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

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

Hint:

Write the overall average as a weighted average of the pass and fail group averages, then solve that equation for the fail fraction.

Answer 1

The Correct Option is A

Step 1: Recognize this as a mixture (alligation) problem.
The overall average $z\%$ is a mixture of two groups: the passed students averaging $x\%$ and the failed students averaging $y\%$. The rule of alligation says the ratio in which the two groups are mixed is the inverse of how far each group's average lies from the overall average.

Step 2: Write the alligation ratio.
\[ \frac{\text{number who passed}}{\text{number who failed}} = \frac{y - z}{z - x} \]
This follows the standard alligation rule: the ratio of the two quantities mixed equals the ratio of the distances of the other quantity's average from the mean, taken in reverse order.

Step 3: Convert the ratio into a fraction of the whole class.
If passed : failed $= (y-z) : (z-x)$, the fraction of the whole class that failed is the failed part divided by the sum of both parts.
\[ \text{fraction failed} = \frac{z-x}{(y-z)+(z-x)} \]

Step 4: Simplify the denominator.
\[ (y-z)+(z-x) = y - x \]
The $z$ terms cancel out, leaving $y-x$.
\[ \text{fraction failed} = \frac{z-x}{y-x} \]

Step 5: Match with the answer choices.
Since $x$, $y$ and $z$ are already percentages, this fraction is directly the percentage who failed, $\dfrac{z-x}{y-x}$. None of the other three options simplify to this expression, since each one pairs the wrong pair of quantities together.

Final Answer:
The percentage of students who failed is $\dfrac{z-x}{y-x}$. \[ \boxed{\dfrac{z-x}{y-x}} \]