The average age of a teacher and three students is 20 years. If all the three students are of same age and the difference between the age of the teacher and each student is 20 years, then what is the age of the teacher?

(a) 25 years
(b) 30 years
(c) 35 years ✓ UPSC's answer
(d) 45 years

Why the answer is (c)

• Let the age of each student be $x$ years and the age of the teacher be $y$ years.

• The average age of the four individuals is 20, so the total sum of their ages is $4 \times 20 = 80$ years.

• This gives the equation: $y + 3x = 80$.

• The problem states the difference between the teacher's age and each student's age is 20 years, so $y - x = 20$, which implies $y = x + 20$.

• Substituting $y$ in the first equation: $(x + 20) + 3x = 80 \Rightarrow 4x + 20 = 80 \Rightarrow 4x = 60 \Rightarrow x = 15$.

• The teacher's age is $y = 15 + 20 = 35$ years, which corresponds to option (c).

Why the other options are wrong

(a) 25 years: Option (a) is incorrect because if the teacher were 25, the students would be 5, making the total age 40 and the average 10, not 20.
(b) 30 years: Option (b) is incorrect because if the teacher were 30, the students would be 10, making the total age 60 and the average 15, not 20.
(d) 45 years: Option (d) is incorrect because if the teacher were 45, the students would be 25, making the total age 120 and the average 30, not 20.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2020, held on 4 October 2020. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.