X and Y can do a piece of work in 20 days and 12 days respectively. X started line work alone and then after 4 days Y joined him till the completion of the work. How long did the work last ?

Options

A. 6 days
B. 10 days
C. 30 days
D. 36 days

Answer

To solve the problem step by step, we will calculate how long the work lasted when X and Y worked together.

### Step 1: Determine the total work
The total work can be calculated based on the time taken by X and Y to complete the work individually.

- X can complete the work in 20 days.
- Y can complete the work in 12 days.

To find the total work in terms of units, we can take the least common multiple (LCM) of their individual work times.

**Total Work (TW) = LCM(20, 12)**

Calculating the LCM:
- The prime factorization of 20 is \(2^2 \times 5^1\).
- The prime factorization of 12 is \(2^2 \times 3^1\).

The LCM is found by taking the highest power of each prime:
- LCM = \(2^2 \times 3^1 \times 5^1 = 60\)

So, the total work is **60 units**.

### Step 2: Calculate the efficiency of X and Y
Next, we calculate the efficiency (work done per day) of X and Y.

- Efficiency of X = Total Work / Days taken by X = \(60 / 20 = 3\) units per day.
- Efficiency of Y = Total Work / Days taken by Y = \(60 / 12 = 5\) units per day.

### Step 3: Calculate the work done by X in the first 4 days
Since X starts the work alone for the first 4 days, we can calculate how much work he completes in that time.

- Work done by X in 4 days = Efficiency of X × Number of days = \(3 \times 4 = 12\) units.

### Step 4: Calculate the remaining work
Now, we find out how much work is left after X has worked alone for 4 days.

- Remaining work = Total Work - Work done by X = \(60 - 12 = 48\) units.

### Step 5: Calculate the time taken by X and Y to complete the remaining work
Now that both X and Y are working together, we can calculate how much work they can do together in one day.

- Combined efficiency of X and Y = Efficiency of X + Efficiency of Y = \(3 + 5 = 8\) units per day.

Now, we can find out how many days it will take for them to complete the remaining work.

- Time taken to complete remaining work = Remaining work / Combined efficiency = \(48 / 8 = 6\) days.

### Step 6: Calculate the total time of work
Finally, we add the time X worked alone to the time taken by both X and Y to complete the remaining work.

- Total time of work = Time worked by X alone + Time worked by X and Y together = \(4 + 6 = 10\) days.

Thus, the total duration of the work is **10 days**.

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