The average of three consecutive odd numbers is 52 more than one third of the largest of these numbers. What is the smallest of the three odd numbers?

Introduction / Context:
This problem uses consecutive odd numbers, averages, and a relation involving a fraction of the largest number. It tests the learner's ability to express consecutive odd numbers algebraically and to work with averages and linear equations. Such questions are common in aptitude tests because they combine simple algebra with careful reading of the given condition.

Given Data / Assumptions:

• There are three consecutive odd numbers.
• Their average is 52 more than one third of the largest number.
• All numbers are odd integers.
• We must find the smallest of the three numbers.

Concept / Approach:
Consecutive odd numbers can be written as x, x + 2, and x + 4, where x is the smallest odd integer. Their average is the sum divided by three. The largest number is x + 4, so one third of the largest is (x + 4) / 3. The condition links these two expressions. We form an equation, solve for x, and then identify the smallest number. This linear equation method is standard for such problems.

Step-by-Step Solution:
Let the three consecutive odd numbers be x, x + 2, and x + 4, where x is odd.Their average is (x + (x + 2) + (x + 4)) / 3.Simplify the numerator: x + x + 2 + x + 4 = 3x + 6.So the average is (3x + 6) / 3 = x + 2.The largest number is x + 4.One third of the largest number is (x + 4) / 3.According to the question, average = (x + 4) / 3 + 52.So we have x + 2 = (x + 4) / 3 + 52.Multiply both sides by 3 to clear the denominator: 3x + 6 = x + 4 + 156.Simplify the right side: x + 4 + 156 = x + 160.So 3x + 6 = x + 160.Subtract x from both sides: 2x + 6 = 160.Subtract 6: 2x = 154.Divide by 2: x = 77.Thus, the smallest odd number is 77.

Verification / Alternative check:
The three consecutive odd numbers are 77, 79, and 81.Average = (77 + 79 + 81) / 3 = 237 / 3 = 79.Largest number is 81, so one third of largest = 81 / 3 = 27.One third of largest plus 52 is 27 + 52 = 79, which matches the average.The condition is satisfied, confirming that 77 is correct.

Why Other Options Are Wrong:
If we start with 75, 79, or 81 as the smallest number, the average and one third of the largest will not differ by exactly 52.Only 77 leads to three consecutive odd numbers that satisfy the given relation.

Common Pitfalls:
Some learners mistakenly take x, x + 1, and x + 2 as consecutive numbers instead of consecutive odd numbers, which breaks the logic.Another common mistake is mishandling the term one third of the largest and writing it as (largest) / 3 incorrectly in the equation.Errors when clearing the denominator by multiplying both sides by 3 can also lead to incorrect answers.

Final Answer:
The smallest of the three consecutive odd numbers is 77.