Question : The average age of 30 students in a class is 14 years, 4 months. After the admission of five new students in the class, the average becomes 13 years and 9 months. The youngest of the five new students is 9 years and 11 months old. The average age of the remaining four new students is:
Option 1: 11 years 2 months
Option 2: 13 years 6 months
Option 3: 12 years 4 months
Option 4: 10 years 4 months
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Correct Answer: 10 years 4 months
Solution :
Using Average = $\frac{\text{Sum of all observations}}{\text{Number of observations}}$
Sum of the ages of 30 students = 30 × 14$\frac{4}{12}$ = 30 ×14$\frac{1}{3}$ = 30 × $\frac{43}{3}$ = 430 years
Sum of age of 35 students after admission = 35 × 13$\frac{9}{12}$ = 35 ×13$\frac{3}{4}$ = 35 × $\frac{55}{4}$ = $\frac{1925}{4}$ years
Total age of 5 students = 35 students – 30 students = $\frac{1925}{4}$ – 430 = $\frac{1925-1720}{4}$ = $\frac{205}{4}$ = $51\frac{1}{4}$ years
i.e., 51 years 3 months
Given: Age of one student = 9 years 11 months
Sum of age of 4 students = 51 years 3 months – 9 years 11 months = 41 years 4 months
So, the average age of the remaining 4 students
= $\frac{\text{41 years 4 months}}{4}$
(Converting years in months we get 496 months)
= $\frac{496}{4}$
= 10 years 4 months
Hence, the correct answer is '10 years 4 months'.
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