Three times the first of three consecutive odd integers is 3 more than twice the third integer. What is the value of the third integer?

Introduction / Context:
This is an algebraic puzzle involving three consecutive odd integers. The relationship between the first and third integers is given in terms of a linear equation. Our job is to translate the wording into algebra, solve for the first odd integer, and then find the third. This type of problem is common in aptitude exams and helps develop equation forming skills.

Given Data / Assumptions:
- There are three consecutive odd integers.
- Let the first integer be some odd number n.
- The second integer is then n + 2 and the third is n + 4.
- Three times the first integer is 3 more than twice the third integer.
- We need to find the third integer.

Concept / Approach:
Consecutive odd integers can be represented as n, n + 2, and n + 4. The condition "three times the first is 3 more than twice the third" becomes a linear equation connecting n and n + 4. Solving this equation gives n, and then the third integer is easily calculated as n + 4.

Step-by-Step Solution:
Step 1: Let the first odd integer be n.Step 2: Then the second odd integer is n + 2 and the third odd integer is n + 4.Step 3: The condition in the problem states that three times the first integer equals three more than twice the third integer.Step 4: Translate this into an equation: 3n = 2(n + 4) + 3.Step 5: Expand the right hand side: 2(n + 4) + 3 = 2n + 8 + 3 = 2n + 11.Step 6: The equation is now 3n = 2n + 11.Step 7: Subtract 2n from both sides to get n = 11.Step 8: Therefore, the first odd integer is 11 and the third odd integer is n + 4 = 11 + 4 = 15.

Verification / Alternative check:
By our solution, the three consecutive odd integers are 11, 13, and 15. Three times the first integer is 3 * 11 = 33. Twice the third integer is 2 * 15 = 30. The statement says that three times the first is 3 more than twice the third, which would require 33 = 30 + 3. This is true, so the integers satisfy the condition given in the problem. Therefore, 15 is confirmed as the correct third integer.

Why Other Options Are Wrong:
Options such as 12, 13, 14, or 17 are either not odd or do not arise as the third member of a set of three consecutive odd integers that satisfy the equation. For instance, if the third integer were 13, then the first would be 9, and three times 9 is 27 while twice 13 plus 3 is 29, which does not match. Similar checks show that these options fail the given condition.

Common Pitfalls:
One common error is to misrepresent consecutive odd integers as n, n + 1, n + 2 instead of n, n + 2, n + 4. Another is to misinterpret the phrase "3 more than twice the third" and write 3n = 3(2(n + 4)) or a similar incorrect expression. Carefully translating the language into an accurate equation is essential for solving such problems correctly.

Final Answer:
The third odd integer is 15.