Question 12

Number 12

At "Harapan Bangsa" High School, which has $800$ students, there are $400$ students who like Mathematics and $500$ students who like Physics.

If $x$ is the maximum number of people who like both and $y$ is the minimum number of people who like both, then the value of $x - y$ is ... people.

$1000$

$800$

$900$

$300$

$100$

Explanation

Given:

Total students ( $S$ ) = $800$ people.
Students who like Mathematics ( $M$ ) = $400$ people.
Students who like Physics ( $F$ ) = $500$ people.

We want to find the maximum value ( $x$ ) and minimum value ( $y$ ) of the intersection of these two sets (students who like both).

Finding Maximum Value x

The number of students who like both will be maximum if the smaller set is a subset of the larger set. Here, the number of students who like Mathematics ( $400$ ) is less than those who like Physics ( $500$ ). So, the maximum value of the intersection is the number of students in the smallest group.

x = 400

Finding Minimum Value y

The number of students who like both will be minimum if the distribution is spread out as much as possible so that the total union is maximum (equal to the total number of students).

Use the set formula: $n(M \cup F) = n(M) + n(F) - n(M \cap F)$ .

To get the minimum intersection, we assume all students ( $800$ ) like at least one of the two subjects (no one dislikes both).

800 = 400 + 500 - y

800 = 900 - y

y = 900 - 800

y = 100

Calculating the Difference

Finally, we calculate the difference between the maximum and minimum values.

x - y = 400 - 100 = 300

So, the value of $x - y$ is $300$ people.

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Number 12

Explanation

Finding Maximum Value x

Finding Minimum Value y

Calculating the Difference