Difficulty: Easy
Correct Answer: 48
Explanation:
Introduction:
This aptitude question uses proportional reasoning. The sides of a triangle are given in a ratio involving fractions, and the total perimeter is known. By converting the fractional ratio to a simpler whole number ratio, we can find the actual side lengths.
Given Data / Assumptions:
Concept / Approach:
Ratios involving fractions can be simplified by multiplying through by the least common multiple (LCM) of the denominators, which converts the ratio into whole numbers. Once we have a:b:c in simplest whole number form, we can write side lengths as ka, kb, kc, use the perimeter to find k, and then deduce each length. The largest of these is the longest side.
Step-by-Step Solution:
Original ratio: 1/2 : 1/3 : 1/4. LCM of denominators 2, 3, 4 is 12. Multiply each term by 12: (1/2)*12 = 6, (1/3)*12 = 4, (1/4)*12 = 3. So the side ratio becomes 6 : 4 : 3. Let the sides be 6k, 4k, and 3k. Perimeter = 6k + 4k + 3k = 13k = 104 cm. Solve for k: k = 104 / 13 = 8. Thus the sides are 6k = 48 cm, 4k = 32 cm, 3k = 24 cm. The longest side is 48 cm.
Verification / Alternative check:
Check that the perimeter is indeed 48 + 32 + 24 = 104 cm. Also, the largest ratio term was 6, and 6k gives 48 cm. This is consistent and respects the original fractional ratio when scaled back down.
Why Other Options Are Wrong:
Values 52, 32, and 26 do not preserve the 6 : 4 : 3 ratio when matched with the other sides under total perimeter 104 cm. For instance, 52 would require a different k and give a perimeter larger than 104. Only 48 cm fits all conditions simultaneously.
Common Pitfalls:
Mistakes include using 1/2, 1/3, 1/4 directly as side lengths or failing to find the correct LCM of denominators. Others sum the fractions incorrectly. Converting to simple integer ratios first is the safest method.
Final Answer:
The length of the longest side is 48 cm.
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