If the sum of two numbers is 55, and the HCF (GCD) and LCM of these numbers are 5 and 120 respectively, then find the sum of the reciprocals of the two numbers. (Express your answer as a simplified fraction.)

Introduction / Context:
This question links the properties of HCF and LCM with reciprocals. Once we determine the two numbers from their sum, HCF, and LCM, we can easily compute the sum of their reciprocals. The key is to use the relationship between HCF, LCM, and the product of the numbers, and to factor out the HCF to obtain a simpler pair of coprime numbers.

Given Data / Assumptions:

• Sum of the two numbers: x + y = 55
• HCF(x, y) = 5
• LCM(x, y) = 120
• We need (1/x) + (1/y)

Concept / Approach:
Let x = 5a and y = 5b because 5 is the HCF. Then gcd(a, b) = 1. The LCM of x and y is 5ab, and we are told:
5ab = 120 ⇒ ab = 24
Also the sum gives:
x + y = 5a + 5b = 55 ⇒ a + b = 11
We need to find coprime integers a and b such that a + b = 11 and ab = 24. Then we can reconstruct x and y and find the sum of reciprocals.

Step-by-Step Solution:
Step 1: From LCM relation: ab = 120 / 5 = 24.Step 2: From sum relation: a + b = 55 / 5 = 11.Step 3: Find integer pairs (a, b) with product 24 and sum 11.Possible factor pairs of 24 are (1, 24), (2, 12), (3, 8), (4, 6).Among these, 3 + 8 = 11 and gcd(3, 8) = 1, so (a, b) = (3, 8) works.Step 4: Therefore, x = 5a = 15 and y = 5b = 40.Step 5: Sum of reciprocals:1/x + 1/y = 1/15 + 1/40 = (40 + 15) / 600 = 55 / 600 = 11 / 120.

Verification / Alternative check:
Check that the numbers 15 and 40 match the original conditions: HCF(15, 40) = 5, LCM(15, 40) = (15 * 40) / 5 = 120, and sum = 15 + 40 = 55. The reciprocal sum 1/15 + 1/40 simplifies to 11/120, confirming the answer.

Why Other Options Are Wrong:
13/125, 7/60, 14/57, 1/5: These fractions do not equal 1/15 + 1/40, and each would correspond to different number pairs that do not satisfy the given HCF, LCM, and sum conditions.

Common Pitfalls:
Attempting to guess x and y directly without using the structured approach with a and b.Forgetting to divide the sum and product by the HCF to move into the coprime pair (a, b).Not simplifying the final fraction when computing the sum of reciprocals.

Final Answer:
11/120