# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-7/10

Exercises: Try Out - Set 7: Real exam simulation to sharpen your skills and build confidence. - Problem 10

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## Exercise 10

### Question

export const metadata = {
  title: "Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

Given an ordered data set <InlineMath math="4, 4, a, b, \text{ and } c" />. The average of these five numbers is <InlineMath math="8" />. The largest data value is known to be <InlineMath math="3" /> times the median value. Which of the following relationships is correct between quantities <InlineMath math="P" /> and <InlineMath math="Q" /> based on the information provided?

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :---: |
| Median | <InlineMath math="4" /> |


### Choices

- [x] $$P > Q$$
- [ ] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$PQ = 32$$
- [ ] Cannot be determined

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/25/2025",
};

The given ordered data set is <InlineMath math="4, 4, a, b, c" />. Since the data is already ordered and there are an odd number of values (<InlineMath math="5" /> values), the median is the <InlineMath math="3" />rd value, which is <InlineMath math="a" />.

The largest data value is <InlineMath math="c" />, and it is <InlineMath math="3" /> times the median, so:
<MathContainer>
  <BlockMath math="c = 3a" />
</MathContainer>

The average of the five numbers is <InlineMath math="8" />:
<MathContainer>
  <BlockMath math="\frac{4 + 4 + a + b + c}{5} = 8" />
  <BlockMath math="8 + a + b + c = 40" />
  <BlockMath math="a + b + c = 32" />
</MathContainer>

Substitute <InlineMath math="c = 3a" /> into the equation:
<MathContainer>
  <BlockMath math="a + b + 3a = 32" />
  <BlockMath math="b + 4a = 32" />
  <BlockMath math="b = 32 - 4a" />
</MathContainer>

Since the data is ordered, <InlineMath math="4 \le a \le b \le c" /> holds true. We will find the bounds for the value of <InlineMath math="a" />.

#### Upper Bound of a
From the inequality <InlineMath math="a \le b" />:
<MathContainer>
  <BlockMath math="a \le 32 - 4a" />
  <BlockMath math="5a \le 32" />
  <BlockMath math="a \le 6.4" />
</MathContainer>

#### Lower Bound of a
From the inequality <InlineMath math="b \le c" />:
<MathContainer>
  <BlockMath math="32 - 4a \le 3a" />
  <BlockMath math="32 \le 7a" />
  <BlockMath math="a \ge \frac{32}{7}" />
  <BlockMath math="a \ge 4.57" />
</MathContainer>

Note also that <InlineMath math="a \ge 4" /> (from the data order). Since <InlineMath math="4.57 > 4" />, the stronger condition is <InlineMath math="a \ge 4.57" />.

#### Conclusion
The range of values for <InlineMath math="a" /> is <InlineMath math="4.57 \le a \le 6.4" />.

The value of <InlineMath math="P" /> is the median (<InlineMath math="a" />) and the value of <InlineMath math="Q" /> is <InlineMath math="4" />.
Since the minimum value of <InlineMath math="a" /> is <InlineMath math="4.57" />, which is greater than <InlineMath math="4" />, it is certain that <InlineMath math="P > Q" />.

Therefore, the correct relationship is <InlineMath math="P > Q" />.


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