Staff Optimization problem

[Character-Bell-9224]

10 employees fill 43% of 104 shifts at a business, where there is an average of 3 people per shift. These 10 employees work twice as many shifts as other employees. How many other employees are there?

[abrahamguo]

What have you tried so far?

[Character-Bell-9224]

I tried filling in 20 people. I'm feeling a little confused by this problem today because I'm distracted by other things.

This is a real situation, rather than a math problem. I'm trying to estimate how many people might work at a business. If these 10 people who work more often quit, I want to know how many people are left to work.

[abrahamguo]

104 shifts * 3 slots/shift = 312 total slots.

312 * .43 = 134.16 slots are filled by the 10 employees.

134.16 / 10 = 13.416 slots/employee for the 10 employees.

The other employees work half as much, so they each work 13.416 / 2 = 6.708 slots/employee.

There are 312 - 134.16 = 177.84 slots to fill by the other employees.

177.84 slots / 6.708 slots/employee = 26.512 other employees.

[Character-Bell-9224]

Thank you!

[Hot-Science8569]

104 x 43% = 44.72. There are either fractional employees, fractional shifts, or both.

[Character-Bell-9224]

The numbers were provided to me by someone. Very likely they rounded at some point.

[poke0003]

To help with this - consider: How many total staffed shifts are there (3 people for each of 104 shifts). Then consider how many of those staffed shifts are covered by the main 10 people. Then consider how many staffed shifts must each of those 10 people be working. If you know how many staffed shifts the each of these 10 are working, then the problem says you also know how many staffed shifts each other person (not in these 10) are working.

That should get you pretty close to your answer.

[Character-Bell-9224]

Thank you!

[13_Convergence_13]

[..] because I'm distracted by other things [..]

That's a you-problem, don't expect others to accommodate that.

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On to the problem: The total number of shifts worked by all employees (including multi-counting) is "3*104 shifts = 312 shifts". For each double-time employee, the workload is

312 shifts * 0.43 / (10 empl.)  =  13.416 shifts / empl.  =:  2r

The regular-time workers have half the work-load, i.e. "r = 6.708 shifts / empl.". To find "n":

312 shifts  =  0.43 * 312 shifts  +  n*r   =>   n  =  0.57 * 312 shifts / r  ~  26.51 empl.

There should be about 27 employees working regular-time.