The average age of 20 students in a group is 21 years. Two students leave the group and one new student joins, after which the average age becomes 20 years. If the age of one student who left the group is 26 years and the age of the student who joined is 20 years, what is the age (in years) of the other student who left the group?

Introduction / Context:
This question explores how changes in group membership affect the average age. When some members leave and others join, the total sum of ages changes. Using averages before and after the change, we can determine the age of an unknown student who left the group.

Given Data / Assumptions:
• Initial number of students = 20.• Initial average age = 21 years.• Two students leave and one new student joins.• New number of students = 19.• New average age = 20 years.• Age of one student who left = 26 years.• Age of the student who joined = 20 years.• Age of the second student who left is unknown and needs to be found.

Concept / Approach:
Average age multiplied by the number of students gives the total sum of ages. We use the initial average to find the initial total sum. After two students leave and one joins, we use the new average to find the new total sum. By subtracting the known leaving and entering ages from the totals, we can isolate and compute the unknown age of the other student who left.

Step-by-Step Solution:
Step 1: Initial total age of 20 students = 20 * 21 = 420 years.Step 2: After two students leave and one joins, the group has 19 students with average age 20 years.Step 3: New total age of 19 students = 19 * 20 = 380 years.Step 4: Let the age of the second student who left be x years.Step 5: The new total age can also be written as: initial total age minus ages of two students who left plus age of new student.Step 6: So 380 = 420 - 26 - x + 20.Step 7: Simplify the right side: 420 - 26 = 394, then 394 + 20 = 414.Step 8: Therefore, 380 = 414 - x.Step 9: Rearranging, x = 414 - 380 = 34.

Verification / Alternative check:
Check with the found value x = 34. Total age of the two students who left = 26 + 34 = 60. So from the original total 420, after removing 60 we get 360. Adding the age of the new student (20) gives 380, which equals 19 * 20, matching the new average. This confirms that the age of the second student who left is 34 years.

Why Other Options Are Wrong:
20, 22, 16: Substituting any of these values for x will not produce the correct new total of 380. The resulting average will differ from 20 years, so they do not satisfy the conditions.

Common Pitfalls:
Students sometimes confuse the direction of change in total age or forget that the group size changes from 20 to 19. Others may incorrectly subtract or add the wrong ages. Carefully organizing the equation for the new total age prevents these mistakes.

Final Answer:
The age of the other student who left the group is 34 years.