Question

A survey shows that 63% of Americans like cheese whereas 76% like apples. If \[x\% \] of the American like both cheese and apples, then
A. \[x = 39\]
B. \[x = 63\]
C. \[39 \le x \le 63\]
D. None of these

Answer 1

Hint: Here, we have to find the value of the variable \[x\] or the percent of Americans who like both cheese and apples. We will substitute the given values in the set formulas. We will then form an inequality and solve it further to get the range of \[x\].

Formula used: If \[A\] and \[B\] are two sets, then set formula is given by\[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\].
Complete step by step solution:
Let \[C\] denote the percentage of people who like cheese and let \[A\] denote the percentage of people who like apples.
Then according to question, we have
\[n\left( C \right) = 63\], \[n\left( A \right) = 76\] and \[n\left( {C \cap A} \right) = x\]
Here \[n\left( C \right)\] means the percentage of Americans who like cheese, \[n\left( A \right)\] means the percentage of Americans who like apples and \[n\left( {C \cap A} \right)\] denotes the percentage of Americans who like both cheese and apples.
Using the set formula, \[n\left( {A \cup B} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)\], for the given data, we get
\[ \Rightarrow n\left( {C \cup A} \right) = n\left( C \right) + n\left( A \right) - n\left( {C \cap A} \right)\]
Now, we will put the values in the formula of the set.
\[ \Rightarrow n\left( {C \cup A} \right) = 63 + 76 - x\]
On further simplification, we get
\[ \Rightarrow n\left( {C \cup A} \right) = 139 - x\] ……. \[\left( 1 \right)\]
We know the value of \[n\left( {C \cup A} \right)\] would be less than or equal to 100 i.e.
\[n\left( {C \cup A} \right) \le 100\]
Substituting the value of \[n\left( {C \cup A} \right)\] from equation \[\left( 1 \right)\] in the above inequality, we get
\[ \Rightarrow 139 - x \le 100\]
Subtracting 139 from sides of inequality, we get
\[\begin{array}{l} \Rightarrow 139 - 139 - x \le 100 - 139\\ \Rightarrow - x \le - 39\end{array}\]
Multiplying \[ - 1\] on both sides, we get
\[ \Rightarrow x \ge 39\]
We also know that \[n(C \cap A) \le n(C)\] and \[n(C \cap A) \le n(A)\] .
Therefore,
\[ \Rightarrow x \le 63\]
Hence, the range of \[x\] is \[39 \le x \le 63\]
Therefore, the correct option is option C.

Note: We need to know the basic property of inequality to find the range of \[x\] here. In mathematics, inequalities are basically used to compare the relative size of two or more values. We need to keep in mind that while multiplying or dividing an inequality by a negative number then the sign of inequality changes. If we don’t change the sign we will get the wrong range of \[x\].