The sum of two numbers is 55 and their H.C.F. and L.C.M. are 5 and 120 respectively. What is the sum of the reciprocals of these two numbers?

Introduction / Context:
This problem again uses the relationship between the H.C.F., L.C.M., and the product of two numbers, but it adds an extra twist by asking for the sum of reciprocals. It tests your ability to manipulate algebraic expressions involving sums, products, and reciprocals, which is a common theme in competitive examinations.

Given Data / Assumptions:

• Let the two positive integers be x and y.
• x + y = 55.
• H.C.F. of x and y is 5.
• L.C.M. of x and y is 120.
• We must find 1/x + 1/y.

Concept / Approach:
We use the key identity x * y = H.C.F. * L.C.M. to find the product x * y. Once we have the sum and product of the two numbers, the sum of the reciprocals can be written as (x + y) / (x * y). This avoids the need to find the numbers explicitly, although we can still find them as a check.

Step-by-Step Solution:
Step 1: Let the numbers be x and y.Step 2: Given x + y = 55.Step 3: Given H.C.F. = 5 and L.C.M. = 120.Step 4: Use the relation x * y = H.C.F. * L.C.M. = 5 * 120 = 600.Step 5: The sum of reciprocals is 1/x + 1/y = (x + y) / (x * y).Step 6: Substitute values: (x + y) / (x * y) = 55 / 600.Step 7: Simplify the fraction 55 / 600 by dividing numerator and denominator by 5.Step 8: 55 ÷ 5 = 11 and 600 ÷ 5 = 120, so the result is 11/120.

Verification / Alternative check:
We can also compute x and y explicitly by solving t^2 - 55t + 600 = 0.The roots are t = 40 and t = 15.Then 1/40 + 1/15 = (15 + 40) / (40 * 15) = 55 / 600 = 11/120. This matches our earlier result, confirming that the sum of the reciprocals is 11/120.

Why Other Options Are Wrong:
Options a (55/601) and b (601/55) are unrelated to the derived formula and represent incorrect manipulations of the sum and product. Option d (120/11) is the reciprocal of the correct answer and corresponds to a common mistake of inverting the fraction incorrectly. The correct direction of the reciprocal transformation must be maintained.

Common Pitfalls:
Some students mistakenly try to take reciprocals of individual numbers guessed from the answer choices rather than using the formula (x + y) / (x * y). Others invert the final fraction by mistake or forget to simplify it. Remember that the sum of reciprocals of two numbers with known sum and product has a direct and simple expression, which avoids unnecessary work.

Final Answer:
The sum of the reciprocals of the two numbers is 11/120.