20 men can do a piece of work in 24 days. After working for 6 days, additional men are employed to finish the work in 21 days from the beginning. Find the number of additional men

each man can do 1/(20*24) = 1/480 of the job in a day.

So, in 6 days, they have done 6*20/480 = 1/4 of the work.

So, if the new men work at the same speed, you need to do 3/4 of the job in 15 days.

m additional men can do 15m/480 = m/32 of the job in 15 days. So, we need

m/32 = 3/4
m = 24

Let's break down the information given:

- 20 men can complete the work in 24 days.
- They have already worked for 6 days.
- The work needs to be completed in a total of 21 days.

First, let's find the rate of work of the 20 men:
20 men can complete the work in 24 days, so their rate of work is 1/24 of the work per day.

If 20 men work for 6 days, they would have completed:
6 days * (1/24 of the work per day) = 1/4 of the work.

So, after 6 days, 1 - 1/4 = 3/4 of the work still needs to be completed.

To finish the remaining 3/4 of the work in 21 days, let's assume we need x more men.

The combined rate of work for the 20 men and the additional x men would be:
(20 + x) men * (1/24 of the work per day) = 3/4 of the work / 21 days

Now, we can set up an equation to solve for x:
(20 + x) / 24 = (3/4) / 21

Cross-multiplying:
21 * (20 + x) = 24 * (3/4)
420 + 21x = 18
21x = 18 - 420
21x = -402
x = -402 / 21
x = -19.14

Since the number of men cannot be negative, we can ignore the negative solution.

Therefore, the number of additional men required to finish the work in 21 days is approximately 19.

To solve this problem, let's break it down step by step.

1. Determine the work rate of the initial set of 20 men:
Since 20 men can complete the work in 24 days, the work rate of each man per day is equal to 1/20.
Therefore, the total work rate of all 20 men combined per day is 20 * (1/20) = 1 unit of work per day.

2. Calculate the work done by the initial set of 20 men in 6 days:
In 6 days, the 20 men will complete 6 * 1 = 6 units of work.

3. Find the remaining work after the initial 6 days:
Since the total work is not mentioned in the problem, let's assume it as "W" units.

After the initial 6 days, there is still W - 6 units of work remaining to be completed.

4. Determine the new work rate required to finish the remaining work in 21 days:
The total work should now be completed in 21 days in total. This means the work rate required is equal to (W - 6) / 21 units of work per day.

5. Calculate the number additional men required:
Let's assume the number of additional men required is "x."
So, the work rate of each man (including the initial 20 men) would be (1/20) + (1/x).

Since the work rate is equal to the work done per day, the equation becomes:

(1/20) + (1/x) = (W - 6) / 21.

To solve this equation, multiply both sides by 20x to eliminate the denominators:

x + 20 = (20x(W - 6)) / 21.

Cross-multiplying and simplifying, we get:

21x + 420 = 20xW - 120x.

Rearranging the equation, we have:

120x - 21x = 420.

99x = 420.

Dividing both sides by 99, we get:

x = 420 / 99 ≈ 4.24.

Since we cannot have a fraction of a person, the number of additional men required would be rounded up to 5.

Therefore, the number of additional men required to finish the work in 21 days is 5.