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 total of 250 patients were treated with type A medicine and a total of 210 patients were treated with type C medicine. 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. 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. 100 patients were treated with exactly three types of medicines. 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 В. 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 type B?

Answer: 340. Solution: Understanding the set :- 1) This set tells us about four medicines - A, B, C, and D, which are tested on 1000 people. 2) These patients were first randomly assigned into two groups of equal size, called treatment group and control group a) Control group - 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 This is a 4-Venn diagram set : In this set we have 4 overlapping areas (A, B, C, D) There will be some people who have taken 1 medicine among A, B, C, or D or any two or any three or all four. we will try to find the exact numbers who belong to each category. —--, Region, Value, Only A, Only B, Only C, Only D, AB only, AC only, AD only, BC only, BD only, CD only, ABC only, ABD only, ACD only, BCD only, ABCD, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, Total C, We are given that, These patients were first randomly assigned into two groups of equal size. => Treatment group = 500 people and, Control group = 500 people Also, The patients in the control group were not treated with any of these medicines => Only treatment group people have been administered with medicines, while the rest 500 have been given the placebo. Based on the given information, we can then make the following table —--, Region, Value, Only A, Only B, Only C, Only D, AB only, AC only, AD only, BC only, BD only, CD only, ABC only, ABD only, ACD only, BCD only, ABCD, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Clue 1 says, i) A total of 250 patients were treated with type A medicine ii) a total of 210 patients were treated with type C medicine. We can directly add this in the diagram. —--, Region, Value, Only A, 25, Only B, Only C, 20, Only D, 10, AB only, AC only, AD only, BC only, BD only, CD only, ABC only, ABD only, ACD only, BCD only, ABCD, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Clue 2, i) 25 patients were treated with type A medicine only. ii) 20 patients were treated with type C medicine only. iii) 10 patients were treated with type D medicine only. —--, Region, Value, Only A, 25, Only B, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, BD only, CD only, 20, ABC only, ABD only, ACD only, BCD only, ABCD, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Clue 3 says, i) 35 patients were treated with type A and type D medicines only. ii) 20 patients were treated with type A and type B medicines only. iii) 30 patients were treated with type A and type C medicines only. iv) 20 patients were treated with type C and type D medicines only. —--, Region, Value, Only A, 25, Only B, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, BD only, CD only, 20, ABC only, ABD only, ACD only, BCD only, ABCD, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Clue 4 says, 100 patients were treated with exactly three types of medicines. => patients treated with - (A + B + C) + (A + C + D) + (A + B + D) + (B + C + D) = 100 —--, Region, Value, Only A, 25, Only B, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, BD only, CD only, 20, ABC only, 40, ABD only, ACD only, 20, BCD only, ABCD, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Clue 5 says, i) 40 patients were treated with medicines of types A, B and C, but not with medicines of type D. ii) 20 patients were treated with medicines of types A, C and D, but not with medicines of type B. => A + B + C = 40 and, A + C + D = 20 —--, Region, Value, Only A, 25, Only B, 20, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, BD only, CD only, 20, ABC only, 40, ABD only, ACD only, 20, BCD only, ABCD, 50, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Clue 6 says, i)50 patients were given all the four types of medicines. ii) 75 patients were treated with exactly one type of medicine. From clue 2, we know Only A = 25 Only c = 20 and, only D = 10 => Only B = 75 - 25 - 20 - 10 = 20 —--, Region, Value, Only A, 25, Only B, 20, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, BD only, CD only, 20, ABC only, 40, ABD only, 30, ACD only, 20, BCD only, 10, ABCD, 50, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Now, lets first take A, We know total of A = 250 Also, we have all values of A except one, of A + B + D Therefore, A + B + D = 250 - (25 - 30 - 20 - 35 - 20 - 40 - 50) = 30 Now, from Clue 4 we know, (A + B + C) + (A + C + D) + (A + B + D) + (B + C + D) = 100 Also, from clue 5 - A + B + C = 40 and, A + C + D = 20 And we have found out - A + B + D = 30 => B + C + D = 100 - (40 + 20 + 30) = 10 —--, Region, Value, Only A, 25, Only B, 20, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, 20, BD only, CD only, 20, ABC only, 40, ABD only, 30, ACD only, 20, BCD only, 10, ABCD, 50, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, Similarly, now, we know total of C = 210 Also, we have all values except one, of B + C Therefore, B + C = 210 - (30 - 40 - 20 - 20 - 10 - 50 - 20) = 20 —--, Region, Value, Only A, 25, Only B, 20, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, 20, BD only, 150, CD only, 20, ABC only, 40, ABD only, 30, ACD only, 20, BCD only, 10, ABCD, 50, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, It is given to us that the total number of people whom medicines were assigned = 500 Now, we have all value except one Firstly, let's take the total of all values - = complete A + 20 + 20 + 10 + 20 + 20 + 10 + (B + D) => 250 + 20 + 20 + 10 + 20 + 20 + 10 + (B + D) = 350 + (B + D) Now, we know total of whole = 500 => 350 + (B + D) = 500 => B + D = 150 —--, Region, Value, Only A, 25, Only B, 20, Only C, 20, Only D, 10, AB only, 20, AC only, 30, AD only, 35, BC only, 20, BD only, 150, CD only, 20, ABC only, 40, ABD only, 30, ACD only, 20, BCD only, 10, ABCD, 50, Also, Total, Value, Treatment Group, 500, Control Group, 500, Total A, 250, Total C, 210, —-- 20 + 20 + 40 + 20 + 50 + 10 + 30 + 150 = 340

Data Interpretation

Venn Diagrams

Medium

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 total of 250 patients were treated with type A medicine and a total of 210 patients were treated with type C medicine.

