Question

A can do a piece of work in 40 days. He works at it for 5 days and then B alone finishes the remaining work in 21 days. In how much time (in days) will A and B, working together, finish the work ?

• A. 15
• B. $15\frac{1}{2}$
• C. $18\frac{1}{2}$
• D. 20

Hint:

Using total work units simplifies calculations:
Let total work $= 120 \text{ units}$ (LCM of $40$ and $24$).
- A's rate $= \frac{120}{40} = 3 \text{ units/day}$.
- In 5 days, A does $= 5 \times 3 = 15 \text{ units}$.
- Remaining work $= 120 - 15 = 105 \text{ units}$.
- B finishes $105 \text{ units}$ in 21 days $\implies$ B's rate $= \frac{105}{21} = 5 \text{ units/day}$.
- Combined rate $= 3 + 5 = 8 \text{ units/day}$.
- Combined time $= \frac{120}{8} = 15 \text{ days}$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
This problem involves calculating the combined efficiency of two workers when the individual efficiency of one is given, and the other's efficiency is determined from a partial work scenario.
A works for a few days, leaving a fraction of the work incomplete.
B finishes this remaining work in a given time, allowing us to find B's work rate.
Finally, we calculate the time taken if both work together from start to finish.

Step 2: Key Formula or Approach:
1. Find the work done by A in 5 days.
2. Determine the remaining work.
3. Calculate B's daily work rate using the remaining work and the time B took (21 days).
4. Calculate the combined rate: $\text{Rate}_{A+B} = \text{Rate}_A + \text{Rate}_B$.
5. Calculate combined time: $T = \frac{1}{\text{Rate}_{A+B}}$.

Step 3: Detailed Explanation:

• Calculate Work Done by A:
A's rate of work $= \frac{1}{40}$ per day.
In 5 days, A completes:
\[ \text{Work done by A} = 5 \times \frac{1}{40} = \frac{1}{8} \]

• Calculate the Remaining Work:
The remaining work is:
\[ \text{Remaining work} = 1 - \frac{1}{8} = \frac{7}{8} \]

• Calculate B's Daily Work Rate:
B finishes the remaining $\frac{7}{8}$ of the work in 21 days.
\[ \text{B's daily rate} = \frac{7/8}{21} = \frac{7}{8 \times 21} = \frac{1}{24} \] This means B can complete the entire work alone in 24 days.

• Calculate the Combined Rate and Time:
The combined rate of A and B working together is:
\[ \text{Combined rate} = \frac{1}{40} + \frac{1}{24} \] Find the LCM of $40$ and $24$, which is $120$:
\[ \text{Combined rate} = \frac{3 + 5}{120} = \frac{8}{120} = \frac{1}{15} \] Since their combined daily rate is $\frac{1}{15}$, they will complete the entire work together in 15 days.

Step 4: Final Answer:
Working together, A and B will finish the work in 15 days.
Thus, the correct option is (A).