CBSE Class 8 Maths: Linear Equations in One Variable - Ex 2.4

Understanding Word Problems in Linear Equations

Word problems are like puzzles where the clues are given in words, and you need to find the missing numbers. The key to solving these is to convert the information given in words into mathematical expressions and then form an equation. This process requires careful reading and understanding of the problem statement. Think of it as a translation task: from English (or Hindi!) to 'Maths-ish'.

Often, there will be an unknown quantity, which we represent with a variable, usually x. The problem will then describe relationships between this unknown and other known quantities, or other unknowns expressed in terms of the first variable. Your goal is to identify these relationships, set up an equality, and then solve for x. For example, if a problem says "one number is five more than another", and you call the first number x, the other number would be x + 5. This careful translation is the most crucial step in tackling Ex 2.4 problems.

Step-by-Step Approach to Solving Exercise 2.4 Problems

1. Step 1: Read and Understand the Problem — Read the problem carefully, not just once, but multiple times. Identify what is known (given information) and what needs to be found (the unknown quantity). Don't rush; a clear understanding is half the battle won.
2. Step 2: Assign a Variable to the Unknown — Choose a variable (like x, y, a, etc.) to represent the main unknown quantity. Sometimes there are multiple unknowns, but they can usually be expressed in terms of one primary variable. For instance, if 'one number is twice another', let the smaller number be x, then the larger one is 2x.
3. Step 3: Translate the Statements into Mathematical Expressions — Convert each part of the word problem into algebraic expressions. Look for keywords that indicate operations: 'sum' means +, 'difference' means -, 'product' means ×, 'quotient' means ÷, 'is' or 'are' or 'was' usually mean =. Build these expressions piece by piece.
4. Step 4: Formulate the Linear Equation — Combine the mathematical expressions from Step 3 to form a linear equation. This equation will represent the relationship stated in the problem, often involving an equality between two expressions. Ensure that both sides of the equation make logical sense according to the problem.
5. Step 5: Solve the Equation — Use the techniques you've learned for solving linear equations (e.g., transposing terms, combining like terms, multiplying/dividing both sides) to find the value of the variable. Remember to perform the same operation on both sides to maintain equality.
6. Step 6: Verify Your Answer — Once you have a solution, substitute the value of the variable back into the original word problem (not just the equation) to see if it satisfies all the conditions. This step is crucial to catch any errors and ensures your answer is correct and makes practical sense.

Worked Examples from NCERT Exercise 2.4

• Example 1: Number Problem Problem: The difference between two numbers is 10. If the larger number is three times the smaller number, find the numbers. Solution: Step 1: Let the smaller number be x. Step 2: Since the larger number is three times the smaller, the larger number is 3x. Step 3: The difference between the two numbers is 10. So, Larger number - Smaller number = 10. Step 4: Formulate the equation: 3x - x = 10. Step 5: Solve the equation: 2x = 10 x = 10 / 2 x = 5 Step 6: The smaller number is 5. The larger number is 3 * 5 = 15. Step 7: Verify: 15 - 5 = 10. The conditions are satisfied. Final Answer: The numbers are 5 and 15.
• Example 2: Age Problem Problem: Rahul's age is 5 years less than twice the age of his sister, Priya. If the sum of their ages is 31 years, find their present ages. Solution: Step 1: Let Priya's present age be x years. Step 2: Rahul's age is 5 years less than twice Priya's age. So, Rahul's age = 2x - 5 years. Step 3: The sum of their ages is 31 years. So, Priya's age + Rahul's age = 31. Step 4: Formulate the equation: x + (2x - 5) = 31. Step 5: Solve the equation: 3x - 5 = 31 3x = 31 + 5 3x = 36 x = 36 / 3 x = 12 Step 6: Priya's age is 12 years. Rahul's age is (2 * 12) - 5 = 24 - 5 = 19 years. Step 7: Verify: 12 + 19 = 31. The conditions are satisfied. Final Answer: Priya's age is 12 years and Rahul's age is 19 years.

