Together X and Y can complete a work in 9 days, Y and Z in 12 days and X and Z in 18 days. Who is the most efficient worker among X, Y and Z?

Options

A. X
B. Y
C. Z
D. Cannot be determined

Explanation

To solve the problem step by step, we will analyze the work done by each pair of workers and then determine the efficiency of each individual worker.

### Step 1: Determine the total work done by each pair
- Let the total work be represented as \( W \).
- The work done by X and Y together in 9 days means their combined work rate is \( \frac{W}{9} \).
- The work done by Y and Z together in 12 days means their combined work rate is \( \frac{W}{12} \).
- The work done by X and Z together in 18 days means their combined work rate is \( \frac{W}{18} \).

### Step 2: Set up equations for the work rates
Let:
- \( x \) = work done by X in one day
- \( y \) = work done by Y in one day
- \( z \) = work done by Z in one day

From the information given, we can set up the following equations:
1. \( x + y = \frac{W}{9} \) (Equation 1)
2. \( y + z = \frac{W}{12} \) (Equation 2)
3. \( z + x = \frac{W}{18} \) (Equation 3)

### Step 3: Express the total work in terms of a common value
To simplify calculations, we can assume \( W = 36 \) (the LCM of 9, 12, and 18). Now substituting \( W \) into the equations:
1. \( x + y = \frac{36}{9} = 4 \) (Equation 1)
2. \( y + z = \frac{36}{12} = 3 \) (Equation 2)
3. \( z + x = \frac{36}{18} = 2 \) (Equation 3)

### Step 4: Solve the equations
Now we can solve these equations step by step:

1. From Equation 1: \( x + y = 4 \) (1)
2. From Equation 2: \( y + z = 3 \) (2)
3. From Equation 3: \( z + x = 2 \) (3)

We can express \( y \) from Equation (1):
- \( y = 4 - x \)

Substituting \( y \) into Equation (2):
- \( (4 - x) + z = 3 \)
- \( z = 3 - (4 - x) = x - 1 \)

Now substituting \( z \) into Equation (3):
- \( (x - 1) + x = 2 \)
- \( 2x - 1 = 2 \)
- \( 2x = 3 \)
- \( x = 1.5 \)

Now substituting \( x \) back to find \( y \) and \( z \):
- From \( y = 4 - x \):
- \( y = 4 - 1.5 = 2.5 \)
- From \( z = x - 1 \):
- \( z = 1.5 - 1 = 0.5 \)

### Step 5: Summary of efficiencies
Now we have the efficiencies of each worker:
- \( x = 1.5 \)
- \( y = 2.5 \)
- \( z = 0.5 \)

### Conclusion
The most efficient worker among X, Y, and Z is Y, with an efficiency of 2.5.