The ratio of work efficiency of a man and a woman is `3:2`. A woman can do a work in 30 days. In how many can 4 men and 4 women do the work?

Answer

To solve the problem, we will follow these steps:

### Step 1: Determine the work done by one woman in one day.
Given that a woman can complete the work in 30 days, we can find the work done by one woman in one day.

\[ \text{Work done by 1 woman in 1 day} = \frac{1}{30} \]

### Step 2: Determine the work done by one man in one day.
The ratio of work efficiency of a man to a woman is given as \(3:2\). This means if a woman does \(\frac{1}{30}\) of the work in a day, then a man does:

\[ \text{Work done by 1 man in 1 day} = \frac{3}{2} \times \frac{1}{30} = \frac{3}{60} = \frac{1}{20} \]

### Step 3: Calculate the work done by 4 men and 4 women in one day.
Now, we can find the total work done by 4 men and 4 women in one day.

\[ \text{Total work done by 4 men in 1 day} = 4 \times \frac{1}{20} = \frac{4}{20} = \frac{1}{5} \]

\[ \text{Total work done by 4 women in 1 day} = 4 \times \frac{1}{30} = \frac{4}{30} = \frac{2}{15} \]

Now, we add the work done by 4 men and 4 women:

\[ \text{Total work done by 4 men and 4 women in 1 day} = \frac{1}{5} + \frac{2}{15} \]

To add these fractions, we need a common denominator. The least common multiple of 5 and 15 is 15.

\[ \frac{1}{5} = \frac{3}{15} \]

So,

\[ \frac{1}{5} + \frac{2}{15} = \frac{3}{15} + \frac{2}{15} = \frac{5}{15} = \frac{1}{3} \]

### Step 4: Determine how many days it takes for 4 men and 4 women to complete the work.
If 4 men and 4 women can complete \(\frac{1}{3}\) of the work in one day, then to complete the entire work, it will take:

\[ \text{Number of days} = \frac{1}{\frac{1}{3}} = 3 \text{ days} \]

### Final Answer:
Thus, 4 men and 4 women can complete the work in **3 days**.

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