The sum of two natural numbers is 55. Their highest common factor (HCF) is 5 and their least common multiple (LCM) is 120. Find the sum of the reciprocals of the two numbers in simplest fractional form.

Introduction / Context:
This question cleverly combines the ideas of HCF, LCM, and reciprocals of numbers. Instead of directly finding the numbers, we can use the relationship between HCF, LCM, and the product of two numbers to compute the sum of their reciprocals in a quick and elegant way. This type of reasoning is useful in higher level arithmetic and algebra problems.

Given Data / Assumptions:

• Let the numbers be x and y.
• x + y = 55.
• HCF(x, y) = 5.
• LCM(x, y) = 120.
• We need the value of 1/x + 1/y.

Concept / Approach:
For two numbers x and y, there is a well known identity: x * y = HCF(x, y) * LCM(x, y). Once we know the product x * y and the sum x + y, we can find the sum of reciprocals easily: 1/x + 1/y = (x + y) / (x * y). We therefore compute the product from the HCF and LCM, and then form the simple fraction with the given sum.

Step-by-Step Solution:
Given HCF = 5 and LCM = 120. Compute product: x * y = 5 * 120 = 600. Given sum: x + y = 55. Sum of reciprocals: 1/x + 1/y = (x + y) / (x * y). Substitute values: 1/x + 1/y = 55 / 600. Simplify the fraction: divide numerator and denominator by 5. 55 / 600 = 11 / 120. Therefore, the sum of the reciprocals is 11/120.

Verification / Alternative check:
We can go further and find x and y explicitly to check. Since HCF is 5, let x = 5a and y = 5b, with HCF(a, b) = 1. The product x * y = 600 implies 25ab = 600, so ab = 24. Also, x + y = 5a + 5b = 55, so a + b = 11. Co prime factors of 24 that sum to 11 are a = 3 and b = 8. Then x = 15 and y = 40. Their reciprocals are 1/15 and 1/40. Sum = (8 + 3) / 120 = 11 / 120, confirming our earlier result.

Why Other Options Are Wrong:
55/601, 601/55, 120/11, and 5/120 do not match the correctly derived expression (x + y) / (x * y). They may look similar or inverted, but they do not come from the established product and sum relationship.

Common Pitfalls:
Some learners attempt to guess x and y without using the HCF and LCM product identity, which can be slow and error prone. Others mistakenly add HCF and LCM or confuse 1/x + 1/y with 1/(x + y). Remember that the key formula for reciprocals is (x + y) / (x * y) and that the product can be obtained directly from HCF and LCM for two numbers.

Final Answer:
11/120