Multiple choice

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 student taking the exam was $z$ $\%$. Find in terms of $x, y$ and $z$ the percentage of students taking the exam who failed ?

  1. $\displaystyle\frac { { \left( z-x \right) }}{{ \left( y-x \right) }}$
  2. $\displaystyle \frac {{ \left( x-z \right) }}{{ \left( y-z \right) }}$
  3. $\displaystyle \frac {{ \left( y-x \right) }}{{ \left( z-y \right) }}$
  4. $\displaystyle \frac {{ \left( y-z \right) }}{{ \left( x-z \right) }}$

  5. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let P be the percentage of students who passed and F be the percentage who failed. Then P + F = 100. The total average is z = (x*P + y*F) / 100. Substituting P = 100 - F, we get 100z = x(100 - F) + yF, which simplifies to 100z = 100x - xF + yF. Solving for F gives F = 100 * (z - x) / (y - x). The option A represents this ratio.

AI explanation

Let the total number of students be N and the number of failed students be F, meaning the passed students are N - F. We can set up the overall average equation: (N - F)x + Fy = Nz. Expanding and rearranging this gives Nx - Fx + Fy = Nz, which simplifies to F(y - x) = N(x - z). Dividing both sides by N(y - x), the percentage of failed students is F divided by N, which equals (x - z) divided by (y - x). Multiplying the numerator and denominator by negative one yields the equivalent fraction (z - x) divided by (y - x).

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