Question

In an examination, the average marks obtained by students who passed was \( x\% \), while the average of those who failed was \( y\% \). The average marks of all students taking the exam was \( z\% \). Find in terms of \( x, y, z \), the percentage of students taking the exam who failed.

• A. \( \frac{x - z}{x - y} \)
• B. \( \frac{z - y}{x - z} \)
• C. \( \frac{z - y}{x - y} \)
• D. \( \frac{y - z}{y - x} \)

Hint:

Apply the weighted average formula carefully and rearrange to isolate the failed student ratio.

Answer 1

The Correct Option is A

Let the number of students who passed be \( a \), and who failed be \( b \).
Then the overall average is: \[ z = \frac{ax + by}{a + b} \Rightarrow z(a + b) = ax + by \Rightarrow az + bz = ax + by \Rightarrow a(z - x) = b(y - z) \Rightarrow \frac{b}{a + b} = \frac{a(z - x)}{(a + b)(y - z)} = \frac{x - z}{x - y} \] So, the percentage of students who failed is: \[ { \frac{x - z}{x - y} } \]