The sides of a triangle are in the ratio 1/2 : 1/3 : 1/4 and its perimeter is 104 cm. What is the length of the longest side (in cm)?

Introduction / Context:
This question involves the concept of side ratios in a triangle and the perimeter. When the sides are given in a certain ratio, we can introduce a common multiplying factor to convert the ratio into actual side lengths. Using the perimeter, which is the sum of all sides, we can determine that factor and hence each side. This is a common style of question in aptitude tests that combines ratios and basic algebra.

Given Data / Assumptions:

• The three sides of the triangle are in the ratio 1/2 : 1/3 : 1/4.
• The perimeter of the triangle is 104 cm.
• We must find the length of the longest side.

Concept / Approach:
If the side lengths are proportional to given ratios, we can write them as a common factor k multiplied by each ratio term. Here, the sides will be:
side 1 = (1/2) * k side 2 = (1/3) * k side 3 = (1/4) * k The perimeter is the sum of all three sides, so we can set up an equation involving k and solve for it. Once k is found, we can compute each side and identify the longest one.

Step-by-Step Solution:
Step 1: Let the common factor be k. Step 2: Express the sides in terms of k: s1 = k/2, s2 = k/3, s3 = k/4. Step 3: Write the perimeter equation: k/2 + k/3 + k/4 = 104. Step 4: Find a common denominator, which is 12. Step 5: Convert each fraction: k/2 = 6k/12, k/3 = 4k/12, k/4 = 3k/12. Step 6: Add these: (6k + 4k + 3k) / 12 = 13k / 12. Step 7: Set 13k / 12 = 104. Step 8: Multiply both sides by 12: 13k = 104 * 12. Step 9: Compute 104 * 12 = 1248, so 13k = 1248. Step 10: Divide by 13: k = 1248 / 13 = 96. Step 11: Now compute the sides: s1 = 96 / 2 = 48, s2 = 96 / 3 = 32, s3 = 96 / 4 = 24. Step 12: The longest side is therefore 48 cm.

Verification / Alternative check:
Verify the perimeter: 48 + 32 + 24 = 80 + 24 = 104 cm, which matches the given perimeter. The ratio check also works: 48 : 32 : 24 simplifies by dividing each term by 96 to (1/2) : (1/3) : (1/4) in fractional form, matching the given ratio when written with a common denominator. This confirms that the values are correct and consistent.

Why Other Options Are Wrong:

• 52, 32, 26, and 24: None of these match the largest side once the perimeter and side ratio are applied correctly. They come from incorrect arithmetic or misinterpretation of the ratio.

Common Pitfalls:
Common errors include adding the ratio terms incorrectly, forgetting to use a common denominator, or equating the sum directly to the perimeter without multiplying by a common factor. Students may also incorrectly assume the largest ratio term corresponds to the largest integer without computing the actual side lengths. Working with a clear algebraic representation and checking both the perimeter and the ratio avoids these mistakes.

Final Answer:
Hence, the length of the longest side of the triangle is 48 cm.