Finding the Maximum Marks in an Exam: A Mathematical Puzzle

Often, in real-life scenarios, we encounter problems that require us to deduce information from given data. One such interesting question involves determining the total marks of an exam based on a student's performance and the passing marks. Let's explore this puzzle in detail.

The Problem Statement

In an examination, a student needs to secure at least 35 marks to pass. However, the student secured 30 marks and failed by 20 marks. This means that the student required an additional 20 marks to pass the exam. Our task is to find out the total marks for the exam. Let's break down the problem and solve it step-by-step.

Solving the Problem

First, let's denote the total marks of the exam as 'a'. According to the problem, the passing marks are 0.36a (35/0.36). The student scored 0.24a (30/0.24) but was short by 18 marks to reach the passing marks. This gives us the following equation:

Equation Setup

0.24a 18 0.36a

Now, let's transpose and solve for 'a':

Transposing and Solving for 'a'

0.24a 18 0.36a

18 0.36a - 0.24a

18 0.12a

a 18 / 0.12 150

Hence, the total marks for the exam are 150. This solution is both logical and consistent with the given information in the problem.

Alternative Methods

Method 1

Another approach involves using a step-by-step calculation to find the total marks:

Passing marks: 36
Student score: 24
Failed by: 18 marks

Therefore, we can set up the equation:

Solving by Equation

36 - 24 18
18 / 12 1.5 (marks per unit)
36 × 1.5 54 (total marks)

Thus, the maximum marks for the exam are 54.

Method 2

A third method involves setting up an equation with a variable 't' for the total marks:

Rearranging and Solving

Pass-mark is 37.5 (36-2412, 12/12*75/200 3/8 t)
200 - 20 3/8 t
So, 180/8 (1/8) t
Therefore, t 180 * 8 / 3 480 (total marks)

Method 3

Using a different approach, we can form the equation:

Passing Marks Calculation

40 of the total marks 60 (marks)
The equation looks like this:
0.4 x 60
Transpose for x
x 60 / 0.4 150 (total marks)

Conclusion

Through various methods, we have solved the exam mark problem in different ways. Whether you use the basic equation method, step-by-step calculation, or a variable-based approach, the total marks in the exam are consistently found to be 150. This problem-solving technique not only helps in understanding the logic behind such questions but also improves critical thinking and mathematical skills.

Related Terms: exam maximum marks, passing marks calculation, mathematical problem solving