If $A : B = \frac{1}{2} : \frac{1}{3}$ and $B : C = \frac{1}{2} : \frac{1}{3}$, then $A : B : C$ is equal to

### Concept & Simplifying Fractional Ratios Ratios expressed as fractions must first be simplified into whole numbers before combining them. You do this by cross-multiplying or multiplying both sides of the ratio by the least common multiple of the denominators. ### Step-by-Step Solution 1. Simplify the first ratio, $A : B = \frac{1}{2} : \frac{1}{3}$: Multiply both fractions by the LCM of $2$ and $3$, which is $6$. $A : B = \left(\frac{1}{2} \times 6\right) : \left(\frac{1}{3} \times 6\right)$ $A : B = 3 : 2$ 2. Simplify the second ratio, $B : C = \frac{1}{2} : \frac{1}{3}$: Using the same logic, multiply by $6$. $B : C = 3 : 2$ 3. Now we have two standard ratios: $A : B = 3 : 2$ $B : C = 3 : 2$ 4. To combine them into $A : B : C$, equalize the common term $B$ (values $2$ and $3$). The LCM is $6$. 5. Scale $A : B$ to make $B = 6$. Multiply by $3$: $A : B = (3 \times 3) : (2 \times 3) = 9 : 6$ 6. Scale $B : C$ to make $B = 6$. Multiply by $2$: $B : C = (3 \times 2) : (2 \times 2) = 6 : 4$ 7. Combine the ratios since the $B$ terms match: $A : B : C = 9 : 6 : 4$ ### Exam Strategy & Shortcut For any ratio in the form $\frac{1}{x} : \frac{1}{y}$, the simplified whole number ratio is always just $y : x$. Thus, $\frac{1}{2} : \frac{1}{3}$ instantly becomes $3 : 2$. Then, use the quick "N" multiplication trick on $3:2$ and $3:2$ $\Rightarrow (3 \times 3) : (2 \times 3) : (2 \times 2) \Rightarrow 9:6:4$. ### Common Pitfall A standard error is attempting to equalize the middle term while the ratios are still in fractional form, which makes the arithmetic complicated and highly error-prone. Always convert to whole numbers first. ### Final Answer Therefore, the correct answer is **$9 : 6 : 4$**.