LEARN MATH STEP BY STEP THROUGH VERY EASY PROCESS

WORD PROBLEM OF VENN DIAGRAM

__WORD PROBLEM__

**Example.- A) Out of 50 students in a class someone like either Chocolate
or Cake or both, 25 like Chocolate while 15 like both. Draw a Venn diagram and
find the number of students who like 1)
Cake, 2) Only Cake and 3) Only Chocolate.**

**Answer) Let decide C = { students who likes Chocolate }, M = { students who like Cake }**

**Then n(C) = 25**

**n (C U M) = number of students who like either chocolate or
milk or both = 50 and n(C ∩ M) = number of students who like both = 15 **

**Now, as per the rules n (C U M) =
n(M) + n(C) - n(C ∩ M)**

**Or, 50 = n(M) + 25 – 15**

**Or n(M) = 40**

**Number of students who like Cake n(M) = 40**

**Number of students who like both Chocolate and Cake n(C ∩ M) =
15**

**The number of students who like only Chocolate = n(M)
- n(C ∩ M) **

** = 40 – 15 **

** = 25**

**And the number of students who like Cake = n(C) - n(C ∩ M)**

** = 25 – 15 **

** = 10**

**If we show this obtained figure in the Venn diagram, then it
looks like -**

**In this diagram -**

**Total students of
class = 50**

**Number of Students
who like Chocolate = 25**

**Number of students
who like Cake = 10**

**Number of students
who like both Chocolate & Cake = 15**

**Example.-B) Out of a class of 120 students, 80 students likes a have math, 70
students like Physics and all like either math or physics or both, find the
number of students who like 1) both math physics & math, 2) only math, 3)
only physics **

**Answer) Let us assume –**

**M = { students who
like Math }, P = { students who like Physics }**

**M U P = { students
who like either math or physics }**

**M ∩ P = { student who like both }**

**As per the given condition, n(M) = 80,
n(P) = 70, n(M U P) = 120**

**Let, n(M ∩ P) = the number of
students who like both = B**

**Then the number of students who like only math = 80 – B**

**And the number of students who like only physics = 70 – B**

**So, the total number of students = 120 = (80 – B) + B + (70 –
B)**

**Or, 120 = 150 – B**

**Or, B = 150
– 120 = 30**

**1) so, the number of students who like both math &
physics = B = 30**

**2) the number of students who like only math = 80 – B
**

** = 80 – 30 = 50**

**3) the number of students who like only physics = 70 – B = 40 **

**
**

**If we show this obtained figure in the Venn
diagram, then it looks like -**

**In this diagram –**

**Total number of
students = 120**

**Number of students
who like Math = 50**

**Number of Students
who like physics = 40**

**Number of students
who like both = 30**

**Example.-C) Out of a group of 2000 people, 250 like to watch Tennis, 500
like to watch football and 150 watch both, find the number of people who like
to watch – 1) only tennis, 2) only
football, 3) either tennis or football
or both, 4) neither tennis nor
football **

**Answer) let us assume –**

**Let, U = { the
entire group }, T = { People who like
to watch tennis }, **

** F = { people who like to watch football }**

**As per the given
condition, n(T) = 250, n(F) = 500, n(T ∩ F) = 150**

**1) the number of people who watch only Tennis =**

** n(T) – n(T ∩ F)
= 250 – 150 = 100**

** **

**2) the number of
people who watch only football =**

** n(F) – n(T ∩ F) **

** = 500
– 150 = 350**

** **

**3) the number of
people who watch either tennis or football or both =**

** n (T U F) = n(T) + n(F) - n(T ∩ F)
**

** = 250 + 500 – 150 = 600**

**4) the number of people who watch neither tennis nor football**

**
**

** n(U) - n (T U F) = 2000 – 600 = 1400**