Question

The results of half yearly and annual examinations are shown in the table given below:

Results	No. of students in section A	No. of students in section B	No. of students in section C	No. of students in section D
Students failed in both exams	28	23	17	27
Students failed in half yearly but passed in annual exam	14	12	8	13
Students passed in half yearly but failed in annual exam	6	17	9	15
Students passed in both exams	64	55	46	76

Which section has the minimum failure rate in half yearly?
(a) A section
(b) B section
(c) C section
(d) D section

Answer 1

Hint: We start solving the problem by finding the total number of students present in each section by adding the no. of students who passed and failed in each category given in the table. We then find the total no. of students failed in half yearly examinations in each section using the data given in the table. We then find the failure rate in half yearly by dividing the total number of students failed in half yearly in a section with the total no. of students present in that section. We then compare results and find the section with minimum failure rate.

Complete step by step answer:
According to the problem, we have given the results of half yearly and annual examinations in the table as shown below and we need to find which section has the minimum failure rate in half yearly examination.

Results	No. of students in section A	No. of students in section B	No. of students in section C	No. of students in section D
Students failed in both exams	28	23	17	27
Students failed in half yearly but passed in annual exam	14	12	8	13
Students passed in half yearly but failed in annual exam	6	17	9	15
Students passed in both exams	64	55	46	76

Let us find the total number of students present in each section.
From the table, we can see that section A has a total of $\left( 28+14+6+64 \right)=112$ students.
Similarly, we can see that section B has a total of $\left( 23+12+17+55 \right)=107$ students.
Similarly, we can see that section C has a total of $\left( 17+8+9+46 \right)=80$ students.
Similarly, we can see that section B has a total of $\left( 27+13+15+76 \right)=131$ students.
Now, let us find the total no. of students failed in half yearly examination in each section.
From the table, we can see that section A has a total of $\left( 28+14 \right)=42$ students failed in half yearly examination.
Similarly, we can see that section B has a total of $\left( 23+12 \right)=35$ students failed in half yearly examinations.
Similarly, we can see that section C has a total of $\left( 17+8 \right)=25$ students failed in half yearly examinations.
Similarly, we can see that section D has a total of $\left( 27+13 \right)=40$ students failed in half yearly examinations.
Now, let us find the failure rate in the half yearly examination for each section.
We know that failure rate = $\dfrac{\text{Total number of students failed in half-yearly examination in the section}}{\text{Total number of students present in that section}}$.
Failure rate for section A = $\dfrac{42}{112}\times 100=37.5\%$.
Failure rate for section B = $\dfrac{35}{107}\times 100=32.7\%$.
Failure rate for section C = $\dfrac{25}{80}\times 100=31.3\%$.
Failure rate for section D = $\dfrac{40}{131}\times 100=30.5\%$.
We can see that section D has a minimum failure rate in half yearly examinations.
∴ Section D has a minimum failure rate in half yearly examinations.
So, the correct answer is “Option d”.

Note: We should not assume that students failed in both exams includes students failed in only half-yearly exams and students failed only in annual exams which is the common mistake many of us can make. We take percentage for the failure rate to get a better view about the results obtained as comparing the decimals will give confusion. Similarly, we can also expect the problems to find the total failure rate of the school and total pass rate of each section and school.