Question : A student's marks were wrongly entered as 65 instead of 45. Due to this, the average marks for the class got increased by $\frac{1}{3}$. The number of students in the class is:
Option 1: 40
Option 2: 20
Option 3: 60
Option 4: 30
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Correct Answer: 60
Solution :
Let $n$ be the total number of students and $x$ be the average when the mark is counted as 45.
The average when the mark is counted as 65 is $x+\frac{1}{3}$
Now, $x$ = $\frac{\text{total marks of (n-1)students + 45}}{n}$
⇒ Total marks of $(n-1)$ students = $nx-45$ --------(i)
Also, $x+\frac{1}{3}$ = $\frac{\text{total marks of (n-1)students + 65}}{n}$
Total marks of $(n-1)$ students = $n(x+\frac{1}{3})-65$ --------(ii)
So, $nx-45$ = $n(x+\frac{1}{3})-65$
⇒ $nx-45$ = $nx + \frac{n}{3} - 65$
⇒ $\frac{n}{3}$ = $20$
⇒ $n$ = $60$
Hence, the correct answer is $60$.
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