20 male labours can finish the work in 30 days and 30 female labours can finish the same work in 24 days. If all of them work together, then in how many days they will finish the same work?

Options

A. A)`(28)/(3)`days
B. B)`(20)/(3)`days
C. C)`(40)/(3)`days
D. D)`(34)/(3)`days

Answer

To solve the problem step by step, we can follow these calculations:

### Step 1: Calculate the total work done in terms of units.
Given that:
- 20 male labours can finish the work in 30 days.

To find the total work in units, we can use the formula:
\[ \text{Total Work} = \text{Number of Labours} \times \text{Days} \]
Thus, for male labours:
\[ \text{Total Work} = 20 \text{ labours} \times 30 \text{ days} = 600 \text{ units} \]

### Step 2: Calculate the efficiency of male labours.
The efficiency of the male labours can be calculated as:
\[ \text{Efficiency of Male Labours} = \frac{\text{Total Work}}{\text{Days}} = \frac{600 \text{ units}}{30 \text{ days}} = 20 \text{ units per day} \]

### Step 3: Calculate the total work done by female labours.
Given that:
- 30 female labours can finish the work in 24 days.

Using the same formula for total work:
\[ \text{Total Work} = 30 \text{ labours} \times 24 \text{ days} = 720 \text{ units} \]

### Step 4: Calculate the efficiency of female labours.
The efficiency of the female labours can be calculated as:
\[ \text{Efficiency of Female Labours} = \frac{\text{Total Work}}{\text{Days}} = \frac{720 \text{ units}}{24 \text{ days}} = 30 \text{ units per day} \]

### Step 5: Calculate the combined efficiency of male and female labours.
Now, we can find the combined efficiency when both male and female labours work together:
\[ \text{Combined Efficiency} = \text{Efficiency of Male Labours} + \text{Efficiency of Female Labours} \]
\[ \text{Combined Efficiency} = 20 \text{ units per day} + 30 \text{ units per day} = 50 \text{ units per day} \]

### Step 6: Calculate the time taken to finish the work together.
Now, we can find the time taken to complete the work when both groups work together:
\[ \text{Time} = \frac{\text{Total Work}}{\text{Combined Efficiency}} \]
Using the total work calculated from either group (let's use the male labours' work):
\[ \text{Time} = \frac{600 \text{ units}}{50 \text{ units per day}} = 12 \text{ days} \]

### Final Answer:
Thus, if all of them work together, they will finish the work in **12 days**.

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