Let's suppose you want to build a house.

To do this, you're going to need people to help build the house. Specifically, you're going to need contractors, engineers, and construction workers.

After weeks of work, they finally finish building the house, and now you owe them a total of $\$85000$.

Given the following data:

1- The number of all employees = 100

2- Each engineer's wage equals $0.2$ times that of a contractor, and each worker's wage equals $0.1$ times that of an engineer.

3- Each contractor's wage = $8500.

How many contractors, engineers, and construction workers do you need, and how many dollars will each of them earn if the wages total to $\$85000$ ?

  • $\begingroup$ I believe we need to know how many of each type of worker there is. Otherwise, there is an infinite number of solutions. $\endgroup$ – Robert S. Sep 10 '18 at 19:00
  • $\begingroup$ @RobertS. The solution is one , it just needs a lucky guess as a key. $\endgroup$ – ben asfan Sep 10 '18 at 19:08
  • $\begingroup$ hi @benasfan I've clarified your question a bit, since you requested so. I'm pretty sure that I haven't changed the meaning of your question, but if I have, you may revert it back to your original. $\endgroup$ – Hugh Sep 10 '18 at 19:10
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    $\begingroup$ I’m voting to close this because of the multiple answers identified by dr jimbob’s answer below. There needs to be more definition to identify one solution, and ought not to rely on “a lucky guess” as Ben keeps mentioning. $\endgroup$ – El-Guest Sep 10 '18 at 20:54
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    $\begingroup$ Voting to reopen, but I will vote to close again because this is a mathematics problem as opposed to a mathematical puzzle. $\endgroup$ – boboquack Sep 11 '18 at 22:03

This problem was edited to be solvable with one solution. However, there may be references in the discussion to my old answer (see edits) when there weren't the constraints 100 total employees or Contractor wage = $8500. We can now easily calculate the salaries from your information:

Contractor = \$8500 = 50w
Engineer = .2 Contractor = .2*\$8500 = \$1700 = 10w
w = Worker = .1 Engineer = .1*\$1700 = \$170

We also now have two other equations, labeling $A$, $B$, $C$ as the number of workers, engineers, and contractors, respectively:

$ A + B + C = 100$
$(A + 10 B + 50 C) \times \$170 = \$85000$

Solving the equations:

$A + 10 B + 50 C = 500$ (divide both sides by \$170)
$A + B + C = 100$
$9B + 49C = 400$ (subtract equations above to cancel out A)

Looking for solutions with positive integer B and C,

we can calculate both sides modulo 9 to eliminate the 9*B term:

$9B + 49C \equiv 400 \mod 9$
$49 C \equiv 400 \mod 9$ (9B disappears when B is an integer mod 9)
$4 C \equiv 4 \mod 9$ (49 mod 9 = 4; 400 mod 9 = 4)
$C \equiv 1 \mod 9$
Thus $C = 1 + 9k$ for some integer $k$, and it must be $k=0$ (as other integer values of k will make either $C$ or $B$ negative).
Then we have $B = (400 - 49)/9 = 351/9 = 39$.
Finally using $A + B + C = 100$, we see that $A = 100 - B - C = 60$

Hence, our solution has:

1 contractor earning \$8500, 39 engineers earning \$66,300 total (\$1700 each), and 60 workers earning $10,200 total (\$170 each).

  • $\begingroup$ I edited the question. now you don't need long time of trials. Though, I made the question, I had spent long time misleaded between lots of probabilities before i could construct it. $\endgroup$ – ben asfan Sep 11 '18 at 18:06

Unless we specify at least a ratio of employees or something, this could be one of the answers.

Contractor will make 50(5X10) times more money than the worker and that an engineer will make 10 times more money than the worker.
Set the salary of a worker to 1 dollar and hire 84940 workers, 1 engineer(10) and 1 contractor(50) for a total of 85000.

  • $\begingroup$ Good start, but with this amout of workers, we would make an invasion to a neighbouring country, not building a house. $\endgroup$ – ben asfan Sep 10 '18 at 19:16
  • $\begingroup$ @benasfan is the goal then to minimize the number of workers used? This solution is valid for the question as it is currently written. $\endgroup$ – El-Guest Sep 10 '18 at 19:17
  • $\begingroup$ @benasfan you’d also need to set at least some of the costs, because otherwise I say one worker earns $85k and all I need is one of him. $\endgroup$ – El-Guest Sep 10 '18 at 19:21
  • $\begingroup$ @El-Guest yes, exactly, the number should be less. $\endgroup$ – ben asfan Sep 10 '18 at 19:26

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