X is 4 times more efficient as Y, then the number of days taken by X and Y are in the ratio :

Options

A. `8:1`
B. `1:5`
C. `6:8`
D. `5:1`

Answer

To solve the problem step by step, we will analyze the information given and apply the relevant mathematical concepts.

### Step-by-Step Solution:

1. **Understanding Efficiency**:
- We know that X is 4 times more efficient than Y. This means if Y's efficiency is considered as 1 unit, then X's efficiency will be 4 units more than Y's efficiency.
- Therefore, if we denote Y's efficiency as \( E_Y = 1 \), then X's efficiency can be expressed as:
\[ E_X = E_Y + 4 \times E_Y = 1 + 4 \times 1 = 5 \]

2. **Establishing the Ratio of Efficiencies**:
- Now we can write the ratio of the efficiencies of X and Y:
\[ \text{Efficiency ratio of X to Y} = E_X : E_Y = 5 : 1 \]

3. **Understanding the Relationship Between Efficiency and Time**:
- Efficiency is inversely proportional to the time taken to complete a work. This means that if one worker is more efficient, they will take less time to complete the same work.
- Therefore, we can express the relationship as:
\[ \text{Time taken by X} \propto \frac{1}{E_X} \quad \text{and} \quad \text{Time taken by Y} \propto \frac{1}{E_Y} \]

4. **Finding the Ratio of Time Taken**:
- Since efficiency and time are inversely proportional, the ratio of the time taken by X and Y will be the inverse of the ratio of their efficiencies:
\[ \text{Time taken by X : Time taken by Y} = \frac{1}{E_X} : \frac{1}{E_Y} = \frac{1}{5} : \frac{1}{1} \]
- This simplifies to:
\[ \text{Time taken by X : Time taken by Y} = 1 : 5 \]

5. **Final Answer**:
- Therefore, the ratio of the number of days taken by X and Y is:
\[ \text{Ratio of days taken by X and Y} = 1 : 5 \]