Exam Tip: Avoiding Common Mistakes in Word Problems

When solving word problems, even small errors can lead to incorrect answers. Here are some common mistakes to watch out for:

• Incorrect Variable Assignment: Ensure your chosen variable x truly represents the quantity you intend. For example, if 'father's age is three times son's age', and you let father's age be x, then son's age would be x/3, not x. Often, it's easier to assign x to the smaller or more fundamental unknown.
• Misinterpreting Keywords: 'Less than' can be tricky. '5 less than x' is x - 5, not 5 - x. Similarly, 'exceeds by' implies subtraction, e.g., 'A exceeds B by 10' means A - B = 10.
• Equation Formulation Errors: Double-check that your equation accurately reflects the entire problem statement. Does it include all given conditions? Are all terms on the correct side of the equality?
• Calculation Mistakes: After formulating the equation, pay close attention to arithmetic. A small addition or subtraction error can derail the entire solution. Work step-by-step and recheck your calculations.
• Forgetting to Answer the Question: Sometimes you solve for x, but the question asks for something else (e.g., 'find the sum of the numbers', not just x). Always read the final question carefully after solving the equation.

Practice Questions with Solutions

• Q: A number is such that when you multiply it by 7 and subtract 5 from the product, you get 16. What is the number? A: Step 1: Let the number be x. Step 2: "Multiply it by 7" gives 7x. Step 3: "Subtract 5 from the product" gives 7x - 5. Step 4: "You get 16" means 7x - 5 = 16. Step 5: Solve the equation: 7x - 5 = 16 7x = 16 + 5 7x = 21 x = 21 / 7 x = 3 Final answer: The number is 3.
• Q: The sum of three consecutive even numbers is 66. Find these numbers. A: Step 1: Let the first even number be x. Step 2: The next consecutive even number will be x + 2. Step 3: The third consecutive even number will be x + 4. Step 4: The sum is 66: x + (x + 2) + (x + 4) = 66. Step 5: Solve the equation: 3x + 6 = 66 3x = 66 - 6 3x = 60 x = 60 / 3 x = 20 Step 6: The numbers are x = 20, x + 2 = 22, x + 4 = 24. Final answer: The three consecutive even numbers are 20, 22, and 24.
• Q: One of the two digits of a two-digit number is three times the other digit. If you interchange the digits and add the resulting number to the original number, you get 88. Find the original number. A: Step 1: Let the unit's digit be x. Then the ten's digit is 3x (assuming the question implies one digit is three times the other, not necessarily the unit's digit). Or let the unit's digit be 3x and ten's digit be x. Let's assume the question implies the digit in the tens place is three times the digit in the units place, so unit's digit is x and ten's digit is 3x. Step 2: Original number = 10 (ten's digit) + (unit's digit) = 10(3x) + x = 30x + x = 31x. Step 3: Interchanged digits: unit's digit becomes 3x, ten's digit becomes x. Step 4: New number = 10 (new ten's digit) + (new unit's digit) = 10(x) + 3x = 10x + 3x = 13x. Step 5: Sum of original and new number is 88: 31x + 13x = 88. Step 6: Solve the equation: 44x = 88 x = 88 / 44 x = 2 Step 7: Original number = 31x = 31 * 2 = 62. Step 8: Check: Unit's digit is 2, ten's digit is 6 (which is 3 times 2). Interchanged number is 26. 62 + 26 = 88. This is correct. Final answer: The original number is 62.
• Q: A's father is 26 years younger than A's grandfather and 29 years older than A. The sum of the ages of all three is 135 years. What is the age of each one of them? A: Step 1: Let A's age be x years. Step 2: A's father's age is 29 years older than A, so Father's age = x + 29 years. Step 3: A's father is 26 years younger than A's grandfather. This means Grandfather's age is 26 years older than Father's age. Grandfather's age = (x + 29) + 26 = x + 55 years. Step 4: The sum of their ages is 135 years: x + (x + 29) + (x + 55) = 135. Step 5: Solve the equation: 3x + 84 = 135 3x = 135 - 84 3x = 51 x = 51 / 3 x = 17 Step 6: A's age = 17 years. A's father's age = 17 + 29 = 46 years. A's grandfather's age = 17 + 55 = 72 years. Step 7: Verify: 17 + 46 + 72 = 135. The conditions are satisfied. Final answer: A's age is 17 years, his father's age is 46 years, and his grandfather's age is 72 years.

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