Two variables
Type  1
Suppose only P ( say who plays cricket) = A
And Only Q ( say who plays hockey) = B
Both P & Q ( plays both cricket and hockey) = A & B = C
P or Q ( cricket or hockey) = At least 1 = A + B + C
P or Q but not both (only cricket or only hockey) = exactly one = A + B
No P or Q ( not cricket not hockey) = D.
A U B = A + B  C
Q. In the Bhilai club all the members participated either in table tennis or the Badminton.320 took part in Badminton & 350 participated in Table Tennis and 220 participated in both.
How many members does the club have ?
Solution :
Let A = 320.
B = 350.
C = 220.
Total members = 320 + 350 – 220 = 450.
Q. At the birthday party of Dev, 55 persons are present. 40 persons decided to hug him & 25 persons decided to shake hands with him.
How many persons chose to both hug him & shake hands with him ?
Solution :
A U B = 55.
A = 40.
B = 25.
C = ?
C = A + B – (A U B )
C = 40 + 25 – 55
C = 10.
Type – 2
 C  D 

A  E  F  K 
B  G  H  L 
Total  I  J  M 
Here
K = E + F.
L= G+H.
I = E+G.
J = F+H.
M = K + L = I + J.
Q. A company manufactures balls.
(i) 60% balls are red in colour & rest are blue.
(ii) 40% of the balls are of size p & rest are size q.
(iii) 20% balls are red & size p.
Find what percent of all balls are blue & size q.
Solution :
As per given data 
 Red (C)  Blue (D) 

p (A)  20% (E)  F  40% (K) 
q (B )  G  H  L 
Toatal  60% (I)  J  100% (M) 
E + G = I
G = I – E = 60 – 20 = 40%.
I + J = M
J = M – I = 100 – 60 = 40%.
E + F = K
F = K – E = 40 – 20 = 20%.
K + L = M
L = M – K = 100 – 40 = 60%.
G + H = L
H = L – G = 60 – 40 = 20%.
H = blue ball of size q.
Balls  red  blue 

p  20  20  40 
q  40  20  60 
total  60  40  100 
Ans = 20%.
Q. 10% of all students are intelligent & have 10 grade points. 75% of students with 8 grade points are intelligent. If 40% of all students are intelligent.
What percent of students have 10 grade points.
Solution :
As per given data :
Students  Intelligent  Not intelligent 

10  10  F  K 
8  75% of X  25% of X  X 
Total  40  J  100 
J + 40 = 100.
J = 60.
10 + 75% of X = 40.
75% of X = 30.
X = 40.
25% of X = 10.
F + 25% of X = J
F + 10 = J
F = 50.
K = 10 + 50 = 60.
Here K = Students who have 10 grade points.
Students  Intelligent  Not intelligent 

10  10  50  60 
8  30  10  40 
Total  40  60  100 
Q. 80% of lights in a certain hostel are switched on.40% of the light that are supposed to be off are actually on and 10% of light that are supposed to be on are actually off.
What % of the lights that are on,are supposed to be off.
Solution :
As per given data –
Light  On  Off 

