In a class, there are 25 students whose average age decreases by 3 months when one student aged 22 years is replaced by a new student. The age of
the new student is:
15 Yrs 9 months
Calculating the New Student's Age Based on Average Age Change
This problem requires us to determine the age of a new student based on how the average age of a class changes when one student is replaced. We know the total number of students, the amount the average age changed, and the age of the student who was replaced.
Analyzing the Problem Statement
Here's a breakdown of the key information given:
- The class has a total of
25students. - When one student leaves and a new one joins, the average age of the class decreases by 3 months.
- The student who left the class was 22 years old.
- We need to find the age of the new student.
Step-by-Step Solution Approach
- Calculate the total reduction in age across all students in the class.
- Use this total age reduction to find the difference between the new student's age and the old student's age.
- Calculate the age of the new student in months.
- Convert the new student's age from months into years and months.
Detailed Calculation Steps
Let \(n\) represent the number of students in the class. We are given \(n = 25\). Let \(\Delta \bar{x}\) represent the change in the average age. Since the average age decreases by 3 months, \(\Delta \bar{x} = -3\) months.
Let \(Age_{old}\) be the age of the student who was replaced, and \(Age_{new}\) be the age of the new student.
The age of the student who left is given as \(Age_{old} = 22\) years.
To perform calculations consistently, let's convert the age of the old student entirely into months:
\[ Age_{old} = 22 \text{ years} \times 12 \text{ months/year} = 264 \text{ months} \]
The relationship between the change in average age and the total change in the sum of ages is:
\[ \Delta \bar{x} = \frac{\text{Sum of Ages}_{new} - \text{Sum of Ages}_{old}}{n} \]
The change in the sum of ages is solely due to the replacement of one student. Thus:
\[ \text{Sum of Ages}_{new} - \text{Sum of Ages}_{old} = Age_{new} - Age_{old} \]
Substituting this into the formula for the change in average age:
\[ \Delta \bar{x} = \frac{Age_{new} - Age_{old}}{n} \]
We can rearrange this to find the total change in the sum of ages:
\[ \text{Total Change in Sum of Ages} = n \times \Delta \bar{x} \]
Plugging in the known values:
\[ \text{Total Change in Sum of Ages} = 25 \times (-3 \text{ months}) = -75 \text{ months} \]
This means the sum of the ages of all students decreased by 75 months. We also know this total change equals \(Age_{new} - Age_{old}\). So:
\[ Age_{new} - Age_{old} = -75 \text{ months} \]
Now substitute the value of \(Age_{old}\) in months:
\[ Age_{new} - 264 \text{ months} = -75 \text{ months} \]
To find the age of the new student, we solve for \(Age_{new}\):
\[ Age_{new} = 264 \text{ months} - 75 \text{ months} \]
\[ Age_{new} = 189 \text{ months} \]
The final step is to convert this age back into years and months. We divide 189 by 12 (the number of months in a year):
\[ \text{Years} = \left\lfloor \frac{189}{12} \right\rfloor = 15 \]
\[ \text{Remaining Months} = 189 \pmod{12} = 9 \]
So, the age of the new student is 15 years and 9 months.
Final Answer
The age of the new student is 15 years 9 months.
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