If $a : b : c = 2 : 3 : 4$, then $\frac{1}{a} : \frac{1}{b} : \frac{1}{c}$ is equal to

### Concept & Reciprocal Ratios To find the ratio of the reciprocals of given numbers, write the reciprocals as fractions and then multiply each fraction by the Least Common Multiple (LCM) of their denominators to clear the fractions and obtain an integer ratio. ### Step-by-Step Solution - **Given:** $a : b : c = 2 : 3 : 4$. - **Form reciprocal ratio:** $\frac{1}{a} : \frac{1}{b} : \frac{1}{c} = \frac{1}{2} : \frac{1}{3} : \frac{1}{4}$. - **Find LCM:** Find the Least Common Multiple of the denominators $2$, $3$, and $4$. The LCM is $12$. - **Clear fractions:** Multiply each term of the ratio by $12$. - First term: $\frac{1}{2} \times 12 = 6$ - Second term: $\frac{1}{3} \times 12 = 4$ - Third term: $\frac{1}{4} \times 12 = 3$ - **Result:** The integer ratio is $6 : 4 : 3$. ### Exam Strategy & Shortcut A fast shortcut to find the reciprocal ratio of $A : B : C$ is to hide one number and multiply the other two. - For $a$: hide 2, multiply $3 \times 4 = 12$. - For $b$: hide 3, multiply $2 \times 4 = 8$. - For $c$: hide 4, multiply $2 \times 3 = 6$. - Ratio is $12 : 8 : 6$, which simplifies to $6 : 4 : 3$. ### Common Pitfall A classic trap is to assume that the reciprocal ratio simply reverses the order of the original numbers (i.e., choosing $4 : 3 : 2$). This is false for three or more numbers. ### Final Answer Therefore, the correct answer is **$6 : 4 : 3$**.