Question

A survey shows that 63% of the americans like cheese, whereas 76% like apples. If x% of the americans like both cheese and apples, then

Answer 1

Answer:

Step-by-step explanation:

Here,

Let the americans who like cheese be denoted by 'C' and who like Apples be denoted by 'A' and the total no. of people be 'U' respectively.

n(U) = 100%

n(C) = 63%

n(A) = 76%

n(C∩A) = x%

n(C∪A)^c = 0%

Now,

n(C∪A) = n(U) - n(C∪A)^c

n(C∪A) = 100% - 0%

n(C∪A) = 100%

Again,

n(C∪A) = n(C) + n(A) - n(C∩A)

100 = 63 + 76 - x

x = 139 - 100

x = 39%

Hence, 39% of Americans like both cheese and apples.

Answer 2

☆ Answer :

• Option 3) 39 ⩽ x ⩽63

___________________________

☆ Given :

• A Survey Concluded

%of Americans who like CHEESE = 63%

% of Americans who like APPLES = 76%

% of Americans who like BOTH = x %

___________________________

☆ To Find :

• % of American who like both i.e. value of x

___________________________

☆ Solution :

Let A and B denote the set of Americans who like Cheese and Apple respectively.

n(A) = 63 and n(B) = 76

According to De-Morgan's Law

⟹n(A U B ) = n(A) + n(B) - n(A∩B)

⟹ n(A U B ) = 63 + 76 - n(A ∩ B)

⟹ n(A ∩ B) = 139 - n(A U B)

But, n(A U B) ⩽ 100

⟹ -n(A U B) ⩾ -100

⟹ 139 - n(A U B) ⩾ 139 - 100 = 39

⟹ 39 ⩽ n(A ∩ B) . . . (i)

Again ,

A ∩ B ⊆ A and A ∩ B ⊆ B

• ∴ n(A ∩ B) ⩽ n(A) = 63

• and n(A ∩ B) ⩽ n(B) = 76

• ∴ n(A ∩ B) ⩽ 63 . . . (ii)

Combining eq (i) And (ii)

• 39 ⩽ n(A ∩ B) ⩽ 63

$\pink{ \huge\underline{\boxed{ \bf 39 \leqslant x \leqslant 63} }}$

$\large\mathfrak{ \text{W}hich \: \: is \: \: the \: \: required \: \: \text{ A}nswer.}$