4 men or 5 women can complete task in 82 days. how many days it will take for 5men and 4 women to complete the same task?

We can solve this problem using the concept of work rates.

### Step 1: Define work rates
Let:
- The work rate of one man be \( r_m \) (work per day).
- The work rate of one woman be \( r_w \) (work per day).

From the problem:
- 4 men complete the task in 82 days. Therefore, the work rate of 4 men combined is \( \frac{1}{82} \) of the task per day.
  \[
  4r_m = \frac{1}{82}
  \]
  Hence,
  \[
  r_m = \frac{1}{4 \times 82} = \frac{1}{328}
  \]
  
- Similarly, 5 women complete the task in 82 days. Therefore, the work rate of 5 women combined is \( \frac{1}{82} \) of the task per day.
  \[
  5r_w = \frac{1}{82}
  \]
  Hence,
  \[
  r_w = \frac{1}{5 \times 82} = \frac{1}{410}
  \]

### Step 2: Calculate combined work rate of 5 men and 4 women
Now, we want to find out how many days it will take for 5 men and 4 women to complete the task. The combined work rate of 5 men and 4 women is:
\[
5r_m + 4r_w = 5 \times \frac{1}{328} + 4 \times \frac{1}{410}
\]
\[
5r_m + 4r_w = \frac{5}{328} + \frac{4}{410}
\]

### Step 3: Solve for total time
The total work rate per day is the sum of these rates, and the total time to complete the task is the reciprocal of this sum:
\[
\text{Total work rate} = \frac{5}{328} + \frac{4}{410}
\]

Let me calculate this for you.It will take 40 days for 5 men and 4 women to complete the task.

Would you like any further details or explanations?

Here are 5 related questions you might find interesting:
1. How would the time change if the number of men and women were increased equally?
2. What if only 3 men and 2 women were assigned to the task?
3. How can you adjust the number of men and women to finish the task in exactly 30 days?
4. How would the work rate change if the women were 20% less efficient than the men?
5. What is the combined work rate of 6 men and 3 women?

**Tip:** Always check if similar problems can be modeled using proportions or work rate equations for faster calculations.

4 men or 5 women can complete a task in 82 days. How many days will it take for 5 men and 4 women to complete the same task?

Learn how to solve a work rate problem where 4 men or 5 women can complete a task in 82 days. Using work rate formulas and proportions, find out how long it will take 5 men and 4 women to finish the same task. This step-by-step solution is ideal for high school students in Grades 9-12.

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