The sides of a triangle are in the ratio of $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$. If the perimeter is 52 cm, then the length of the smallest side is

### Concept & Fractional Ratios When a ratio is presented in fractions, it is mathematically much cleaner to convert it into a ratio of whole integers by multiplying all terms by the Least Common Multiple (LCM) of their denominators. ### Step-by-Step Solution * **Given:** The ratio of the sides is $\frac{1}{2} : \frac{1}{3} : \frac{1}{4}$ and the total perimeter is 52 cm. * **Convert Ratio to Integers:** Find the LCM of the denominators 2, 3, and 4, which is 12. Multiply each fraction in the ratio by 12: $(\frac{1}{2} \times 12) : (\frac{1}{3} \times 12) : (\frac{1}{4} \times 12)$ This simplifies to a clean integer ratio of $6 : 4 : 3$. * **Set up Equation:** Let the physical sides be $6x$, $4x$, and $3x$. $\text{Perimeter} = 6x + 4x + 3x = 13x$ $13x = 52$ $x = \frac{52}{13} = 4$ * **Find Smallest Side:** The smallest side mathematically corresponds to the smallest ratio part, which is 3. $\text{Smallest side} = 3x = 3 \times 4 = 12\text{ cm}$. ### Exam Strategy & Shortcut Once you have formulated the integer ratio $6:4:3$, simply sum the parts ($6+4+3=13$). Divide the total perimeter by this sum ($52 / 13 = 4$). To find the specific smallest side, just multiply this base multiplier by the smallest ratio component ($4 \times 3 = 12$). ### Common Pitfall Attempting to work directly with the fractions to solve for the perimeter without normalizing them first. This method is much slower and highly prone to arithmetic mistakes involving common denominators. ### Final Answer Therefore, the correct answer is **12 cm**.