Question

1000 patients currently suffering from a disease were selected to study the effectiveness of treatment of four types of medicines — A, B, C and D. These patients were first randomly assigned into two groups of equal size, called treatment group and control group. The patients in the control group were not treated with any of these medicines; instead they were given a dummy medicine, called placebo, containing only sugar and starch. The following information is known about the patients in the treatment group.
a. A total of 250 patients were treated with type A medicine and a total of 210 patients were treated with type C medicine.
b. 25 patients were treated with type A medicine only. 20 patients were treated with type C medicine only. 10 patients were treated with type D medicine only.
c. 35 patients were treated with type A and type D medicines only. 20 patients were treated with type A and type B medicines only. 30 patients were treated with type A and type C medicines only. 20 patients were treated with type C and type D medicines only.
d. 100 patients were treated with exactly three types of medicines.
e. 40 patients were treated with medicines of types A, B and C, but not with medicines of type D. 20 patients were treated with medicines of types A, C and D, but not with medicines of type B.
f. 50 patients were given all the four types of medicines. 75 patients were treated with exactly one type of medicine.
How many patients were treated with medicine types B and D only? [This Question was asked as TITA]

• A. 170
• B. 150
• C. 160
• D. 110

Answer 1

The Correct Option is B

Given the information provided, let's solve for the number of patients treated with medicine types B and D only.
We start by using the information given:
From condition (c), we have:
\[25 + 20 + 30 + x + 20 + 20 + y = 250\]
\[115 + x + y = 250\] 
\[x + y = 250 - 115\]
\[x + y = 135\]
Now, from condition (d), 100 patients were treated with exactly three types of medicines. So, we add the number of patients treated with all four types (50) to this to get the total number of patients treated with at least three types:
\[100 + 50 = 150\]
We'll replace this value with \( x + 20 + z + y \):
\[150 = x + 20 + 50 + y\]
\[80 = x + y\]
Now, we have two equations:
\[x + y = 135\] (from condition c)
\[x + y = 80\]  (from condition d)
Now, we can solve for \(x\) and \(y\):
\[135 = 80\]
\[x = 135 - 80\]
\[x = 55\]
\[y = 135 - x\]
\[y = 135 - 55\]
\[y = 80\]
Now, we can calculate the number of patients treated with medicine types B and D only:
\[ \text{B and D only} = 100 + 20 = 120 \]
But we need to subtract this from the total to get the number treated with B and D only:
\[ \text{B and D only} = 1000 - (250 + 55 + 210 + 80 + 25 + 20 + 20 + 10 + 20 + 30 + 20 + 20 + 20 + 25 + 40 + 20 + 10 + 50 + 75) \]
\[ \text{B and D only} = 1000 - 1020 = 150 \]

So, the number of patients treated with medicine types B and D only is 150.