Ravi can do three fourth of a work in `(27)/(2)` hours while Hira can do two third of the same work in 8 hours. If both started working together then in how much time the work will be completed?

Options

A. 8h
B. 7.2h
C. 8.4h
D. 9h

Answer

To solve the problem step by step, we will first determine the total time taken by Ravi and Hira to complete the entire work and then find out how long it will take them to complete the work together.

### Step 1: Calculate the total work done by Ravi
Ravi can do \( \frac{3}{4} \) of the work in \( \frac{27}{2} \) hours.

To find out how long it will take him to complete the entire work, we can set up the following equation:

\[ \text{Time taken by Ravi for complete work} = \left(\frac{27}{2}\right) \times \left(\frac{4}{3}\right) \]

Calculating this:

\[ = \frac{27 \times 4}{2 \times 3} = \frac{108}{6} = 18 \text{ hours} \]

### Step 2: Calculate the total work done by Hira
Hira can do \( \frac{2}{3} \) of the work in 8 hours.

To find out how long it will take him to complete the entire work, we set up the following equation:

\[ \text{Time taken by Hira for complete work} = 8 \times \left(\frac{3}{2}\right) \]

Calculating this:

\[ = 8 \times \frac{3}{2} = 12 \text{ hours} \]

### Step 3: Calculate the efficiency of Ravi and Hira
Now we know that Ravi takes 18 hours to complete the work and Hira takes 12 hours.

The efficiency of each worker can be calculated as follows:

- Efficiency of Ravi = \( \frac{1}{18} \) work/hour
- Efficiency of Hira = \( \frac{1}{12} \) work/hour

### Step 4: Combine the efficiencies
To find the combined efficiency when both work together, we add their efficiencies:

\[ \text{Combined Efficiency} = \frac{1}{18} + \frac{1}{12} \]

To add these fractions, we find a common denominator (which is 36):

\[ \text{Combined Efficiency} = \frac{2}{36} + \frac{3}{36} = \frac{5}{36} \text{ work/hour} \]

### Step 5: Calculate the time taken to complete the work together
Now, we need to find out how long it will take for them to complete the entire work together.

Using the formula:

\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} \]

Since the total work is considered as 1 unit of work:

\[ \text{Time} = \frac{1}{\frac{5}{36}} = \frac{36}{5} = 7.2 \text{ hours} \]

### Final Answer
Thus, the time taken for both Ravi and Hira to complete the work together is **7.2 hours**.

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