Supposed on  E  10% of K  K 
Supposed off  40% of L  H  L 
total  80  J  100 
40% of L = 0.4 L
10% of K = 0.1 K
H = 60% of L = 0.6 L
E = 90% of K = 0.9 K
J =100 – 80 = 20.
0.9 K + 0.4 L = 80.
0.1 K + 0.6 L = 20.
9 K + 4 L = 800.
K + 6 L = 200.
K / (4800 + 800) = L / (800 – 1800) = (1) / (54 – 4)
K / (4000) = L /(1000) = (1) / 50.
K = 80.
L = 20.
 On  Off  Total 
Suppose On  72  8  80 
Supposed Off  8  12  20 
Total  80  20  100 
Our question is, what percent of swiches are on supposed to be off.
So required percentage = ( 8 / 80 ) x 100%
= 10%.
3 Variables
Suppose 3 variables are X,Y & Z.
X or Y or Z = A+B+C+D+E+F+G
Exactly one among X,Y,Z = a+b+c.
Exactly two among = d+e+f.
At least one among = a+b+c+d+e+f+g.
At least two among = d+e+f+g
At least one
A U B U C = A+B+C+ (A&B&C) (A&B+B&C+A&C)
At least two
(A&B )+(B&C)+(A&C) – 2(A&B&C)
Exactly one
A+B+C+3(A&B&C) – 2(A&B + B&C + C&A)
Exactly two
(A&B )+(B&C)+(C&A) – 3 (A&B&C)
Q. In a CAT mock paper there were 3 sections. Out of them 33 cleared the cutoff in section 1, 34 studnets cleared the cut off in section 2 and 32 cleared the cutoff in section 3. 10 cleared cut off of section 1 & 2 , 9 cleared cut off of in section 2 & 3, 8 cleared the cut off in section 1 &3.The number of people who cleared each section alone was equal and was equal 21 for each section.
1. How many cleared all 3 sections.
2. How many cleared only one of 3 sections.
3. The ratio of number of students clearing the cut off in one or more of sections to number of students clearing the cutoff in section 1 alone.
Solution :
Section 1 = 33
Section 2 =34
Section 3 =32.
33 = 21+x+(10x)+(8x).
x = 6.
1. 6
2. 21+21+21 = 63.
3. (21+21+21+6+4+3+2)/21 = 78/21.
Q. BCCI conducted a meeting.100 people attended the meeting out of which 60 like Virat Kohli,70 like Dhoni & 80 like Rohit Sharma.
1. Find the minimum & maximum number of people who like all three players.
Solution :
A+B+C+D = 100.
A= people who like only 1 player.
B= people who like 2 players.
C= people who like all 3 players.
D= people who don’t like any player.
Given A+2B+3C = 210.
To get C(max) we need to take A=0,B=0,D=0.
3 C = 210.
C = 70.
But people who like virat is only 60, so if we consider these 60 people must like all players.
Hence maximum number of people who like all 3 players is 60.
To get minimum no of people who like all 3 players.
Let suppose people who like Rohit(80) are person no 1 to person 80.(1 – 80)
People who like Dhoni(70) are person no 31 to 100.(31 – 100)
People who like Virat (60) are (1 to 30) & (81100) & rest 10 persons from [31,80].
So minimum 10 people are there who like all 3 players.
2. Find the minimum and maximum number of person who like exactly two players.
Solution:
To get maximum numbers of person who like exactly two players:
Rohit(80) & Dhoni(70) hence 70.
Rest 20 people can also like exactly two players.
Total =70+20 = 90.
Minimum number of person who like exactly two people 0.
3. Find the maximum and minimum number of people who like exactly one player.
Minimum no of people who like exactly one player = 0.
Maximum no of people who like exactly 1 player
2 x Amax = Cmax
Amax = 60 / 2 = 30.
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Two variables
Type  1
Suppose only P ( say who plays cricket) = A
And Only Q ( say who plays hockey) = B
Both P & Q ( plays both cricket and hockey) = A & B = C
P or Q ( cricket or hockey) = At least 1 = A + B + C
P or Q but not both (only cricket or only hockey) = exactly one = A + B
No P or Q ( not cricket not hockey) = D.
A U B = A + B  C
Q. In the Bhilai club all the members participated either in table tennis or the Badminton.320 took part in Badminton & 350 participated in Table Tennis and 220 participated in both.
How many members does the club have ?
Solution :
Let A = 320.
B = 350.
C = 220.
Total members = 320 + 350 – 220 = 450.
Q. At the birthday party of Dev, 55 persons are present. 40 persons decided to hug him & 25 persons decided to shake hands with him.
How many persons chose to both hug him & shake hands with him ?
Solution :
A U B = 55.
A = 40.
B = 25.
C = ?
C = A + B – (A U B )
C = 40 + 25 – 55
C = 10.
Type – 2
 C  D 

A  E  F  K 
B  G  H  L 
Total  I  J  M 
Here
K = E + F.
L= G+H.
I = E+G.
J = F+H.
M = K + L = I + J.
Q. A company manufactures balls.
(i) 60% balls are red in colour & rest are blue.
(ii) 40% of the balls are of size p & rest are size q.
(iii) 20% balls are red & size p.
Find what percent of all balls are blue & size q.
Solution :
As per given data 
 Red (C)  Blue (D) 

p (A)  20% (E)  F  40% (K) 
q (B )  G  H  L 
Toatal  60% (I)  J  100% (M) 
E + G = I
G = I – E = 60 – 20 = 40%.
I + J = M
J = M – I = 100 – 60 = 40%.
E + F = K
F = K – E = 40 – 20 = 20%.
K + L = M
L = M – K = 100 – 40 = 60%.
G + H = L
H = L – G = 60 – 40 = 20%.
H = blue ball of size q.
Balls  red  blue 

p  20  20  40 
q  40  20  60 
total  60  40  100 
Ans = 20%.
Q. 10% of all students are intelligent & have 10 grade points. 75% of students with 8 grade points are intelligent. If 40% of all students are intelligent.
What percent of students have 10 grade points.
Solution :
As per given data :
Students  Intelligent  Not intelligent 

