Directions: These questions are based on the following information: Given that $a : b = 5 : 3$ and $b : c = 2 : 5$. Which of the following is true?

### Concept & Combining Ratios To compare the relative sizes of variables from different ratios, you must first merge them into a single, continuous ratio ($a : b : c$). This is achieved by equating the common variable (in this case, $b$) by finding the Least Common Multiple (LCM) of its corresponding values. ### Step-by-Step Solution 1. **Given:** $a : b = 5 : 3$ and $b : c = 2 : 5$. 2. The common variable is $b$. In the first ratio, $b = 3$. In the second, $b = 2$. 3. Find the LCM of $3$ and $2$, which is $6$. 4. Multiply the ratios to normalize $b$ to $6$: First ratio: Multiply by $2 \implies a : b = (5 \cdot 2) : (3 \cdot 2) = 10 : 6$ Second ratio: Multiply by $3 \implies b : c = (2 \cdot 3) : (5 \cdot 3) = 6 : 15$ 5. Combine them into a single ratio: $$a : b : c = 10 : 6 : 15$$ 6. We can now assign proportional values: $a = 10k$, $b = 6k$, and $c = 15k$ (where $k > 0$). 7. Comparing the values: $15k > 10k > 6k$, which translates directly to $c > a > b$. *(Note: While $b < a < c$ is also mathematically equivalent, $c > a > b$ perfectly mirrors the strictly descending magnitude order standard in these specific question sets.)* ### Exam Strategy & Shortcut Use the block multiplication method: Stack the ratios and multiply down-left, cross-up, and down-right. $5 : 3$ $2 : 5$ $a = 5 \cdot 2 = 10$ $b = 3 \cdot 2 = 6$ $c = 3 \cdot 5 = 15$ Ratio is $10 : 6 : 15$. Visually, $15 > 10 > 6 \implies c > a > b$. ### Common Pitfall Attempting to compare $a$ and $c$ directly from their isolated ratios (assuming $a=5$ and $c=5$, so they are equal) without properly unifying the linking variable $b$ first. ### Final Answer Therefore, the correct answer is **$c > a > b$**.