# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-5/6

Exercises: Try Out - Set 5: Real exam simulation to sharpen your skills and build confidence. - Problem 6

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## Exercise 6

### Question

export const metadata = {
  title: "Problem 6",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

Given the following data:
<InlineMath math="2, 3, 5, 7, 8, 9, 11" />

Two identical natural numbers are added to the data, resulting in a new data set with an average of <InlineMath math="9" />.

| <InlineMath math="P" /> | <InlineMath math="Q" /> |
| :---: | :---: |
| The difference between the median of the new data and the mean of the new data | <InlineMath math="-12" /> |


### Choices

- [x] $$P > Q$$
- [ ] $$P < Q$$
- [ ] $$P = Q$$
- [ ] $$P + Q = 1$$
- [ ] The information provided is not sufficient to decide one of the options.

### Answer & Explanation

export const metadata = {
  title: "Solution to Problem 6",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/23/2025",
};

We will find the value of the added numbers, determine the new median, and then compare the values of <InlineMath math="P" /> and <InlineMath math="Q" />.

#### Finding the Added Numbers

Let the two added natural numbers be <InlineMath math="x" />.
The initial data consists of <InlineMath math="7" /> numbers: <InlineMath math="2, 3, 5, 7, 8, 9, 11" />.
Sum of initial data:
<MathContainer>
  <BlockMath math="2 + 3 + 5 + 7 + 8 + 9 + 11 = 45" />
</MathContainer>

After adding <InlineMath math="2" /> numbers of value <InlineMath math="x" />, the count of data becomes <InlineMath math="7 + 2 = 9" />.
The new average is given as <InlineMath math="9" />.

<MathContainer>
  <BlockMath math="\bar{x}_{new} = \frac{\text{Sum of Initial Data} + 2x}{\text{Count of New Data}}" />
  <BlockMath math="9 = \frac{45 + 2x}{9}" />
  <BlockMath math="81 = 45 + 2x" />
  <BlockMath math="2x = 81 - 45" />
  <BlockMath math="2x = 36" />
  <BlockMath math="x = 18" />
</MathContainer>

So, the added number is <InlineMath math="18" />.

#### Determining the New Median

The new data after adding <InlineMath math="18" /> and <InlineMath math="18" /> is:
<InlineMath math="2, 3, 5, 7, 8, 9, 11, 18, 18" />

The data is already sorted. Since the number of data points <InlineMath math="n = 9" /> (odd), the median is the <InlineMath math="\frac{9+1}{2} = 5" />-th data point.

<MathContainer>
  <BlockMath math="\text{Median} = \text{5th Data} = 8" />
</MathContainer>

#### Calculating P Value

The value of <InlineMath math="P" /> is defined as the difference between the new median and the new mean.
The new mean is known to be <InlineMath math="9" />.

<MathContainer>
  <BlockMath math="P = \text{Median} - \text{Mean}" />
  <BlockMath math="P = 8 - 9" />
  <BlockMath math="P = -1" />
</MathContainer>

#### Comparing P and Q

Given <InlineMath math="Q = -12" />.

<MathContainer>
  <BlockMath math="P = -1" />
  <BlockMath math="Q = -12" />
</MathContainer>

Since <InlineMath math="-1 > -12" />, then:

<MathContainer>
  <BlockMath math="P > Q" />
</MathContainer>

#### Conclusion

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


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