10  10  F  K 
8  75% of X  25% of X  X 
Total  40  J  100 
J + 40 = 100.
J = 60.
10 + 75% of X = 40.
75% of X = 30.
X = 40.
25% of X = 10.
F + 25% of X = J
F + 10 = J
F = 50.
K = 10 + 50 = 60.
Here K = Students who have 10 grade points.
Students  Intelligent  Not intelligent 

10  10  50  60 
8  30  10  40 
Total  40  60  100 
Q. 80% of lights in a certain hostel are switched on.40% of the light that are supposed to be off are actually on and 10% of light that are supposed to be on are actually off.
What % of the lights that are on,are supposed to be off.
Solution :
As per given data –
Light  On  Off 

Supposed on  E  10% of K  K 
Supposed off  40% of L  H  L 
total  80  J  100 
40% of L = 0.4 L
10% of K = 0.1 K
H = 60% of L = 0.6 L
E = 90% of K = 0.9 K
J =100 – 80 = 20.
0.9 K + 0.4 L = 80.
0.1 K + 0.6 L = 20.
9 K + 4 L = 800.
K + 6 L = 200.
K / (4800 + 800) = L / (800 – 1800) = (1) / (54 – 4)
K / (4000) = L /(1000) = (1) / 50.
K = 80.
L = 20.
 On  Off  Total 
Suppose On  72  8  80 
Supposed Off  8  12  20 
Total  80  20  100 
Our question is, what percent of swiches are on supposed to be off.
So required percentage = ( 8 / 80 ) x 100%
= 10%.
3 Variables
Suppose 3 variables are X,Y & Z.
X or Y or Z = A+B+C+D+E+F+G
Exactly one among X,Y,Z = a+b+c.
Exactly two among = d+e+f.
At least one among = a+b+c+d+e+f+g.
At least two among = d+e+f+g
At least one
A U B U C = A+B+C+ (A&B&C) (A&B+B&C+A&C)
At least two
(A&B )+(B&C)+(A&C) – 2(A&B&C)
Exactly one
A+B+C+3(A&B&C) – 2(A&B + B&C + C&A)
Exactly two
(A&B )+(B&C)+(C&A) – 3 (A&B&C)
Q. In a CAT mock paper there were 3 sections. Out of them 33 cleared the cutoff in section 1, 34 studnets cleared the cut off in section 2 and 32 cleared the cutoff in section 3. 10 cleared cut off of section 1 & 2 , 9 cleared cut off of in section 2 & 3, 8 cleared the cut off in section 1 &3.The number of people who cleared each section alone was equal and was equal 21 for each section.
1. How many cleared all 3 sections.
2. How many cleared only one of 3 sections.
3. The ratio of number of students clearing the cut off in one or more of sections to number of students clearing the cutoff in section 1 alone.
Solution :
Section 1 = 33
Section 2 =34
Section 3 =32.
33 = 21+x+(10x)+(8x).
x = 6.
1. 6
2. 21+21+21 = 63.
3. (21+21+21+6+4+3+2)/21 = 78/21.
Q. BCCI conducted a meeting.100 people attended the meeting out of which 60 like Virat Kohli,70 like Dhoni & 80 like Rohit Sharma.
1. Find the minimum & maximum number of people who like all three players.
Solution :
A+B+C+D = 100.
A= people who like only 1 player.
B= people who like 2 players.
C= people who like all 3 players.
D= people who don’t like any player.
Given A+2B+3C = 210.
To get C(max) we need to take A=0,B=0,D=0.
3 C = 210.
C = 70.
But people who like virat is only 60, so if we consider these 60 people must like all players.
Hence maximum number of people who like all 3 players is 60.
To get minimum no of people who like all 3 players.
Let suppose people who like Rohit(80) are person no 1 to person 80.(1 – 80)
People who like Dhoni(70) are person no 31 to 100.(31 – 100)
People who like Virat (60) are (1 to 30) & (81100) & rest 10 persons from [31,80].
So minimum 10 people are there who like all 3 players.
2. Find the minimum and maximum number of person who like exactly two players.
Solution:
To get maximum numbers of person who like exactly two players:
Rohit(80) & Dhoni(70) hence 70.
Rest 20 people can also like exactly two players.
Total =70+20 = 90.
Minimum number of person who like exactly two people 0.
3. Find the maximum and minimum number of people who like exactly one player.
Minimum no of people who like exactly one player = 0.
Maximum no of people who like exactly 1 player
2 x Amax = Cmax
Amax = 60 / 2 = 30.
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