# Shortcuts in Venn Diagram-1 (Least value )

Questions to find: minimum percentage of interception / Least value Venn Diagram

According to a survey, at least 70% of people like apples, at least 75% like bananas and at least 80% like cherries. What is the minimum percentage of people who like all three?

Shortcut formula

2* Minimum percentage = Total(A)+ Total(B)+ Total(c) – 100 – ( 100- Total(A)) – (100- Total(B)) – (100- Total(c))

So here in this example:

2* Minimum percentage = 70+ 75+80 -100 -(30) – (25) – (20)

2* Minimum percentage = 225-100- 75

2* Minimum percentage = 50

Minimum percentage = 25%

Derivation of the shortcut:

70% eat apples means: it includes those who eat

Only Apple -OA
Apples, and Bananas - AB
Apples and Cherry - AC
Apples, Bananas and Cherry =ABC

75% eat Bananas means: it includes those who eat

Only Bananas -OB
Bananas and Apple - AB
Bananas and Cherry - BC
Apples, Bananas and Cherry =ABC

Total in the set=100% = OA + OB + OC + AB+AC+BC+ABC
70 = Only OA + AB + AC + ABC
75 = Only OB + AB + BC + ABC
80 = Only OC + BC + AC + ABC

70+75+80= OA + OB + OC + 2AB+2AC+2BC+3ABC
225= 100 + AB + AC + BC + 2ABC .........eq1

Now remaining
30% eat = BC
25% Eat = AC
20% Eat = AB

30+25+20 = AB + AC + BC = 75...........eq2

Substitute this in eq1

225= 100 + 75 + 2ABC
225-175 = 2 ABC
ABC = 25