If $A : B : C = 2 : 3 : 4$, then the ratio $\frac{A}{B} : \frac{B}{C} : \frac{C}{A}$ is equal to

### Concept & Proportional Ratios To find a complex ratio, substitute the parts of the initial ratio directly as values. When fractions appear in a ratio, multiply all terms by the Least Common Multiple (LCM) of the denominators to convert them into integers. ### Step-by-Step Solution 1. **Given:** $A : B : C = 2 : 3 : 4$. 2. We can assume the values to be $A = 2k$, $B = 3k$, and $C = 4k$. Since we are finding another ratio, the constant $k$ will cancel out, so we can simply use $A = 2$, $B = 3$, and $C = 4$. 3. Substitute these values into the required ratio $\frac{A}{B} : \frac{B}{C} : \frac{C}{A}$: $$\frac{2}{3} : \frac{3}{4} : \frac{4}{2}$$ $$\frac{2}{3} : \frac{3}{4} : 2$$ 4. To eliminate the denominators (3 and 4), find their LCM, which is 12. 5. Multiply each term by 12: $$\left(\frac{2}{3} \cdot 12\right) : \left(\frac{3}{4} \cdot 12\right) : (2 \cdot 12)$$ $$8 : 9 : 24$$ ### Exam Strategy & Shortcut Always substitute the ratio components directly as raw numbers (ignoring variables) for speed. Calculate the LCM of denominators in your head and multiply across in one step. ### Common Pitfall A frequent error is forgetting to multiply the whole number part (the third term, 2) by the LCM, resulting in $8 : 9 : 2$ instead of $8 : 9 : 24$. Always multiply every term across the entire ratio. ### Final Answer Therefore, the correct answer is **$8 : 9 : 24$**.