Difficulty: Medium
Correct Answer: 11/120
Explanation:
Introduction:
This question combines number theory with fractions. It tests your understanding of the relationship between two numbers, their H.C.F., their L.C.M., and how to find the sum of their reciprocals. Such questions commonly appear in competitive exams to check conceptual clarity.
Given Data / Assumptions:
Concept / Approach:
Let the numbers be a and b. We know the identity: a * b = HCF(a, b) * LCM(a, b) Once we find the product a * b and know a + b, we can compute the sum of reciprocals: 1/a + 1/b = (a + b) / (a * b) This removes the need to explicitly determine a and b, making the calculation straightforward.
Step-by-Step Solution:
Step 1: Let the two numbers be a and b. Step 2: Given HCF(a, b) = 5 and LCM(a, b) = 120. Step 3: Compute the product: a * b = 5 * 120 = 600. Step 4: The sum a + b is given as 55. Step 5: Sum of reciprocals is 1/a + 1/b = (a + b) / (a * b). Step 6: Substitute values: 1/a + 1/b = 55 / 600. Step 7: Simplify the fraction: 55 / 600 = 11 / 120.
Verification / Alternative check:
We can also find the actual numbers. Since a * b = 600 and a + b = 55, the numbers are roots of x^2 - 55x + 600 = 0. Solving, we get x = 40 and x = 15. Then 1/40 + 1/15 = (3 + 8) / 120 = 11 / 120, which matches our earlier result.
Why Other Options Are Wrong:
55/601: This uses an incorrect denominator and does not follow from the product relation. 120/11: This is the reciprocal of the correct answer, not the required value. 601/55: This is a large number with no basis in the calculations. 5/120: This is a simplified incorrect fraction and does not match the computed sum of reciprocals.
Common Pitfalls:
A typical error is to try to compute the two numbers directly without using the identity for product. Another mistake is inverting the final fraction or mixing up numerator and denominator in the sum of reciprocals formula.
Final Answer:
The sum of the reciprocals of the two numbers is 11/120.
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