Reasoning: How to Solve Comparison-Based Inequality with Two or More Variables

Introduction This topic typically carries 20-30 marks in competitive exams, and mastering it is a must because it tests your ability to analyze complex situations and make logical conclusions.

WHAT YOU NEED TO KNOW FIRST To solve comparison-based inequality questions with two or more variables, you need to have a basic understanding of:

1. Direction Chart: A simple chart that helps you determine the direction of the inequality (i.e., increasing or decreasing).
2. BODMAS: A mnemonic that reminds you to follow the order of operations (Brackets, Orders, Division, Multiplication, Addition, and Subtraction).
3. Sitting Arrangement Conventions: Basic rules for arranging people or objects in a specific order.

CRYSTAL‑CLEAR METHOD (Step-by-Step) To solve comparison-based inequality questions with two or more variables, follow these steps:

1. Read the question carefully: Understand what is being asked and what the variables represent.
2. Identify the inequality: Determine the type of inequality (i.e., greater than, less than, greater than or equal to, less than or equal to).
3. Create a direction chart: Draw a simple chart to visualize the direction of the inequality.
4. Analyze the variables: Examine each variable and its relationship to the inequality.
5. Use BODMAS: Apply the order of operations to simplify the expression.
6. Compare the values: Determine which value is greater or lesser than the other.
7. Draw a conclusion: Based on your analysis, answer the question.

WORKED EXAMPLES

Example 1 – Easy Question: In a class of 20 students, the average height of boys is 160 cm, and the average height of girls is 150 cm. If the average height of the entire class is 155 cm, what is the ratio of boys to girls in the class?

Step-by-Step Reasoning:

1. Read the question carefully: Understand that we need to find the ratio of boys to girls.
2. Identify the inequality: None, but we need to find the ratio.
3. Create a direction chart: Not necessary for this example.
4. Analyze the variables: The average height of boys is 160 cm, and the average height of girls is 150 cm.
5. Use BODMAS: Not necessary for this example.
6. Compare the values: The average height of the entire class is 155 cm, which is between the average heights of boys and girls.
7. Draw a conclusion: Since the average height of the entire class is between the average heights of boys and girls, the ratio of boys to girls must be 1:1.

What we learned: To solve this type of question, we need to analyze the variables and compare their values to draw a conclusion.

Example 2 – Medium Question: In a company, the average salary of employees is $50,000 per year. If the average salary of managers is $80,000 per year, and the average salary of non-managers is $40,000 per year, what is the ratio of managers to non-managers in the company?

Step-by-Step Reasoning:

1. Read the question carefully: Understand that we need to find the ratio of managers to non-managers.
2. Identify the inequality: None, but we need to find the ratio.
3. Create a direction chart: Not necessary for this example.
4. Analyze the variables: The average salary of managers is $80,000 per year, and the average salary of non-managers is $40,000 per year.
5. Use BODMAS: Not necessary for this example.
6. Compare the values: The average salary of managers is higher than the average salary of non-managers.
7. Draw a conclusion: Since the average salary of managers is higher than the average salary of non-managers, the ratio of managers to non-managers must be less than 1:1.

What we learned: To solve this type of question, we need to analyze the variables and compare their values to draw a conclusion.

Example 3 – Exam-Style Question: In a school, the average score of students in a math test is 80%. If the average score of students who studied for 5 hours or more is 90%, and the average score of students who studied for less than 5 hours is 70%, what is the ratio of students who studied for 5 hours or more to students who studied for less than 5 hours?

Step-by-Step Reasoning:

1. Read the question carefully: Understand that we need to find the ratio of students who studied for 5 hours or more to students who studied for less than 5 hours.
2. Identify the inequality: None, but we need to find the ratio.
3. Create a direction chart: Not necessary for this example.
4. Analyze the variables: The average score of students who studied for 5 hours or more is 90%, and the average score of students who studied for less than 5 hours is 70%.
5. Use BODMAS: Not necessary for this example.
6. Compare the values: The average score of students who studied for 5 hours or more is higher than the average score of students who studied for less than 5 hours.
7. Draw a conclusion: Since the average score of students who studied for 5 hours or more is higher than the average score of students who studied for less than 5 hours, the ratio of students who studied for 5 hours or more to students who studied for less than 5 hours must be greater than 1:1.

What we learned: To solve this type of question, we need to analyze the variables and compare their values to draw a conclusion.

Common Mistakes

MISTAKE → WHY IT HAPPENS → CORRECT APPROACH

1. Ignoring the direction chart: Why it happens: Students may not create a direction chart or may not use it correctly. Correct approach: Create a direction chart and use it to visualize the inequality.
2. Not analyzing the variables: Why it happens: Students may not examine each variable and its relationship to the inequality. Correct approach: Analyze each variable and its relationship to the inequality.
3. Not using BODMAS: Why it happens: Students may not apply the order of operations to simplify the expression. Correct approach: Use BODMAS to simplify the expression.
4. Not comparing the values: Why it happens: Students may not compare the values of the variables. Correct approach: Compare the values of the variables to draw a conclusion.
5. Not drawing a conclusion: Why it happens: Students may not draw a conclusion based on their analysis. Correct approach: Draw a conclusion based on your analysis.

EXAM TRAPS

Trap → How to Spot it → How to Avoid it

1. Trick questions: How to spot it: Questions that seem simple but have a hidden twist. How to avoid it: Read the question carefully and look for any hidden twists.
2. Misleading information: How to spot it: Information that seems relevant but is actually irrelevant. How to avoid it: Analyze the information carefully and eliminate any irrelevant information.
3. Complex expressions: How to spot it: Expressions that seem complex but can be simplified. How to avoid it: Use BODMAS to simplify the expression.

TIME‑SAVING SHORTCUTS

1. Elimination trick: Eliminate any options that are clearly incorrect based on your analysis.
2. Diagram hack: Use a diagram to visualize the inequality and make it easier to analyze.
3. Variable analysis: Analyze each variable and its relationship to the inequality to make it easier to compare the values.

1‑MINUTE RECAP Hey there, student! Tomorrow is the big day, and I know you're feeling nervous. But don't worry, I've got your back. To solve comparison-based inequality questions with two or more variables, just remember:

• Read the question carefully and understand what is being asked.
• Identify the inequality and create a direction chart if necessary.
• Analyze the variables and compare their values to draw a conclusion.
• Use BODMAS to simplify the expression if necessary.
• Eliminate any options that are clearly incorrect based on your analysis.
• Use a diagram to visualize the inequality and make it easier to analyze.

Just follow these steps, and you'll be able to solve these questions with ease. Remember, practice makes perfect, so make sure to practice these questions before the exam. Good luck, and I'll see you on the other side!