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 the student taking the exam were $z\%$. Find in terms of x, y, z the percentage of students taking the exam who failed?
A). $\dfrac{{\left( {z - x} \right)}}{{\left( {y - x} \right)}}$
B). $\dfrac{{\left( {x - z} \right)}}{{\left( {y - z} \right)}}$
C). $\dfrac{{\left( {y - x} \right)}}{{\left( {z - y} \right)}}$
D). $\dfrac{{\left( {y - z} \right)}}{{\left( {x - z} \right)}}$
E). None of these

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Hint: We need to consider the pass, fail and total with some variables like b, c, and a respectively so that based on the given question we know that $b \times x + c \times y = a \times z$ and also $a = b + c$. On considering these equations we can now be able to find the value of $\dfrac{c}{a}$ which is the percentage of students taking the exam who failed.

Complete step-by-step solution:
Given, in an examination, the average marks obtained by students who passed was $x\%$
The average of those who failed was $y\%$.
The average marks of student taking the exam were $z\%$
The percentage of students taking the exam who failed =?
Once we have calculated the decimal values of each percentage for each given sample size, then we need to add these decimal values together and divide the total number by the total sum of both sample sizes. Then we need to multiply this value by 100 to get the average percentage.
Let pass = b, fail = c and total =a
Thus, $b \times x + c \times y = a \times z$ ………………...… (1)
$ \Rightarrow a = b + c$
$ \Rightarrow ax = bx + cx$ ……………..… (2)
From (1) and (2), we have
We will get, $c (y – x) = a (z – x)$
$ \Rightarrow \dfrac{c}{a} = \dfrac{{z - x}}{{y - x}}$
The percentage of students taking the exam who failed is $\dfrac{{z - x}}{{y - x}}$
Hence, option A is correct.

Note: We need to consider some steps to calculate the average percentage into decimals. They are 1. Convert the percentages into decimals 2. Determine the number that each decimal represents 3. Add the numbers together 4.Add the sample sizes together 5. Calculate the percentage average.