# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-6/18

Exercises: Try Out - Set 6: Real exam simulation to sharpen your skills and build confidence. - Problem 18

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## Exercise 18

### Question

export const metadata = {
  title: "Problem 18",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

In a district, a survey was conducted on <InlineMath math="300" /> teenagers. There are <InlineMath math="135" /> people who participate in sports activities, and <InlineMath math="255" /> people who participate in nature lover activities. If <InlineMath math="A" /> is the minimum number of teenagers who participate in both and <InlineMath math="B" /> is the maximum number of teenagers who participate in both. Then the difference between <InlineMath math="B" /> and <InlineMath math="A" /> is ... people.


### Choices

- [ ] $$90$$
- [ ] $$80$$
- [ ] $$70$$
- [ ] $$65$$
- [x] $$45$$

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 18",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given:
*   Total teenagers (<InlineMath math="S" />) = <InlineMath math="300" />
*   Sports activities (<InlineMath math="O" />) = <InlineMath math="135" />
*   Nature lover activities (<InlineMath math="C" />) = <InlineMath math="255" />

We are asked to find the difference between the maximum value (<InlineMath math="B" />) and the minimum value (<InlineMath math="A" />) of the intersection of these two sets (<InlineMath math="O \cap C" />).

#### Finding Minimum Value A
The minimum intersection occurs when the union of the two sets is maximized, which equals the total universe (<InlineMath math="300" />). This means everyone participates in at least one activity.
<MathContainer>
  <BlockMath math="n(O \cup C) = n(O) + n(C) - n(O \cap C)" />
  <BlockMath math="300 = 135 + 255 - A" />
  <BlockMath math="300 = 390 - A" />
  <BlockMath math="A = 390 - 300" />
  <BlockMath math="A = 90" />
</MathContainer>

#### Finding Maximum Value B
The maximum intersection occurs when the smaller set is a subset of the larger set. In this case, all sports participants also participate in nature lover activities.
<MathContainer>
  <BlockMath math="B = \min(n(O), n(C))" />
  <BlockMath math="B = \min(135, 255)" />
  <BlockMath math="B = 135" />
</MathContainer>

#### Calculating the Difference
<MathContainer>
  <BlockMath math="B - A = 135 - 90" />
  <BlockMath math="B - A = 45" />
</MathContainer>

Thus, the difference between <InlineMath math="B" /> and <InlineMath math="A" /> is <InlineMath math="45" /> people.


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