https://file.aiquickdraw.com/mathbot/file/1726369157553my88kbov.jpg

Let's analyze the problem in the image.

In part (c), we are given the ratio of three currencies \( A : B : C = 6 : 7 : 8 \), and we need to find the equations that represent the relationships between the quantities. 

We can express these relationships using proportional equations:
1. \( \frac{A}{B} = \frac{6}{7} \) and \( \frac{A}{C} = \frac{6}{8} = \frac{3}{4} \)
2. \( \frac{B}{C} = \frac{7}{8} \)

Now, let's write these out in terms of equalities:
- \( A = \frac{6}{7}B \) or \( B = \frac{7}{6}A \)
- \( A = \frac{3}{4}C \) or \( C = \frac{4}{3}A \)
- \( B = \frac{7}{8}C \) or \( C = \frac{8}{7}B \)

Now, check each option given:
- **Option A**: \( B = 120 \cdot C \) → Incorrect
- **Option B**: \( B = \frac{1}{120} \cdot C \) → Incorrect
- **Option C**: \( C = 120 \cdot A \) → Incorrect
- **Option D**: \( C = 120 \cdot B \) → Incorrect
- **Option E**: \( A = 140 \cdot C \) → Incorrect
- **Option F**: \( A = 7 \cdot B \) → Incorrect
- **Option G**: \( B = \frac{7}{6} \cdot A \) → Correct (This matches the ratio \( \frac{B}{A} = \frac{7}{6} \)).

So, the correct answer is **G**.

Would you like more details on proportional relationships or how to solve similar problems? Here are some related questions:

1. What does it mean for two quantities to be proportional?
2. How can we express the relationship between three variables using ratios?
3. How do we convert a ratio into an equation?
4. How would changing the ratio affect the proportionality equations?
5. What are some real-life examples of proportional relationships?

**Tip**: When working with ratios, always try converting them into fractions to simplify your comparisons.

Assume that the ratio of currency A to currency B to currency C is 6:7:8 and that these currencies are proportional to each other. Find equations that represent relationships between pairs of these quantities.

In this problem, you will learn how to set up and solve proportional equations based on the ratio of currencies A, B, and C, given as 6:7:8. This exercise covers algebraic manipulations of ratios and is suitable for students in grades 8-10.

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