Four consecutive even integers are such that the sum of the reciprocals of the first two is 11 divided by 60. What is the reciprocal of the third largest number among these four even integers?

Introduction / Context:
This question examines understanding of consecutive even integers, reciprocals and the ability to set up and solve a rational equation. It also checks whether the learner can interpret the phrase third largest correctly when the numbers are ordered. Such problems are common in number system and algebra sections of competitive exams.

Given Data / Assumptions:
- We have four consecutive even integers. - The sum of the reciprocals of the first two integers is 11 / 60. - We are asked to find the reciprocal of the third largest integer among the four.

Concept / Approach:
- Let the four consecutive even integers be n, n + 2, n + 4 and n + 6, where n is an even integer. - The first two integers are n and n + 2, so their reciprocals are 1 / n and 1 / (n + 2). - Use the given sum of reciprocals to form an equation and solve for n. - Once n is known, list the four integers in increasing order and identify the third largest integer, then compute its reciprocal.

Step-by-Step Solution:
Step 1: Write the condition on reciprocals: 1 / n + 1 / (n + 2) = 11 / 60. Step 2: Combine the fractions on the left side: ( (n + 2) + n ) / ( n * (n + 2) ) = (2n + 2) / ( n * (n + 2) ). Step 3: Factor the numerator as 2(n + 1), so the expression becomes 2(n + 1) / ( n * (n + 2) ). This equals 11 / 60. Step 4: Cross multiply: 2(n + 1) * 60 = 11 * n * (n + 2). Step 5: Simplify the left side to 120(n + 1). Expand the right side as 11(n^2 + 2n) = 11n^2 + 22n. Step 6: Set up the equation 120n + 120 = 11n^2 + 22n. Step 7: Rearrange to standard quadratic form: 11n^2 + 22n - 120n - 120 = 0 which simplifies to 11n^2 - 98n - 120 = 0. Step 8: Factor the quadratic. It can be written as (n - 10)(11n + 12) = 0. Step 9: The solutions are n = 10 or n = -12 / 11. Since n must be an integer and we are dealing with positive even integers in a typical aptitude context, we take n = 10. Step 10: The four consecutive even integers are therefore 10, 12, 14 and 16. Step 11: In increasing order, the smallest is 10 and the largest is 16. The third largest is 12 (the numbers from smallest to largest are 10, 12, 14, 16, so largest is 16, second largest is 14, third largest is 12). Step 12: The reciprocal of 12 is 1 / 12.

Verification / Alternative check:
Check the original condition with n = 10. Then 1 / 10 + 1 / 12 = (6 + 5) / 60 = 11 / 60, which matches the given value. The arithmetic is consistent, confirming that the chosen n is correct and that 1 / 12 is the required reciprocal.

Why Other Options Are Wrong:
Option B (2 / 14): This simplifies to 1 / 7, corresponding to a number 7 which is not part of the even sequence 10, 12, 14, 16. Option C (1 / 14): This is the reciprocal of 14, which is the second largest integer, not the third largest. Option D (2 / 13): This is not even a simple reciprocal of an integer and does not correspond to any of the four integers directly. Option E (1 / 16): This is the reciprocal of the largest number, whereas the question explicitly asks for the third largest.

Common Pitfalls:
- Misinterpreting third largest as third number instead of counting from the largest end. - Making algebraic mistakes when simplifying the rational equation. - Forgetting that the integers are even and consecutive, which restricts valid values of n.

Final Answer:
The reciprocal of the third largest integer in the sequence is 1/12.