"A project is normally finished in x days", but this time new workers are coming in after y days. What should I base my solution on?

Question

I am copying the following example from a textbook.

Example: A project can be completed by 150 workers in 40 days. But the project manager brought in 30 more workers after 16 days. In how many days will the remaining work be finished?

Solution: (Number of workers is inversely proportional to the working days)

Number of workers = 150
150 workers can finish the work in = 40 days
Remaining days available = 40 days - 16 days = 24 days
Total workers now = 150 + 30 = 180 workers

Suppose that 180 workers the work in x days.

Workers      Days
150  |       /|\ 24 *************************
180 \|/       |  x

150/180 = x/24
180x = 150*24
x = (150*24)/180 = 20 days

Hence 180 workers will finish the work in 20 days

WHAT I DON'T UNDERSTAND:

I don't agree to the assumption that 150 workers are completing it in 24 days. What is the link between the ability of 150 workers to *normally* finish the project in 40 days (which is the given information in the question) with the introduction of 30 more workers in this particular scenerio ?

I think their introduction after the first 16 days, as well as the remaining part of the work, is irrelevant to the problem we are solving, simply because they have not metioned anything about the amount of work. So we are not thinking about it all.

I think the question is more like "if 150 workers finish normally finish this work in 40 days, how many days will 180 workers take"?

I think the solution should be based on:

 Workers      Days
  150  |      /|\ 40 **************************
  180 \|/      |  x

QUESTION

Am I right? Am I wrong? Please tell me the reason.

Answer 1

We assume each worker can do the same amount of work per day and that all work is interchangeable. If $150$ workers can complete the project in $40$ days, each worker does $\frac 1{40\cdot 150}=\frac 1{6000}$ of the project per day. The amount of work is $1$ project. Certainly the $16$ days with $150$ workers are relevant, because that tells you how much of the project is remaining after that time. The rate of work increases because of the extra staff.

Another way to approach the answer would be that the $150$ workers finish $\frac {16 \cdot 150}{6000}=\frac 25$ of the job in the first $16$ days. The remaining $\frac 35$ of the job will take the $180$ workers $d$ days, where $\frac 35=\frac {180}{3600}d$ and we find $d=20$ and the whole job is done in $16+20=36$ days.