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 find value of x .

Answer

To solve the problem, we will use the principle of inclusion-exclusion in set theory. Let's denote:

- \( N(A) \): the percentage of Americans who like cheese = 63%
- \( N(B) \): the percentage of Americans who like apples = 76%
- \( N(A \cap B) \): the percentage of Americans who like both cheese and apples = \( x \% \)
- \( N(A \cup B) \): the percentage of Americans who like either cheese or apples or both = 100%

According to the principle of inclusion-exclusion, we have:

\[ N(A \cup B) = N(A) + N(B) - N(A \cap B) \]

Substituting the known values into the equation:

\[ 100 = 63 + 76 - x \]

Now, let's simplify this equation step by step:

1. Combine the percentages of those who like cheese and apples:
\[ 63 + 76 = 139 \]

2. Substitute this back into the equation:
\[ 100 = 139 - x \]

3. Rearranging the equation to isolate \( x \):
\[ x = 139 - 100 \]

4. Calculate the value of \( x \):
\[ x = 39 \]

Thus, the percentage of Americans who like both cheese and apples is \( \boxed{39\%} \).