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Task: A team of reapers/mowers got a task to reap two meadows; acreage of the first meadow was twice as big as the second one. The whole team spent half the day reaping the first meadow; For the second half of the same day, the team split into two equal groups. The first group was still reaping the first meadow and finsished by the end of the day (the whole meadow was reaped). The second group went to reap the second(smaller) meadow, but they didn't finish by the end of the day. On the next day the smaller meadow was reaped by only one reaper who needed the whole day to finish the job (to fully reap the second/smaller meadow). How many reapers was in that team ?

possible answer: - not mine - (n - number of reapers, d - day, x - acreage of the smaller meadow):

For bigger meadow: $\frac{1}{2}d \cdot n + \frac{1}{2}d \cdot \frac{1}{2}n = 2x$ //half the day by the whole team + half the day by half the team

For smaller meadow: $ \frac{1}{2}d \cdot \frac{1}{2}n+d = x$ //half the day by half the team + one day by one reaper

rest is easy and doesn't need an explanation.

my approach: I just wanted to transform the information that bigger meadow was reaped by the certain number of people in one day: $(n+\frac{1}{2}n) \rightarrow 1$day $\rightarrow 2x$, in order to find out number of reapers necessary to reap x (smaller meadow) in one day and a half. Something like: $n+\frac{1}{2}n=2x \Rightarrow x = \frac{n+\frac{1}{2}n}{2}$ subsequently find Z (number of people necessary to reap x in 1.5 day) based on below proportion:

$\frac{n+\frac{1}{2}n}{2} - 1 d$

$Z - \frac{3}{2} d$

$Z * 1d=(\frac{n+\frac{1}{2}}{2}) * (\frac{3}{2} d)$

and next, equate Z to the second information, that smaller meadow needed 1.5 day and $\frac{1}{2}n + 1$ reapers. Thus: $Z=\frac{1}{2}n + 1$ -> $(\frac{n+\frac{1}{2}}{2}) * (\frac{3}{2} d)=\frac{1}{2}n + 1$

Wrap up my (wrong) approach: I wanted to transform the first information (about the bigger meadow) in such a way that expresses number of people needed to reap x acreage in 1.5 day and then equate it to information about smaller meadow. But I'm wrong, and I would appreciate help why my train of thought is wrong

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  • $\begingroup$ Your post is very thorough, and you’ve made a wonderful effort to format it $\ddot\smile$ Here’s a MathJax tutorial to help you learn more tricks! $\endgroup$ Sep 24 '17 at 22:11
  • $\begingroup$ I re-wrote my answer to show where your mistakes were. Your approach was correct but there were a couple of mistakes. $\endgroup$ Sep 25 '17 at 15:17
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Your first statement that the bigger meadow was reaped by $n+\frac{n}{2}$ workers in one day is not correct. That would only be true if $\frac{3}{2}n$ workers worked a full day to reap the larger field. But $n$ workers work $\frac{1}{2}$ day and $\frac{n}{2}$ workers work for $\frac{1}{2}$ day to reap the larger field. Therefore $n\cdot\frac{1}{2}+\frac{n}{2}\cdot\frac{1}{2}=\frac{3}{4}n$ workers reap the larger field in one day.

The same error is made later when you say that it takes $\frac{n}{2}+1$ workers to reap the smaller field. In fact it takes $\frac{n}{2}$ workers $\frac{1}{2}$ day and one worker one day to clear the smaller field, or $\frac{n}{2}\cdot\frac{1}{2}+1\cdot1=\frac{n}{4}+1$ workers.

Taking these two errors into account your equation should be

$$ \frac{\frac{3}{4}n}{2}=\frac{n}{4}+1 $$

giving the correct solution $n=8$.

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