Question

600 students took the test on physics and chemistry. \(35 \%\) students failed in Physics and \(45 \%\) students failed in chemistry and \(40 \%\) of those who passed in chemistry also passed in Physics, then how many students failed in both : (a) 162 (b) 138 (c) 60 (d) none of thesev

Short answer

Answer: 210 students.

Step-by-step solution

Determine the number of students who failed in Physics and Chemistry separately

The total number of students who took the test is 600. 35% of students failed in Physics: \(0.35 \times 600 = 210\) 45% of students failed in Chemistry: \(0.45 \times 600 = 270\)

Find the number of students who passed in Physics and Chemistry

We are given that 40% of students, who passed in Chemistry, also passed in Physics. The number of students who passed in Chemistry is the total students minus those who failed in Chemistry: \(600 - 270 = 330\) So, 40% of 330 students also passed in Physics: \(0.4 \times 330 = 132\)

Determine the number of students who passed in Physics only

We know that 132 students passed in both Physics and Chemistry. We need to find the students who passed in Physics only. The total number of Physics-failures is 210, so the number of students who passed in Physics is the total students minus those who failed in Physics: \(600 - 210 = 390\) So, the students who passed in physics only are: \(390 - 132 = 258\)

Find the number of students who failed in both Physics and Chemistry

We have found the number of students who passed in Physics only (258) and those who passed in both Physics and Chemistry (132). Now, we can find the total number of students who passed in either Physics or Chemistry: \(258 + 132 = 390\) Since there are 600 students in total, we can find the number of students who failed in both Physics and Chemistry by subtracting the students who passed in either subject from the total students: \(600 - 390 = 210\) The number of students who failed in both Physics and Chemistry is 210. So, the correct answer is: (d) None of these

Key concepts

Understanding Set Theory Through Venn Diagrams

Set theory is a fundamental concept in mathematics that deals with the collection of objects or elements, referred to as 'sets.' Venn diagrams, which are simple graphical representations, can be incredibly effective tools for visualizing relationships between different sets, especially when it comes to solving problems involving elements that belong to multiple sets.

Take the example of students taking tests in two subjects, like Physics and Chemistry. Imagine two circles overlapping each other. One circle represents all students who took the test in Physics, and the other all those who took it in Chemistry. The overlapping area represents students who have dealings with both subjects – in this case, students who possibly failed both. Here's where set theory comes into play: we are essentially dealing with subsets (e.g., students failing each subject) and trying to identify the intersections (students failing both) and unions (students failing at least one).

Investigating our exercise, we first categorize the students into subsets based on the given percentages and then proceed to calculate those in the intersections. This approach simplifies complex relationships between sets, making it easier to comprehend and solve the underlying problem.

Cracking Percentage Calculations

Percentage calculations are a crucial component of quantitative aptitude, a skill that is extensively applied in fields such as finance, data analysis, and academic assessments. The very notion of percentage is about understanding parts per hundred, which enables us to easily compare and comprehend proportions and ratios.

In the provided exercise, identifying what proportion of students failed in each subject is pivotal. For example, calculating that 35% of 600 students failed in Physics involves a straightforward percentage calculation: \(0.35 \times 600 = 210\). This tells us the number of students represented by that percentage. Understanding how to convert percentages to actual values is not only key to solving such problems but is also essential in making informed decisions based on quantitative data in real-life situations.

When you improve your acumen in percentage calculations, you also enhance your ability to quickly estimate and make mental calculations, which are highly valuable for efficiency in various academic or professional tasks.

Developing Quantitative Aptitude

Quantitative aptitude encompasses the ability to handle numerical and logical reasoning problems efficiently. This proficiency is vital across many disciplines and is often assessed in standardized tests, employment examinations, and in school curriculums. It includes understanding numerical information, applying basic mathematical concepts, and developing problem-solving strategies.

In our example problem on student test results, quantitative aptitude steps in as we translate percentages into raw numbers, use basic arithmetic to determine the number of students who passed in both subjects, and systematically deduce who failed both tests. It demands careful analysis, logical reasoning, and when necessary, an ability to manipulate algebraic expressions to reach the solution.

Quantitative aptitude is not just about performing calculations; it's also about interpreting data correctly, spotting patterns, and making logical deductions, all of which are vital skills for academic success and beyond. By engaging with problems like these and practicing regularly, one can substantially boost their numerical reasoning capabilities, opening doors to mastering more complex quantitative challenges.