Four students are sitting in a row. The average age of the last three students is 20 years, and the average age of the first three students is 21 years. If the age of the first student is 26 years, what is the age (in years) of the last student?

Introduction / Context:
This question examines reasoning with overlapping averages. Four students are arranged in a row, and we are given the average ages of the first three and of the last three. Additionally, the age of the first student is known. The aim is to determine the age of the last student. Solving this requires translating the average information into equations and then eliminating the middle students to isolate the last student's age.

Given Data / Assumptions:

• There are four students in a row with ages a, b, c and d in order.
• Average age of the last three (b, c, d) = 20 years.
• Average age of the first three (a, b, c) = 21 years.
• Age of the first student a = 26 years.
• We must find the age of the last student d.

Concept / Approach:
From each average, we can find a total. The total age of the last three is 3 * 20 and of the first three is 3 * 21. By writing these totals as sums involving a, b, c and d, we can form two linear equations. Subtracting the equation for the first three from the equation for the last three eliminates b and c and leaves an expression involving only a and d. Substituting the known value of a then gives d directly.

Step-by-Step Solution:
Step 1: Represent the average of the last three students. (b + c + d) / 3 = 20. So b + c + d = 3 * 20 = 60. Step 2: Represent the average of the first three students. (a + b + c) / 3 = 21. So a + b + c = 3 * 21 = 63. Step 3: Use the known value of a. a = 26, so 26 + b + c = 63. Thus b + c = 63 − 26 = 37. Step 4: Use b + c in the equation for the last three. From b + c + d = 60 and b + c = 37, we get d = 60 − 37 = 23.
Verification / Alternative check:
We can test with a concrete example. Let b + c = 37; pick any b and c that add to 37, for example b = 17 and c = 20. Then a = 26, d = 23 from the calculation. Average of first three (26, 17, 20) = (26 + 17 + 20) / 3 = 63 / 3 = 21. Average of last three (17, 20, 23) = (17 + 20 + 23) / 3 = 60 / 3 = 20. Both conditions are satisfied, confirming that d = 23 is consistent.
Why Other Options Are Wrong:
If d were 24, the sum b + c + d would be 37 + 24 = 61, giving an average of 61 / 3, which is more than 20. If d were 29 or 21 or 37, similar checks would show averages that do not match 20 and 21 as required.
Common Pitfalls:
One common error is to average the two given averages or to treat them as independent rather than recognizing the shared students b and c. Another is to forget to multiply the averages by 3 when forming equations for total ages, which leads to incorrect sums and a wrong value of d.
Final Answer:
The age of the last student is 23 years.