A survey shows that `63%` of the Americans like cheese where as `76%` like apples. If `x%` of the Americans like both cheese and apples then find x.

Options

A. `x=39`
B. `x=63`
C. `39 le x le 63`
D. none of these

Answer

To solve the problem, we will use the principle of inclusion-exclusion for sets. Let's break down the solution step by step.

### Step 1: Define the Sets
Let:
- Set A = the set of Americans who like cheese.
- Set B = the set of Americans who like apples.

From the problem, we know:
- |A| = 63% (the percentage of Americans who like cheese)
- |B| = 76% (the percentage of Americans who like apples)

### Step 2: Understand the Union of Sets
We are interested in finding the percentage of Americans who like both cheese and apples, which we denote as |A ∩ B| = x%.

The total percentage of Americans is 100%, so we can express the union of sets A and B as:
\[ |A ∪ B| = |A| + |B| - |A ∩ B| \]

### Step 3: Substitute the Known Values
We know that:
- |A| = 63% = 0.63
- |B| = 76% = 0.76
- |A ∪ B| = 100% = 1

Now substituting these values into the union formula:
\[ 1 = 0.63 + 0.76 - x \]

### Step 4: Solve for x
Now, we can rearrange the equation to solve for x:
\[ x = 0.63 + 0.76 - 1 \]
\[ x = 1.39 - 1 \]
\[ x = 0.39 \]

### Step 5: Convert x to Percentage
To express x as a percentage, we multiply by 100:
\[ x = 0.39 \times 100 = 39\% \]

### Final Answer
Thus, the percentage of Americans who like both cheese and apples is **39%**.

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