The sum of three consecutive odd integers is 93. What is the value of the middle odd integer?

Introduction / Context:
This question explores simple algebra using consecutive odd integers. Consecutive odd numbers follow a fixed pattern with a constant difference of 2 between them. By expressing these numbers in terms of a single variable, we can form an equation based on the given sum and solve it. This type of question is a staple in aptitude exams because it connects number patterns with algebraic expressions.

Given Data / Assumptions:

• The three numbers are consecutive odd integers.
• Their total sum is 93.
• The difference between any two successive odd integers is 2.
• We are asked specifically for the middle integer.

Concept / Approach:
Let the middle odd integer be M. Then the previous odd integer is M − 2 and the next odd integer is M + 2. These three numbers form a simple arithmetic sequence. Their sum is (M − 2) + M + (M + 2). This expression simplifies neatly and allows us to solve for M directly. This approach avoids guessing and works for any similar problem involving sums of consecutive odd or even numbers.

Step-by-Step Solution:
Let the middle odd integer be M. Then the three consecutive odd integers are M − 2, M and M + 2. Their sum is (M − 2) + M + (M + 2). Simplify: (M − 2) + M + (M + 2) = 3M. Given that the sum is 93, we have 3M = 93. Divide both sides by 3: M = 93 / 3 = 31. Therefore, the middle odd integer is 31.

Verification / Alternative check:
Once we obtain M = 31, the three consecutive odd integers are 29, 31 and 33. Their sum is 29 + 31 + 33 = 93. This matches the sum given in the question, so the solution is correct. You can also reason that the average of three numbers is their sum divided by 3. Since they are equally spaced, the middle value must equal this average, and 93 / 3 = 31 gives the same result directly.

Why Other Options Are Wrong:

• 33: If 33 were the middle number, the others would be 31 and 35, and the sum would be 99, not 93.
• 29: As the middle number, the sequence would be 27, 29 and 31, summing to 87.
• 27: This would give 25, 27 and 29, with a sum of only 81.
• 35: This would give 33, 35 and 37, which sum to 105.

Common Pitfalls:
Learners sometimes misidentify the pattern and use M, M + 1 and M + 2, which are consecutive integers, not consecutive odd integers. Another error is to forget that the sum of three numbers is three times their average, which is a quick shortcut here. Always check whether the difference between the numbers should be 1, 2, or some other fixed amount based on the wording of the question.

Final Answer:
The middle odd integer is 31.