A can do a work in 12 days and B can do it in 18 days. They started working together. But A leaves 3 days before the work gets over. For how many days did the two work together ?

Options

A. 6
B. 5
C. 7
D. `7.2`

Answer

To solve the problem, we need to determine how many days A and B worked together before A leaves 3 days before the work is completed. Here’s a step-by-step solution:

### Step 1: Determine the total work
A can complete the work in 12 days, and B can complete it in 18 days. To find the total work, we can use the least common multiple (LCM) of the two times.

**Calculation:**
- LCM of 12 and 18 is 36.

**Hint:** The LCM gives us a common measure of total work that both A and B can complete.

### Step 2: Calculate the work done by A and B in one day
Now, we calculate how much work A and B can do in one day.

**Calculation:**
- Work done by A in one day = Total work / Days taken by A = 36 / 12 = 3
- Work done by B in one day = Total work / Days taken by B = 36 / 18 = 2

**Hint:** The work done in one day is found by dividing the total work by the number of days each person takes to complete the work.

### Step 3: Calculate the total work done by B alone in the last 3 days
Since A leaves 3 days before the work is completed, we need to calculate how much work B does alone in those 3 days.

**Calculation:**
- Work done by B in 3 days = Work done by B in one day × 3 = 2 × 3 = 6

**Hint:** To find the work done by one person over multiple days, multiply the daily work by the number of days.

### Step 4: Calculate the remaining work after B's contribution
Now, we need to find out how much work was done by A and B together before A left.

**Calculation:**
- Total work = 36
- Work done by B in the last 3 days = 6
- Remaining work done by A and B together = Total work - Work done by B = 36 - 6 = 30

**Hint:** Subtract the work done by B alone from the total work to find out how much work was completed while A was still present.

### Step 5: Calculate the total work done together in one day
Now, we calculate how much work A and B do together in one day.

**Calculation:**
- Total work done together in one day = Work done by A + Work done by B = 3 + 2 = 5

**Hint:** The combined daily work rate is the sum of the individual daily work rates.

### Step 6: Calculate the number of days they worked together
Finally, we can find out how many days A and B worked together by dividing the remaining work by their combined daily work.

**Calculation:**
- Days worked together = Remaining work / Combined daily work = 30 / 5 = 6

**Hint:** To find the number of days worked together, divide the remaining work by the total work done together in one day.

### Conclusion
A and B worked together for **6 days**.