Question

In a class of 20 students, one student aged 18 years is replaced by a new student, and the average age of the class falls by 2 months. What is the age of the new student?

• A. 14 years
• B. 14 years 4 months
• C. 14 years 6 months
• D. 14 years 8 months

Hint:

Introduce a variable for the original (unknown) average age and set up one equation connecting the old and new averages and totals.

Answer 1

The Correct Option is D

Step 1: Let the original average age of the 20-student class (with the 18-year-old) be $A$ years, so the total age $=20A$.
Step 2: Let $x$ be the new student's age. New total $=20A-18+x$, and the new average is $A-\frac{2}{12}=A-\frac{1}{6}$ years (since 2 months $=\frac{1}{6}$ year).
Step 3: $\dfrac{20A-18+x}{20}=A-\dfrac{1}{6}$.
Step 4: $20A-18+x=20A-\dfrac{20}{6}=20A-\dfrac{10}{3}$, so $x=18-\dfrac{10}{3}=\dfrac{54-10}{3}=\dfrac{44}{3}$ years.
Step 5: $\dfrac{44}{3}$ years $=14\dfrac{2}{3}$ years $=14$ years $8$ months (since $\frac{2}{3}$ year $=8$ months).
\[\boxed{14\text{ years }8\text{ months}}\]