# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5/11

Exercises: Try Out - Set 5: Real exam simulation to sharpen your skills and build confidence. - Problem 11

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## Exercise 11

### Question

export const metadata = {
  title: "Problem 11",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

There are <InlineMath math="700" /> students on a campus, <InlineMath math="350" /> students choose sports activities, and <InlineMath math="450" /> students choose arts activities. If <InlineMath math="P" /> is the maximum number of students who participate in both and <InlineMath math="Q" /> is the minimum number of students who participate in both, then the difference between <InlineMath math="P" /> and <InlineMath math="Q" /> is ... people.


### Choices

- [ ] $$350$$
- [x] $$250$$
- [ ] $$150$$
- [ ] $$100$$
- [ ] $$75$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 11",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given:
- Total students <InlineMath math="= 700" />.
- Students choosing sports <InlineMath math="= 350" />.
- Students choosing arts <InlineMath math="= 450" />.

We want to find the value of <InlineMath math="P" /> (maximum intersection) and <InlineMath math="Q" /> (minimum intersection).

1. **Finding <InlineMath math="P" /> (Maximum Intersection)**
   The maximum number of students participating in both occurs if one set is a subset of the other (or as many members of the smaller set as possible are included in the larger set).
   <MathContainer>
     <BlockMath math="P = \min(350, 450) = 350 \text{ students}" />
   </MathContainer>

2. **Finding <InlineMath math="Q" /> (Minimum Intersection)**
   The minimum number of students participating in both occurs if the distribution of students is as spread out as possible so that the union of the two sets is maximized (equal to the total universe, which is <InlineMath math="700" />).
   Using the Inclusion-Exclusion Principle:
   <MathContainer>
     <BlockMath math="\text{Intersection} = (\text{Sports} + \text{Arts}) - \text{Total}" />
     <BlockMath math="Q = (350 + 450) - 700" />
     <BlockMath math="Q = 800 - 700 = 100 \text{ students}" />
   </MathContainer>

3. **Finding the Difference Between <InlineMath math="P" /> and <InlineMath math="Q" />**
   <MathContainer>
     <BlockMath math="P - Q = 350 - 100 = 250 \text{ students}" />
   </MathContainer>

Thus, the difference between <InlineMath math="P" /> and <InlineMath math="Q" /> is <InlineMath math="250" /> students.


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