Working for 9 hours a day. X can finish a task in 3 days, Y can finish three times of the same task in 8 days, and Z can finish five times of the same task in 12 days. Working together, in how many hours will they complete the task?

Options

A. 12
B. 8
C. 10
D. 9

Answer

To solve the problem step by step, we will first determine the total work done by each person (X, Y, and Z) in terms of a common unit and then find out how long it will take for them to complete the task together.

### Step 1: Calculate the work done by each person

1. **Work done by X**:
- X can finish a task in 3 days working 9 hours a day.
- Total hours worked by X = 3 days × 9 hours/day = 27 hours.
- Therefore, the total work done by X = 1 task.

2. **Work done by Y**:
- Y can finish three times the same task in 8 days working 9 hours a day.
- Total hours worked by Y = 8 days × 9 hours/day = 72 hours.
- Since Y completes 3 tasks in that time, the work done by Y for 1 task = 72 hours / 3 = 24 hours.

3. **Work done by Z**:
- Z can finish five times the same task in 12 days working 9 hours a day.
- Total hours worked by Z = 12 days × 9 hours/day = 108 hours.
- Since Z completes 5 tasks in that time, the work done by Z for 1 task = 108 hours / 5 = 21.6 hours.

### Step 2: Calculate the work rates of X, Y, and Z

- Work rate of X = 1 task / 27 hours = 1/27 tasks per hour.
- Work rate of Y = 1 task / 24 hours = 1/24 tasks per hour.
- Work rate of Z = 1 task / 21.6 hours = 1/21.6 tasks per hour.

### Step 3: Combine their work rates

To find the combined work rate of X, Y, and Z:
\[ \text{Combined work rate} = \frac{1}{27} + \frac{1}{24} + \frac{1}{21.6} \]

### Step 4: Find a common denominator and add the fractions

The least common multiple (LCM) of 27, 24, and 21.6 can be calculated. However, for simplicity, we can convert these fractions to a common denominator:

1. Convert each fraction:
- \( \frac{1}{27} = \frac{24 \times 1}{24 \times 27} = \frac{24}{648} \)
- \( \frac{1}{24} = \frac{27 \times 1}{27 \times 24} = \frac{27}{648} \)
- \( \frac{1}{21.6} = \frac{30 \times 1}{30 \times 21.6} = \frac{30}{648} \)

Now, add them:
\[ \text{Combined work rate} = \frac{24 + 27 + 30}{648} = \frac{81}{648} = \frac{1}{8} \text{ tasks per hour} \]

### Step 5: Calculate the time taken to complete 1 task

If they work together at a rate of \( \frac{1}{8} \) tasks per hour, the time taken to complete 1 task is:
\[ \text{Time} = \frac{1 \text{ task}}{\frac{1}{8} \text{ tasks/hour}} = 8 \text{ hours} \]

### Conclusion

Thus, working together, X, Y, and Z will complete the task in **8 hours**.

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