$25$ men are employed to do a work

Question

$25$ men are employed to do a work, which they could finish it in $20$ days but the drop off by $5$ men at the end of every $10$ days. In what time will the work be completed?

My Attempt In $20$ days, $25$ men can do $1$ work.

In $1$ day, $25$ men can do $\frac {1}{20}$ work.

In $1$ day, $1$ man can do $\frac {1}{20\times 25}$ work.

Now how can I proceed further?

Answer 1

As you've said in one day a man can complete $\frac{1}{500}$ of the job.

The amount of work done every 10 days is then given by $\frac{1}{500}\times10\times(30-5t)$ where $t$ is how many lots of 10 days have passed (starting from $t=1$).

Adding this from $t=1$ to $t=n$ gives:

$$\sum_{t=1}^n\frac{1}{500}\times10(30-5t)=\sum_{t=1}^n\frac{3}{5}-\frac{1}{10}t$$

$$=\frac{3}{5}n-\frac{1}{20}n(n+1)$$

$$=\frac{11}{20}n-\frac{1}{20}n^2$$

Then we need to solve this equal to $1$ to complete the job.

$$1=\frac{11}{20}n-\frac{1}{20}n^2$$

$$n^2-11n+20=0$$

$$n=\frac{11\pm\sqrt{11^2-4\times1\times20}}{2}$$

$$n=\frac{11\pm\sqrt{41}}{2}$$

$$n\approx2.3,8.7$$

So the job will get completed in between 20 and 30. To work out exactly we need to do the individual sums.

First ten days: $\frac{1}{500}\times10\times25=\frac{1}{2}$

Second ten days: $\frac{1}{500}\times10\times20=\frac{2}{5}$, total so far: $\frac{9}{10}$.

Remaining: $\frac{1}{10}$. Days needed for this: $\frac{1}{500}\times15\times n=\frac{1}{10}$ gives $n=\frac{10}{3}$ so $3$ and $\frac{1}{3}$ days.

So the total is $23$ and $\frac{1}{3}$ days.

Answer 2

Hint

It will be much easier if you think of the composite unit man-days.

The work requires $25\times 20 = 500$ man-days

If they drop off $5$ men every $10$ days, you just have to solve as follows:

In the first $10$ days, $25\times 10 = 250$ man-days of work is done

In the next $10$ days, a further $20\times 10 = 200$ man-days of work is done, and so on.

Can you set up an equation, and solve ?

Added

If you can't, hover over the next portion

After working for $20$ days, $450$ man-days of work have been done, and $15$ men are available to complete the residual $50$ man-days of work, thus further days needed = $\dfrac{50}{15}$

Now total up the days taken

Answer 3

In the first 10 days, half of work is done. In the next 10 days, 80$\%$ of the rest work is done by 20 men. Now, you have 15 men with 50 days work for a man that means $50/15$ days for 15 men. So the work is done in $20+10/3$ days.