# Nakafa Framework: LLM

URL: https://nakafa.com/en/exercises/high-school/snbt/quantitative-knowledge/try-out/set-9/10

Exercises: Try Out - Set 9: 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: "12/26/2025",
};

If the ratio of the volumes of two cubes is <InlineMath math="1 : 27" />, then the ratio of the surface areas of the two cubes is...


### Choices

- [ ] $$1 : 3$$
- [ ] $$1 : 4$$
- [ ] $$1 : 8$$
- [x] $$1 : 9$$
- [ ] $$2 : 3$$

### Answer & Explanation

export const metadata = {
  title: "Solution for Problem 10",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "12/26/2025",
};

#### Finding the Side Length Ratio

The volume of a cube (<InlineMath math="V" />) is directly proportional to the cube of its side length (<InlineMath math="s" />). Given that the volume ratio is <InlineMath math="1 : 27" />.

<MathContainer>
  <BlockMath math="\frac{V_1}{V_2} = \left(\frac{s_1}{s_2}\right)^3" />
  <BlockMath math="\frac{1}{27} = \left(\frac{s_1}{s_2}\right)^3" />
  <BlockMath math="\frac{s_1}{s_2} = \sqrt[3]{\frac{1}{27}}" />
  <BlockMath math="\frac{s_1}{s_2} = \frac{1}{3}" />
</MathContainer>

So, the ratio of the side lengths of the two cubes is <InlineMath math="1 : 3" />.

#### Finding the Surface Area Ratio

The surface area of a cube (<InlineMath math="L" />) is directly proportional to the square of its side length.

<MathContainer>
  <BlockMath math="\frac{L_1}{L_2} = \frac{6s_1^2}{6s_2^2}" />
  <BlockMath math="\frac{L_1}{L_2} = \left(\frac{s_1}{s_2}\right)^2" />
  <BlockMath math="\frac{L_1}{L_2} = \left(\frac{1}{3}\right)^2" />
  <BlockMath math="\frac{L_1}{L_2} = \frac{1}{9}" />
</MathContainer>

Thus, the ratio of the surface areas of the two cubes is <InlineMath math="1 : 9" />.


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