A team of physician has determined that of all the athletes who were treated for minor back pain, 72% responded to an analgesic, 59% responded to a muscle relaxant, and 44% responded to both forms of treatment.
a) What percent of the athletes who were treated responded to the muscle relaxant but not to the analgesic?
(b) What percent of the athletes who were treated did not respond to either form of treatment?
(a) 59-44 = 15
(b) 72+59-44 = 87, so 13%
(a) Well, it looks like we have some math to do! To figure out the percentage of athletes who responded to the muscle relaxant but not to the analgesic, we can subtract the percentage of athletes who responded to both treatments from the percentage of athletes who responded to the muscle relaxant.
So, 59% responded to the muscle relaxant, and 44% responded to both. If we subtract 44% from 59%, we get 15%.
So, approximately 15% of the athletes responded to the muscle relaxant but not to the analgesic. Poor analgesic, apparently it just didn't do the trick for them!
(b) Now, let's find out the percentage of athletes who didn't respond to either form of treatment. To do this, we need to subtract the percentage of athletes who responded to either treatment (or both) from 100%.
So, we have 72% responding to the analgesic, 59% responding to the muscle relaxant, and 44% responding to both. If we add 72% and 59%, we get 131%. Uh-oh, looks like we went over 100% by 31%!
Now, we subtract 131% from 100%, which gives us -31%. But since we can't have a negative percentage, we'll round it down to 0%.
So, approximately 0% of the athletes didn't respond to either treatment. They must have superpowers or something!
To answer these questions, we can use the concept of set theory and the information provided.
Let's define:
A: Athletes who responded to analgesic
B: Athletes who responded to muscle relaxant
a) To find the percentage of athletes who responded to the muscle relaxant but not to the analgesic, we need to find the percentage of athletes in set B but not in set A.
To find this, we can subtract the percentage of athletes who responded to both forms of treatment (44%) from the percentage of athletes who responded to the muscle relaxant (59%):
59% - 44% = 15%
Therefore, 15% of the athletes responded to the muscle relaxant but not to the analgesic.
b) To find the percentage of athletes who did not respond to either form of treatment, we need to find the percentage of athletes not in sets A or B.
We can use the formula: 100% - (percentage of athletes in A + percentage of athletes in B - percentage of athletes in both A and B).
So, the percentage of athletes who did not respond to either form of treatment is:
100% - (72% + 59% - 44%) = 13%
Therefore, 13% of the athletes did not respond to either form of treatment.
To answer these questions, we can use set theory and logic. Let's break down the given information and work through it step by step.
Let's assume that the total number of athletes treated for minor back pain is 100 (this is an arbitrary assumption and not given in the question).
Now, let's label some sets to represent the different groups of athletes:
A = Athletes who responded to the analgesic
B = Athletes who responded to the muscle relaxant
A ∩ B = Athletes who responded to both forms of treatment
According to the given information:
The percentage of athletes who responded to the analgesic (A) is 72%.
The percentage of athletes who responded to the muscle relaxant (B) is 59%.
The percentage of athletes who responded to both forms of treatment (A ∩ B) is 44%.
(a) What percent of the athletes who were treated responded to the muscle relaxant but not to the analgesic?
We want to find the percentage of athletes in the set B who are not in the set A. To do this, we need to subtract the percentage of athletes who responded to both treatments from the percentage of athletes who responded to the muscle relaxant.
Percentage in set B but not in set A = Percentage in set B - Percentage in (A ∩ B)
= 59% - 44%
= 15%
So, 15% of the athletes who were treated responded to the muscle relaxant but not to the analgesic.
(b) What percent of the athletes who were treated did not respond to either form of treatment?
To find the percentage of athletes who did not respond to either form of treatment, we need to subtract the percentage of athletes who responded to any of the treatments (A ∪ B) from 100% (the total).
Percentage not responding to either form of treatment = 100% - Percentage in (A ∪ B)
= 100% - (Percentage in A + Percentage in B - Percentage in (A ∩ B))
= 100% - (72% + 59% - 44%)
= 100% - 87%
= 13%
So, 13% of the athletes who were treated did not respond to either form of treatment.
I hope this explanation helps! Let me know if you have any further questions.