Expert: Josh - 3/2/2005

Question
32 treatments a day can be performed
each treatemnt gets performed on a person
60% of our people get one treatment per day
40% of our people get eactly 2 treatments per day

we can only perform 32 treatments a day

how many people will be seen per day, total
how many for one treatement
how many for two treatements  

Answer
Hi Michael,

Each treatment received by each person occupies one of thirty two time slots. If a person receives 2 treatements, that counts as a double (i.e., that session occupies two time slots).

The general rule is to let the unknown quantity be something and we usually represent this as "x".

Here, let the total number of clients be X.
Assume that "N" people receive only 1 treatment.
From the information given, N accounts for 60% of X.
Relatively speaking, the remaining group of people, who receive two treatments (given by X-N, "X less N people") accounts for 40% of X. How do we balance things out?

Total time occupied by people in the single treatment group is N = 0.6*X [time slots].

Total time occupied by people in the two treatment group is (X-N)*2 or (0.4*X)*2 [time slots]. We multiply by 2 because each person receives two treatments.

Their sum cannot exceed 32 time slots. So, we have
0.6*X+0.8*X<=32, which gives
1.4X<=32,
X<=32/1.4=22.85

Since people come in whole, not in parts, the maximum number of people that the salon can accomodate is 22 people. If the ratio between the two groups of people is exact, then, the largest possible number of clients is 20.
Because 40% of 22 is not an integer, neither is 40% of 21.

So, the total number of clients is X=20. Of which, 60% receive one treatment (12 people occupy 1 time slot each), 40% receive two treatments (8 people each occupy 2 time slots).

Of the 32 time slots, only 12*1+8*2=28 spots are actually filled. Is this okay?