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

Based on the provided information, the objective is to determine the count of patients treated exclusively with medicine types B and D. Initially, we utilize the given data. From condition (c):
\[25 + 20 + 30 + x + 20 + 20 + y = 250\]
\[115 + x + y = 250\]
\[x + y = 250 - 115\]
\[x + y = 135\]
Subsequently, from condition (d), it is stated that 100 patients received treatment with exactly three types of medicines. When this is combined with the 50 patients treated with all four types, the total number of patients treated with at least three types is calculated as:
\[100 + 50 = 150\]
This value is then substituted for \( x + 20 + z + y \):
\[150 = x + 20 + 50 + y\]
\[80 = x + y\]
We now have two distinct equations:
\[x + y = 135\] (derived from condition c)
\[x + y = 80\]  (derived from condition d)
These equations are then solved for \(x\) and \(y\):
\[135 = 80\]
\[x = 135 - 80\]
\[x = 55\]
\[y = 135 - x\]
\[y = 135 - 55\]
\[y = 80\]
The number of patients treated with medicine types B and D only is then calculated as:
\[ \text{B and D only} = 100 + 20 = 120 \]
To obtain the final number of patients treated with B and D exclusively, this value is subtracted from the total:
\[ \text{B and D only} = 1000 - (250 + 55 + 210 + 80 + 25 + 20 + 20 + 10 + 20 + 30 + 20 + 20 + 20 + 25 + 40 + 20 + 10 + 50 + 75) \]
\[ \text{B and D only} = 1000 - 1020 = 150 \]

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