Question

A and B can complete a work in 12 days and 18 days respectively. They start together, but A leaves after 4 days. How many more days will B take to finish the remaining work?

Hint:

The LCM method is generally the fastest way to solve Time \& Work problems as it avoids fractions and simplifies calculations.

Answer 1

Correct Answer: 8

Step 1: Understanding the Question:
The problem is a standard 'Time and Work' scenario. We need to calculate the portion of work completed by A and B together, find the remaining work, and then calculate the time B alone will take to finish it.
Step 2: Key Formula or Approach:
The most efficient method is the LCM method. We assume a total amount of work that is the Least Common Multiple (LCM) of the days taken by each person. This allows us to work with integer rates.
1. Total Work = LCM(Time taken by A, Time taken by B).
2. Work Rate = Total Work / Time taken.
3. Time = Work / Rate.
Step 3: Detailed Explanation:
Part A: Calculate Individual Work Rates
Let the total work be the LCM of 12 and 18.
\[ \text{Total Work} = \text{LCM}(12, 18) = 36 \text{ units} \] Now, we can find the rate of work for A and B.
Rate of A = \(\frac{36 \text{ units}}{12 \text{ days}} = 3\) units/day.
Rate of B = \(\frac{36 \text{ units}}{18 \text{ days}} = 2\) units/day.
Part B: Calculate Work Done Together
A and B work together for 4 days. Their combined rate is:
Combined Rate = Rate of A + Rate of B = \(3 + 2 = 5\) units/day.
Work done in 4 days = Combined Rate \(\times\) Days = \(5 \times 4 = 20\) units.
Part C: Calculate Remaining Work
Remaining Work = Total Work - Work Done = \(36 - 20 = 16\) units.
Part D: Calculate Time for B to Finish
This remaining work of 16 units is completed by B alone.
Time taken by B = \(\frac{\text{Remaining Work}}{\text{Rate of B}} = \frac{16 \text{ units}}{2 \text{ units/day}} = 8\) days.
Step 4: Final Answer:
B will take 8 more days to finish the remaining work.