Working efficiency of A is 20% more than that of B, who can complete a work 'X' in 36 days.
B and C together started to complete the work 'X' and after 10 days they both left the work and then remaining work is done by A alone in 15 days.
A and C together started to complete another work 'Y' and after working for 12 days they both left the work and Remaining work is done by B alone in 16 days. If D first completed work 'X' and then completed work 'Y' in total 38 days.
It is given that efficiency of all, in completing work 'X' and work 'Y' is same.
A person E starts the work 'X' and leave after 12 days, then B and C complete the remaining work in 8 days. What is the ratio of number of days taken by A and E together to complete the work 'X' to the number of days taken by D, B and C together to complete the both work 'X' and 'Y'.

Options

A. `3:5`
B. `5:3`
C. `8:7`
D. `1:2`

Answer

To solve the problem step by step, let's break it down into manageable parts:

### Step 1: Determine the efficiencies of A and B
Given that the efficiency of A is 20% more than that of B, if we let the efficiency of B be \( E_B = 5 \), then:
\[ E_A = E_B + 0.2 \times E_B = 5 + 1 = 6 \]
Thus, the efficiencies are:
- Efficiency of A = 6
- Efficiency of B = 5

### Step 2: Calculate the total work for work X
B can complete work X in 36 days. Therefore, the total work \( W_X \) can be calculated as:
\[ W_X = E_B \times \text{Time taken by B} = 5 \times 36 = 180 \]

### Step 3: Find the efficiency of C
B and C together worked for 10 days, and A completed the remaining work in 15 days. The total work done by A in 15 days is:
\[ \text{Work done by A} = E_A \times \text{Time taken by A} = 6 \times 15 = 90 \]
Thus, the work done by B and C together in 10 days is:
\[ W_X - \text{Work done by A} = 180 - 90 = 90 \]
Let the efficiency of C be \( E_C \). Therefore:
\[ (E_B + E_C) \times 10 = 90 \]
Substituting \( E_B = 5 \):
\[ (5 + E_C) \times 10 = 90 \implies 5 + E_C = 9 \implies E_C = 4 \]

### Step 4: Calculate the total work for work Y
A and C together worked for 12 days, and B completed the remaining work in 16 days. The total work done by A and C in 12 days is:
\[ \text{Work done by A and C} = (E_A + E_C) \times 12 = (6 + 4) \times 12 = 120 \]
Thus, the remaining work done by B in 16 days is:
\[ W_Y - 120 = E_B \times 16 \implies W_Y - 120 = 5 \times 16 = 80 \implies W_Y = 200 \]

### Step 5: Calculate the efficiency of D
D completed both works X and Y in 38 days:
\[ \text{Total work} = W_X + W_Y = 180 + 200 = 380 \]
Let the efficiency of D be \( E_D \):
\[ E_D \times 38 = 380 \implies E_D = \frac{380}{38} = 10 \]

### Step 6: Determine the efficiency of E
E starts work X and leaves after 12 days. The remaining work is completed by B and C in 8 days. The total work done by B and C in 8 days is:
\[ (E_B + E_C) \times 8 = (5 + 4) \times 8 = 72 \]
Thus, the work done by E in 12 days is:
\[ W_X - 72 = 180 - 72 = 108 \]
So, the efficiency of E can be calculated as:
\[ E \times 12 = 108 \implies E = \frac{108}{12} = 9 \]

### Step 7: Calculate the ratio of days taken by A and E together to complete work X
The combined efficiency of A and E is:
\[ E_A + E_E = 6 + 9 = 15 \]
The time taken by A and E together to complete work X is:
\[ \text{Time} = \frac{W_X}{E_A + E_E} = \frac{180}{15} = 12 \text{ days} \]

### Step 8: Calculate the time taken by D, B, and C together to complete both works X and Y
The combined efficiency of D, B, and C is:
\[ E_D + E_B + E_C = 10 + 5 + 4 = 19 \]
The total work for both X and Y is 380, so the time taken is:
\[ \text{Time} = \frac{W_X + W_Y}{E_D + E_B + E_C} = \frac{380}{19} = 20 \text{ days} \]

### Step 9: Calculate the final ratio
The required ratio of the number of days taken by A and E together to complete work X to the number of days taken by D, B, and C together to complete both works X and Y is:
\[ \text{Ratio} = \frac{12}{20} = \frac{3}{5} \]

### Final Answer
The ratio of the number of days taken by A and E together to complete work X to the number of days taken by D, B, and C together to complete both works X and Y is \( \frac{3}{5} \).