# Word problem

A new drug was tested on a group of 200 patients with high blood pressure. By the end of the trial period, 90 patients showed improvement from the drug, 35 patients developed side effects, and 95 patients showed no improvement and had no side effects. How many patients showed improvement and had side effects?

I first tired to solve it by using a venn diagram but that got me know where because I didn't know what to do. Then, I thought that I would try to find the percentages of each of the given categories over 200, like 90 patients showed improvement from the drug/200 total patients to get 45% and then I did the same for the other two and found 17.5% developed side effects and 47.5% showed no improvement and no side effects but I was at a loss of what to do next.

Thank you!

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cepheid
Staff Emeritus
Gold Member
The Venn diagram approach is a good way to visualize it. You draw two circles that are partially overlapping, one on the left, one on the right. Then you draw a third circle that partially overlaps with the right fence, but not with the left one. Imagine you've actually constructed overlapping circular fences just like that, and you literally place the people within the fences. The leftmost fence encloses everyone who showed improvement. The middle fence encloses everyone who developed side effects. The rightmost fence encloses everyone who did not show improvement. The reason why the leftmost fence and the rightmost fence DON'T overlap is because these two possibilities are mutually exclusive. You can't improve and not improve at the same time. It's either one or the other.

The overlapping regions represent people who are simultaneously enclosed by two of the fences, because two of the outcomes happen to be true for these people. For example the overlapping region between the leftmost fence and the middle fence represents those people who both showed improvement and developed side effects.

The non-overlapping areas of the fences contain people who only had one of the three possible outcomes.

EDIT: Finally, you know that the total number of people must be 200.

The percentage method is not useful because it is possible for a person to experience two out of the three possible outcomes. So your percentages aren't going to add up to 100.

I designed a venn diagram like you said and tried placing the numbers in the circles but I still do not understand what to do next. I know that 95 patients did NOT show improvement and NO side effects. So if I subtract that amount from 200, I am left with 105 people that showed side effects and did show improvement. And this is where I am stuck again.

cepheid
Staff Emeritus
Gold Member
I designed a venn diagram like you said and tried placing the numbers in the circles but I still do not understand what to do next. I know that 95 patients did NOT show improvement and NO side effects. So if I subtract that amount from 200, I am left with 105 people that showed side effects and did show improvement. And this is where I am stuck again.
This subtraction is wrong, because (improvement & side effects) and (no improvement & no side effects) are not the only two possibilities.

I've attached an image illustrating the situation. The numbers outside of the circles indicate the total number of people within the circle pointed to by the arrow. The numbers inside the circles indicate the number of people in the smallest region surrounding that number. Although this diagram only contains information that was given to you in the problem, I've done a couple of the important steps for you. The grey shaded regions are not allowed, because somebody having side effects either improved or he didn't. There is no third possibility.

http://img130.imageshack.us/img130/2139/venn.th.png [Broken]

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That you for your diagram! That's what I was doing wrong since the beginning. I did not have the shaded region, I was trying to make it account for something.

I was able to come up with the final answer! thank you for all of your help! :]

But now looking at the problem, one thing that I do not understand is how the right most circle can be 95? Doesn't it account for those patients that did not show improvement and did not have any side effects? if so then how can it be in union with those that did have side effects?

cepheid
Staff Emeritus