Question

In an examination a candidate got \(30 \%\) marks and failed by 30 marks. If the passing marks are \(60 \%\) of the total marks, then the maximum marks will be : (a) 450 (b) 600 (c) 300 (d) 100

Short answer

The candidate would have passed if they had achieved 60% of the maximum marks. Answer: The maximum marks in the examination are (d) 100.

Step-by-step solution

Analyze the given information

The candidate got 30% marks and failed by 30 marks. Let the marks obtained by the candidate be y. So, y = 0.30 * x The candidate failed by 30 marks, which means that if they had 30 more marks, they would have passed: y + 30 = passing marks We also know that the passing marks are 60% of the total marks: 0.60 * x = passing marks

Set up an equation

We can connect the equations of marks obtained by the candidate and the passing marks: y + 30 = 0.60 * x Substitute the value of y from the first equation: 0.30 * x + 30 = 0.60 * x

Solve for x

Subtract 0.30 * x from both sides of the equation: 30 = 0.30 * x Divide by 0.30 to get the maximum marks (x): x = 30 / 0.30 x = 100 The maximum marks in the examination are 100. So, the correct option is (d) 100.

Key concepts

Percentage Calculation

Percentage calculation is an essential skill, particularly when dealing with problems related to exams or financial matters. It involves working with ratios and proportions to find the part of a whole. For instance, in the given problem, the candidate scored 30% of the total marks. Assuming the total marks are 'x', this can be expressed as 0.30 times 'x', or simply 0.30x. Understanding how to convert a percentage to a decimal is crucial here, as it simplifies the arithmetic operations that follow.
The concept can become trickier when you need to backtrack from a known percentage to find the original amount. In our case, we know the candidate was 30 marks short of the passing score, which is 60% of the total. To solve problems like these, setting up a clear equation, as shown in the solution, is key. Practicing percentage calculations helps not only in exams but also in real-life scenarios like calculating tips, discounts, and interest rates.

Exam Score Analysis

Analyzing exam scores involves more than just looking at the marks received; it requires understanding the required percentage to pass and how close or far a student is from this threshold. In our example, a candidate's failure despite earning 30% indicates a gap between achievement and the passing criterion, which is another layer of percentage analysis.
When we analyze exam scores, we consider the relationships between obtained marks, passing marks, and total possible marks. The key to solving this type of problem is identifying those relations using algebraic equations. Once the data is translated into an equation, it's a step-by-step process to find out the exact numbers, whether it's the total marks or what score was needed to pass the exam. This level of analysis is critical for educators who need to make decisions based on student performance and for students aiming to improve their academic standing.

Algebraic Equations

Algebraic equations are powerful tools for representing real-world problems in a solvable mathematical format. They allow us to find unknown values by setting up relationships between known quantities. In this exercise, we used algebra to connect the dots between the candidate's obtained marks and the passing criteria.
With the knowledge that the candidate scored 30% and was 30 marks short of the passing score (which is 60%), we created an equation to represent this scenario. Algebraic manipulation, such as substituting and solving for the unknown variable, brought us to the conclusion. By practicing setting up and solving algebraic equations, students can apply these skills not only to exams but also to various situations where unknown variables are involved, from budgeting to engineering.