** **

**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