Question

4 In an examination \(70 \%\) of the candidates passed in History and \(50 \%\) in Geography and \(20 \%\) students failed in both the subjects. If 500 students passed in both the subjects, then how many candidates appeared for the exam? (a) 1,000 (b) 1,500 (c) 2,500 (d) none of these

Short answer

a) 1000 b) 1100 c) 1200 d) none of these Answer: d) none of these (1250 candidates appeared for the examination)

Step-by-step solution

Identify the Known Information

We know that: - \(70 \%\) of students passed in History - \(50 \%\) of students passed in Geography - \(20 \%\) of students failed in both subjects - 500 students passed in both subjects.

Use the Inclusive-Exclusive Principle

We will use the Inclusive-Exclusive principle, which helps to find the total number of elements in two sets without counting any element twice. In our case, the two sets are the students who passed in History and Geography. The formula for the Inclusive-Exclusive principle is: Total = A + B - (A ∩ B) where: - Total is the total number of candidates who appeared for the exam - A is the number of students who passed in History - B is the number of students who passed in Geography - A ∩ B is the number of students who passed in both subjects (500 students)

Calculate the Total Number of Passed Students

Before applying the Inclusive-Exclusive principle, let's find the total number of students who passed in both History and Geography individually. Let X be the total number of candidates who appeared for the exam. - Number of students who passed in History: \(0.7X\) - Number of students who passed in Geography: \(0.5X\) Now, let's use the Inclusive-Exclusive Principle: Total = (Number of students who passed in History) + (Number of students who passed in Geography) - (Number of students passed in both subjects) \(X = 0.7X + 0.5X - 500\)

Solve the Equation for the Total Number of Candidates

Now, let's solve the equation for X: \(X = 0.7X + 0.5X - 500\) \(X - 0.7X - 0.5X = -500\) \(0.3X = -500\) Now, we know that \(20 \%\) of students failed in both subjects. So, the left-hand side of our equation should represent the \(80 \%\) of students. Therefore, divide by 0.8 on both sides to find the total number of candidates who appeared for the exam: \(X = \frac{-500}{0.3 \cdot 0.8}\) \(X = \frac{-500}{0.24}\) \(X = \frac{-1250 \cdot 2}{-3 \cdot 2}\) \(X = 1250\) So, 1250 candidates appeared for the exam, which means the correct answer is (d) none of these.

Key concepts

Quantitative Aptitude

Quantitative aptitude is a crucial area for competitive exams that typically measures a person's ability to handle numerical data, understand relationships among numbers, and to draw appropriate conclusions. It involves arithmetic, algebra, geometry, and other areas of mathematics.

When faced with questions like the given examination problem, it is essential to have a thorough grasp of percentages and how they can be combined or separated to solve problems related to quantities. For instance, understanding how to convert percentages into fractions can be pivotal in dividing or combining sets of data represented by percentages.

In the original exercise, students must navigate percentages efficiently to calculate the total number of candidates. This requires not just mathematical skill but also logical reasoning to apply the inclusive-exclusive principle correctly. Effective quantitative aptitude calls for this combined analytical approach, allowing students to decode intricate problems in structured steps.

Set Theory

Set theory is a fundamental concept in mathematics that deals with the collection of objects, known as elements or members of a set. It is widely used in various fields of study, including logic, computer science, and statistics.

In the context of exam problem solving, understanding the basics of set theory, such as the notion of subsets, intersections (denoted as \(A \cap B\)), and unions (denoted as \(A \cup B\)), is critical. Specifically, the inclusive-exclusive principle, also known as the principle of addition, leverages these set theory concepts to calculate the total number of unique elements across multiple sets.

The principle essentially states that to find the total number of elements in combined sets, one should sum the number of elements in each set separately and then subtract the number of elements that are counted in both sets (the intersection). This principle avoids double counting and is exemplified in the shared exercise by considering students passing in History (set A), Geography (set B), and both subjects (set \(A \cap B\)).

Exam Problem Solving

Exam problem solving is a skill that involves analyzing the question, identifying the relevant information, and using appropriate methods to find the solution. It's crucial for students to not only understand the mathematical concepts but also to follow a structured approach when solving exam problems.

Firstly, breaking down the problem into smaller, more manageable parts, as seen in the step-by-step solution for the inclusive-exclusive principle question, aids comprehension. Secondly, it's important to interpret the meaning behind the given data correctly.

Improving on the provided solution could involve seeking clarity about the terms used (e.g., understanding 'failed in both subjects' means they are not included in either passing group) and being meticulous about converting percentages into actual figures that relate to the total number of students. By doing so, a student can avoid common mistakes and be confident in their methodology, ensuring the execution of the solution is both calculated and precise.