B works three times more slowly than A and C together. C can do it twice slowly than A and B. if they work together, the work is completed in 10 days. In how many days they will do it separately?

Options

A. 24, 40, 30
B. 40, 30, 24
C. 40,24, 20
D. 30, 20, 40

Answer

To solve the problem step by step, we will define the efficiencies of A, B, and C based on the information given and then calculate the time taken by each to complete the work separately.

### Step 1: Define the relationship between A, B, and C
- Let the efficiency of A be \( a \) (units of work per day).
- According to the problem, B works three times more slowly than A and C together. Therefore, if A and C together can do \( a + c \) work, then B's efficiency can be expressed as:
\[ b = \frac{1}{3}(a + c) \]

### Step 2: Define C's efficiency
- The problem states that C works twice as slowly as A and B together. Therefore, if A and B together can do \( a + b \) work, then C's efficiency can be expressed as:
\[ c = \frac{1}{2}(a + b) \]

### Step 3: Express B's efficiency in terms of A and C
- From the equation for B, we can substitute \( c \) from the previous step:
\[ b = \frac{1}{3}(a + \frac{1}{2}(a + b)) \]
Simplifying this gives:
\[ b = \frac{1}{3}(a + \frac{1}{2}a + \frac{1}{2}b) = \frac{1}{3}(\frac{3}{2}a + \frac{1}{2}b) \]
Multiplying through by 3 to eliminate the fraction:
\[ 3b = \frac{3}{2}a + \frac{1}{2}b \]
Rearranging gives:
\[ \frac{5}{2}b = \frac{3}{2}a \implies b = \frac{3}{5}a \]

### Step 4: Substitute B's efficiency back into C's equation
- Now substituting \( b \) back into C's equation:
\[ c = \frac{1}{2}(a + \frac{3}{5}a) = \frac{1}{2}(\frac{8}{5}a) = \frac{4}{5}a \]

### Step 5: Calculate total efficiency
- Now we have:
- \( a \) (efficiency of A)
- \( b = \frac{3}{5}a \) (efficiency of B)
- \( c = \frac{4}{5}a \) (efficiency of C)

- The total efficiency of A, B, and C together is:
\[ a + b + c = a + \frac{3}{5}a + \frac{4}{5}a = a + \frac{7}{5}a = \frac{12}{5}a \]

### Step 6: Work done in 10 days
- According to the problem, A, B, and C together complete the work in 10 days. Therefore, the total work done is:
\[ \text{Total Work} = \text{Total Efficiency} \times \text{Time} = \frac{12}{5}a \times 10 = 24a \]

### Step 7: Calculate individual times
- Now we can find the time taken by each to complete the work separately:
- Time taken by A:
\[ \text{Time}_A = \frac{\text{Total Work}}{\text{Efficiency of A}} = \frac{24a}{a} = 24 \text{ days} \]
- Time taken by B:
\[ \text{Time}_B = \frac{24a}{\frac{3}{5}a} = 24 \times \frac{5}{3} = 40 \text{ days} \]
- Time taken by C:
\[ \text{Time}_C = \frac{24a}{\frac{4}{5}a} = 24 \times \frac{5}{4} = 30 \text{ days} \]

### Final Answer
- Therefore, the time taken by A, B, and C to complete the work separately is:
- A: 24 days
- B: 40 days
- C: 30 days