 = 1/18.
- Time taken together = 1 / (Combined Work Rate) = 1 / (1/18) = 18 days.
This matches the information given in the question, confirming our calculation is correct.
| Individual/Group | Efficiency (units/day) | Time Taken (days) | Total Work (units) |
|---|---|---|---|
| B | 1 | 72 | $1 \times 72 = 72$ |
| A | 3 | 24 | $3 \times 24 = 72$ |
| A and B together | $1 + 3 = 4$ | 18 | $4 \times 18 = 72$ |
Revision Table: Time and Work Formulas
| Concept | Formula | Explanation |
|---|---|---|
| Work Done | Work = Efficiency × Time | Total work completed is the product of the rate of work (efficiency) and the time spent. |
| Efficiency | Efficiency = $\frac{\text{Work}}{\text{Time}}$ | Efficiency is the amount of work done per unit of time. |
| Time Taken | Time = $\frac{\text{Work}}{\text{Efficiency}}$ | The time needed to complete a task is the total work divided by the efficiency. |
| Combined Efficiency (A and B) | $E_{A+B} = E_A + E_B$ | When multiple people work together, their efficiencies are added. |
| Individual Time from Combined (for 2 people A & B) | If A takes 'a' days and B takes 'b' days, then time together = $\frac{ab}{a+b}$ | Useful for relating individual times to combined time. |
Additional Information: Solving Time and Work Problems
Time and work problems are common in quantitative aptitude. Understanding the inverse relationship between efficiency and time is crucial. If a person is more efficient, they take less time to complete the same amount of work. Conversely, if someone is less efficient, they take more time.
- Always define a unit of work or efficiency based on the given information. Assuming a variable or a simple number (like 1 or the LCM of time periods) can make calculations easier.
- Remember that when people work together, their rates of work (efficiencies) are combined, usually by adding them.
- The total work remains constant regardless of who does it or how many people are involved.
This problem used the efficiency approach. Another common approach involves calculating the fraction of work done per day. If B takes 'x' days, B does 1/x of the work per day. If A takes 'y' days, A does 1/y per day. Since A is 3 times as efficient as B, A takes 1/3 the time B takes, so $y = x/3$. Together they do $1/x + 1/(x/3) = 1/x + 3/x = 4/x$ of the work per day. Since they finish in 18 days, they do 1/18 of the work per day. Thus, $4/x = 1/18$, which gives $x = 4 \times 18 = 72$ days. Both methods yield the same correct answer.
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