There were 24 students in a class. One student who was 18 years old left the class and was replaced by a new student. As a result, the average age of the class was lowered by 1 month. What is the age, in years, of the new student?

Introduction / Context:
This question deals with averages and how they change when one data point is replaced by another. It is a standard example from the topic of average age and is often seen in quantitative aptitude exams. The key idea is to understand how total age is affected by the exit of one student and the entry of a new one, and how this translates into a change in the average.

Given Data / Assumptions:

• Initially, there are 24 students in the class.
• One student aged 18 years leaves the class.
• A new student joins the class in place of the student who left.
• The average age of the class decreases by 1 month, which is 1 / 12 of a year.
• We must find the age of the new student in years.

Concept / Approach:
The average age is the total age divided by the number of students. Since the number of students remains 24, the change in the average gives us direct information about the change in total age. A decrease of 1 month in average age means a decrease of 1 month multiplied by 24 in the total age. This decrease is caused by replacing the 18 year old student with the new student. We equate the change in total age to the difference in ages of the leaving and entering students to solve for the unknown age.

Step-by-Step Solution:
Step 1: Let the original average age of the 24 students be A years. Step 2: Then the original total age of the class is 24A years. Step 3: When the 18 year old student leaves and the new student of age x years joins, the number of students remains 24. Step 4: The new average age becomes A minus 1 month, that is A - 1 / 12 years. Step 5: The new total age of the class is 24 * (A - 1 / 12) = 24A - 2 years, because 24 * (1 / 12) = 2. Step 6: The new total age is also equal to the old total age minus 18 plus x, that is 24A - 18 + x. Step 7: Equate these two expressions: 24A - 18 + x = 24A - 2. Step 8: Cancel 24A from both sides to obtain -18 + x = -2. Step 9: Add 18 to both sides to get x = -2 + 18 = 16. Step 10: Therefore, the age of the new student is 16 years.

Verification / Alternative check:
We can interpret the result directly. Since the average decreased by 1 month for 24 students, the total age decreased by 2 years. This decrease must come from replacing the 18 year old student with the new student, so the new student must be 2 years younger than the old one. Thus the new student age is 18 - 2 = 16 years, which matches our algebraic solution. This verifies that the answer is consistent.

Why Other Options Are Wrong:
Option A (14 years) would represent a decrease of 4 years for the one student change, leading to a drop in average of 4 / 24 years, which is greater than 1 month.
Option B (15 years) would give a 3 year difference and thus a larger decrease in the average than the stated 1 month.
Option D (17 years) would produce only a 1 year change, causing a smaller drop in average than required, so it is also incorrect.

Common Pitfalls:
Students often forget to convert months into years, or they mistakenly multiply by the wrong number of students when computing the change in total age. Another frequent error is to treat the new average as A + 1 / 12 instead of A - 1 / 12. Carefully converting all units to years and writing out the total age equations step by step helps avoid these issues.

Final Answer:
The age of the new student is 16 years.