20 men and 20 women can complete half of the work in 20 days working with alternatively. If 10 men and 20 boys together can complete the whole work in 10 days, then find the number of days required by 20 boys alone complete the Remaining work?

To solve this problem, we will first determine the work rates for the men, women, and boys. Then we will use these rates to find how long it will take for 20 boys to complete the remaining work.

### Step-by-Step Solution:

1. **Determine the work rate of men and women:**

   We know that 20 men and 20 women working alternately for 20 days complete half of the work. This means each group works for 10 days:

   - 20 men work for 10 days.
   - 20 women work for 10 days.

   Let \(M\) be the amount of work done by 1 man in 1 day, and \(W\) be the amount of work done by 1 woman in 1 day.

   Total work done by 20 men in 10 days:
   \[
   20 \times M \times 10 = 200M
   \]

   Total work done by 20 women in 10 days:
   \[
   20 \times W \times 10 = 200W
   \]

   Since half of the work is completed:
   \[
   200M + 200W = \frac{1}{2} \text{ (Total Work)}
   \]
   \[
   400(M + W) = 1 \text{ (Total Work)}
   \]
   \[
   M + W = \frac{1}{400}
   \]

2. **Determine the work rate of men and boys:**

   Given that 10 men and 20 boys can complete the whole work in 10 days, let \(B\) be the amount of work done by 1 boy in 1 day.

   Total work done by 10 men and 20 boys in 10 days:
   \[
   (10M + 20B) \times 10 = 1 \text{ (Total Work)}
   \]
   \[
   100M + 200B = 1
   \]

   Substitute \(M = \frac{1}{400} - W\):
   \[
   100 \left( \frac{1}{400} - W \right) + 200B = 1
   \]
   \[
   \frac{1}{4} - 100W + 200B = 1
   \]
   \[
   -100W + 200B = \frac{3}{4}
   \]
   \[
   -W + 2B = \frac{3}{400}
   \]
   \[
   W = 2B - \frac{3}{400}
   \]

3. **Find the work rate of boys:**

   Substitute \(W\) back into \(M + W = \frac{1}{400}\):

   \[
   M + \left(2B - \frac{3}{400}\right) = \frac{1}{400}
   \]
   \[
   M + 2B = \frac{4}{400} = \frac{1}{100}
   \]

   Substitute \(M = \frac{1}{400} - W\):
   \[
   \frac{1}{400} - \left(2B - \frac{3}{400}\right) + 2B = \frac{1}{100}
   \]

Now, solve for number of days boys would need for the task.

20 men and 20 women can complete half of the work in 20 days working alternatively. If 10 men and 20 boys together can complete the whole work in 10 days, then find the number of days required by 20 boys alone to complete the remaining work?

Find out how to solve a complex time and work problem where 20 men and 20 women complete half the work, and 10 men with 20 boys complete the whole work. Learn to calculate how many days 20 boys alone need to finish the remaining work using algebraic equations and work rate formulas. Suitable for students in Grades 10-12.

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