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.

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.

100 patients were treated with exactly three types of medicines.

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 В.

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 type B?

Entered answer:

✅ Correct Answer: 340

Slide 1/13

Understanding the set :-

This set tells us about four medicines - A, B , C, and D, which are tested on 1000 people.

These patients were first randomly assigned into two groups of equal size, called treatment group and control group

a) Control group - 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

This is a 4-Venn diagram set :

In this set we have 4 overlapping areas (A, B, C, D)

There will be some people who have taken 1 medicine among A, B, C, or D or any two or any three or all four.

we will try to find the exact numbers who belong to each category.

Region	Value
Only A
Only B
Only C
Only D
AB only
AC only
AD only
BC only
BD only
CD only
ABC only
ABD only
ACD only
BCD only
ABCD

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A
Total C

We are given that,

These patients were first randomly assigned into two groups of equal size.

=> Treatment group = 500 people

and, Control group = 500 people

Also, The patients in the control group were not treated with any of these medicines

=> Only treatment group people have been administered with medicines, while the rest 500 have been given the placebo.

Based on the given information, we can then make the following table

Region	Value
Only A
Only B
Only C
Only D
AB only
AC only
AD only
BC only
BD only
CD only
ABC only
ABD only
ACD only
BCD only
ABCD

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Clue 1 says,

i) A total of 250 patients were treated with type A medicine

ii) a total of 210 patients were treated with type C medicine.

We can directly add this in the diagram.

Region	Value
Only A	25
Only B
Only C	20
Only D	10
AB only
AC only
AD only
BC only
BD only
CD only
ABC only
ABD only
ACD only
BCD only
ABCD

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Clue 2,

i) 25 patients were treated with type A medicine only.

ii) 20 patients were treated with type C medicine only.

iii) 10 patients were treated with type D medicine only.

Region	Value
Only A	25
Only B
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only
BD only
CD only	20
ABC only
ABD only
ACD only
BCD only
ABCD

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Clue 3 says,

i) 35 patients were treated with type A and type D medicines only.

ii) 20 patients were treated with type A and type B medicines only.

iii) 30 patients were treated with type A and type C medicines only.

iv) 20 patients were treated with type C and type D medicines only.

Region	Value
Only A	25
Only B
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only
BD only
CD only	20
ABC only
ABD only
ACD only
BCD only
ABCD

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Clue 4 says,

100 patients were treated with exactly three types of medicines.

=> patients treated with -

(A + B + C) + (A + C + D) + (A + B + D) + (B + C + D) = 100

Region	Value
Only A	25
Only B
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only
BD only
CD only	20
ABC only	40
ABD only
ACD only	20
BCD only
ABCD

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Clue 5 says,

i) 40 patients were treated with medicines of types A, B and C, but not with medicines of type D.

ii) 20 patients were treated with medicines of types A, C and D, but not with medicines of type B.

=> A + B + C = 40

and, A + C + D = 20

Region	Value
Only A	25
Only B	20
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only
BD only
CD only	20
ABC only	40
ABD only
ACD only	20
BCD only
ABCD	50

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Clue 6 says,

i)50 patients were given all the four types of medicines.

ii) 75 patients were treated with exactly one type of medicine.

From clue 2 , we know

Only A = 25

Only c = 20

and, only D = 10

=> Only B = 75 - 25 - 20 - 10 = 20

Region	Value
Only A	25
Only B	20
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only
BD only
CD only	20
ABC only	40
ABD only	30
ACD only	20
BCD only	10
ABCD	50

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Now, lets first take A,

We know total of A = 250

Also, we have all values of A except one, of A + B + D

Therefore, A + B + D = $250 - (25 - 30 - 20 - 35 - 20 - 40 - 50) = 30$

Now, from Clue 4 we know,

(A + B + C) + (A + C + D) + (A + B + D) + (B + C + D) = 100

Also, from clue 5 - A + B + C = 40

and, A + C + D = 20

And we have found out - A + B + D = 30

=> B + C + D = $100 - (40 + 20 + 30) = 10$

Region	Value
Only A	25
Only B	20
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only	20
BD only
CD only	20
ABC only	40
ABD only	30
ACD only	20
BCD only	10
ABCD	50

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

Similarly, now,

we know total of C = 210

Also, we have all values except one, of B + C

Therefore, B + C = $210 - (30 - 40 - 20 - 20 - 10 - 50 - 20) = 20$

Region	Value
Only A	25
Only B	20
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only	20
BD only	150
CD only	20
ABC only	40
ABD only	30
ACD only	20
BCD only	10
ABCD	50

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

It is given to us that the total number of people whom medicines were assigned = 500

Now, we have all value except one

Firstly, let's take the total of all values -

= complete A + $20 + 20 + 10 + 20 + 20 + 10 +$ (B + D)

=> $250 + 20 + 20 + 10 + 20 + 20 + 10 +$ (B + D)

= $350$ + (B + D)

Now, we know total of whole = 500

=> $350$ + (B + D) = $500$

=> B + D = $150$

Region	Value
Only A	25
Only B	20
Only C	20
Only D	10
AB only	20
AC only	30
AD only	35
BC only	20
BD only	150
CD only	20
ABC only	40
ABD only	30
ACD only	20
BCD only	10
ABCD	50

Also,

Total	Value
Treatment Group	500
Control Group	500
Total A	250
Total C	210

$20 + 20 + 40 + 20 + 50 + 10 + 30 + 150 = 340$

Question 2

The number of patients who were treated with medicine types B, C and D, but not type A was:

Question 3

How many patients were treated with medicine types B and D only?

Question 4

The number of patients who were treated with medicine type D was:?

How many patients were treated with medicine type B?

Solution

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