A boy and his mother together can do a piece of work in 24 days while the day, working with his father, can complete it in 18 days. If the mother and father together take 36 days to complete the same work, then find the time in which they can complete the work if all the three work together. Also find the time they would take to complete the job, if they work individually.

Answer

To solve the problem, we will first determine the efficiencies of the boy, mother, and father based on the information provided. We will then calculate how long it will take for them to complete the work together and individually.

### Step 1: Determine the total work
We will assume the total work is represented in units. The least common multiple (LCM) of the days taken by each pair will give us the total work.

- Boy and Mother together can complete the work in 24 days.
- Boy and Father together can complete the work in 18 days.
- Mother and Father together can complete the work in 36 days.

**LCM of 24, 18, and 36:**
- The prime factorization is:
- 24 = 2^3 × 3^1
- 18 = 2^1 × 3^2
- 36 = 2^2 × 3^2
- The LCM will take the highest power of each prime:
- LCM = 2^3 × 3^2 = 8 × 9 = 72

So, the total work is 216 units (since we will multiply LCM by 3).

### Step 2: Calculate individual efficiencies
Now, we can calculate the efficiencies of each pair:

1. **Boy and Mother:**
- Work done in one day = Total work / Days = 216 / 24 = 9 units/day

2. **Boy and Father:**
- Work done in one day = Total work / Days = 216 / 18 = 12 units/day

3. **Mother and Father:**
- Work done in one day = Total work / Days = 216 / 36 = 6 units/day

### Step 3: Set up equations for individual efficiencies
Let:
- Efficiency of Boy = B
- Efficiency of Mother = M
- Efficiency of Father = F

From the above calculations, we can set up the following equations:
1. B + M = 9 (Equation 1)
2. B + F = 12 (Equation 2)
3. M + F = 6 (Equation 3)

### Step 4: Solve the equations
We will add all three equations:
- (B + M) + (B + F) + (M + F) = 9 + 12 + 6
- 2B + 2M + 2F = 27
- B + M + F = 27 / 2 = 13.5 (This is the total efficiency of all three working together)

Now, we can find each individual's efficiency:
- From Equation 1: M = 9 - B
- Substitute M in Equation 3: (9 - B) + F = 6
- F = 6 - 9 + B
- F = B - 3

Now substitute F in Equation 2:
- B + (B - 3) = 12
- 2B - 3 = 12
- 2B = 15
- B = 7.5 units/day

Now substitute B back to find M and F:
- M = 9 - 7.5 = 1.5 units/day
- F = 7.5 - 3 = 4.5 units/day

### Step 5: Calculate time taken when all three work together
Total efficiency when all three work together:
- Total efficiency = B + M + F = 7.5 + 1.5 + 4.5 = 13.5 units/day

Now, calculate the time taken to complete the work together:
- Time = Total work / Total efficiency = 216 / 13.5 = 16 days

### Step 6: Calculate individual times
1. **Time taken by Boy alone:**
- Time = Total work / Boy's efficiency = 216 / 7.5 = 28.8 days

2. **Time taken by Mother alone:**
- Time = Total work / Mother's efficiency = 216 / 1.5 = 144 days

3. **Time taken by Father alone:**
- Time = Total work / Father's efficiency = 216 / 4.5 = 48 days

### Final Answers:
- Time taken by Boy, Mother, and Father together = 16 days
- Time taken by Boy alone = 28.8 days
- Time taken by Mother alone = 144 days
- Time taken by Father alone = 48 days