X and Y can complete a work in 10 days, Y and Z in 12 days and X and Z in 15 days. What will be the ratio of efficiency of X, Y and Z respectively?

Options

A. a. 10 : 12 : 15
B. b. `6 : 5 : 4`
C. c. `5 : 7 : 3`
D. d. `5 : 6 : 4`

Answer

To solve the problem of finding the ratio of the efficiencies of X, Y, and Z, we can follow these steps:

### Step 1: Determine the work done by pairs
Let the total work be represented as 1 unit of work.

- **X and Y together can complete the work in 10 days.**
- Therefore, their combined work rate is \( \frac{1}{10} \) units/day.

- **Y and Z together can complete the work in 12 days.**
- Their combined work rate is \( \frac{1}{12} \) units/day.

- **X and Z together can complete the work in 15 days.**
- Their combined work rate is \( \frac{1}{15} \) units/day.

### Step 2: Set up equations for individual efficiencies
Let the efficiencies of X, Y, and Z be represented as \( x, y, \) and \( z \) respectively.

From the combined work rates, we can write the following equations:

1. \( x + y = \frac{1}{10} \) (Equation 1)
2. \( y + z = \frac{1}{12} \) (Equation 2)
3. \( x + z = \frac{1}{15} \) (Equation 3)

### Step 3: Solve the equations
We can solve these equations step by step.

- **From Equation 1:**
\[ y = \frac{1}{10} - x \]

- **Substituting \( y \) in Equation 2:**
\[ \left(\frac{1}{10} - x\right) + z = \frac{1}{12} \]
Rearranging gives:
\[ z = \frac{1}{12} - \left(\frac{1}{10} - x\right) \]
\[ z = x + \left(\frac{1}{12} - \frac{1}{10}\right) \]

- **Finding a common denominator (60):**
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{10} = \frac{6}{60} \]
Thus,
\[ z = x + \left(\frac{5}{60} - \frac{6}{60}\right) = x - \frac{1}{60} \]

- **Substituting \( z \) in Equation 3:**
\[ x + \left(x - \frac{1}{60}\right) = \frac{1}{15} \]
\[ 2x - \frac{1}{60} = \frac{4}{60} \]
\[ 2x = \frac{4}{60} + \frac{1}{60} = \frac{5}{60} \]
\[ x = \frac{5}{120} = \frac{1}{24} \]

### Step 4: Find \( y \) and \( z \)
- **Using \( x \) to find \( y \):**
\[ y = \frac{1}{10} - \frac{1}{24} \]
Finding a common denominator (120):
\[ y = \frac{12}{120} - \frac{5}{120} = \frac{7}{120} \]

- **Using \( y \) to find \( z \):**
\[ z = \frac{1}{12} - y = \frac{1}{12} - \frac{7}{120} \]
Finding a common denominator (120):
\[ z = \frac{10}{120} - \frac{7}{120} = \frac{3}{120} = \frac{1}{40} \]

### Step 5: Write the efficiencies
Now we have:
- \( x = \frac{1}{24} \)
- \( y = \frac{7}{120} \)
- \( z = \frac{1}{40} \)

### Step 6: Find the ratio of efficiencies
To find the ratio of efficiencies \( x:y:z \):
- Convert to a common denominator (120):
- \( x = \frac{5}{120} \)
- \( y = \frac{7}{120} \)
- \( z = \frac{3}{120} \)

Thus, the ratio of efficiencies is:
\[ x:y:z = 5:7:3 \]

### Final Answer
The ratio of the efficiencies of X, Y, and Z is \( 5:7:3 \).

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