Question

Five workers have been hired to complete a job. If one additional worker is hired, they could complete the job $8$ days earlier. If the job needs to completed $28$ days earlier, how many additional workers should be hired?

Answer 1

Let \( d \) be the number of days it takes for the original 5 workers to complete the job. The work done by the 5 workers in \( d \) days can be expressed in terms of a single worker's rate of work.

Let the work done by one worker in one day be \( w \). Therefore, the total amount of work \( W \) needed to complete the job is:

\[
W = 5 \times d \times w
\]

If one additional worker is hired, making a total of 6 workers, the amount of time they take to complete the job is \( d - 8 \) days. The work done by the 6 workers in this time is:

\[
W = 6 \times (d - 8) \times w
\]

Since both expressions equal the same amount of work, we can set them equal to each other:

\[
5d \cdot w = 6(d - 8) \cdot w
\]

Assuming \( w \neq 0 \) (since the workers are doing some work), we can divide both sides by \( w \):

\[
5d = 6(d - 8)
\]

Expanding the right side:

\[
5d = 6d - 48
\]

Now, isolate \( d \):

\[
5d - 6d = -48
\]

\[
-d = -48
\]

\[
d = 48
\]

Now that we know it takes 5 workers 48 days to complete the job, we can calculate the total amount of work:

\[
W = 5 \times 48 \times w = 240w
\]

Next, we find out how many additional workers are needed to finish the job 28 days earlier than the original timeline, meaning the job must be completed in \( 48 - 28 = 20 \) days.

Let \( x \) be the total number of workers needed to complete the job in 20 days. The equation for the work done by \( x \) workers in \( 20 \) days is:

\[
W = x \times 20 \times w
\]

Setting this equal to the total work \( W \):

\[
240w = x \times 20 \times w
\]

Again, assuming \( w \neq 0 \), we can divide both sides by \( w \):

\[
240 = 20x
\]

Now, solve for \( x \):

\[
x = \frac{240}{20} = 12
\]

Since we initially had 5 workers, the number of additional workers needed is:

\[
12 - 5 = 7
\]

Thus, the number of additional workers that should be hired is:

\[
\boxed{7}
\]