Exercise 2.4 — More Applications

Applying Linear Equations to Real-Life Problems

Exercise 2.4 is where linear equations stop being abstract and start solving real situations. Instead of being handed a ready-made equation, you are given a word problem — a description of a real scenario — and your job is to form the equation yourself, then solve it. This is one of the most important skills tested in Class 8 board exams across CBSE, Telangana, and Andhra Pradesh syllabi, and it appears repeatedly in higher classes too.

The Standard Approach to Word Problems

Every word problem in this exercise follows a reliable four-step method:

• Assign a variable — choose a letter (usually x) to represent the unknown quantity. Be specific: "Let x be the units digit" or "Let x be Reshma's present age."
• Translate the condition into an equation — convert the key relationship described in the problem into a mathematical equation.
• Solve the equation — use transposition to find the value of x.
• Interpret the answer — if x is just one part of the answer (e.g., a digit, a ratio part, or an age), calculate the remaining unknowns and state the final result clearly.

Types of Problems in This Exercise

The eight problems cover a wide variety of real-life contexts, which is exactly what makes this exercise valuable for exam preparation.

Geometry and parallel lines (Q1): When two lines l and m are parallel and cut by a transversal, alternate exterior angles are equal. Setting 2x + 15° = 3x − 10° and solving gives x = 25. This problem neatly connects linear equations with properties of parallel lines.

Number-based problems (Q2, Q5): In Q2, "eight times a number reduced by 10 equals six times the number plus 4" translates directly to 8x − 10 = 6x + 4, giving x = 7. In Q5, tripling a number and adding 2 gives the same result as subtracting it from 50 — this becomes 3x + 2 = 50 − x, so x = 12.

Two-digit number problem (Q3): This is a classic problem type. The digits sum to 9, so if the units digit is x, the tens digit is (9 − x) and the number itself is 10(9 − x) + x. Subtracting 27 reverses the digits, leading to the equation whose solution gives x = 3 and the original number as 63.

Ratio and parts problem (Q4): A number split into two parts in the ratio 5:3, where one part exceeds the other by 10. Let the parts be 5x and 3x. Then 5x = 3x + 10 gives x = 5, so the parts are 25 and 15, and the number is 40.

Age problems (Q6, Q7): Age problems are among the most frequently asked in board exams. In Q6, Mary is currently twice her sister's age, but in 5 years she will be only 2 years older — setting up 2x + 5 = x + 7 gives Mary's sister as 2 years and Mary as 4 years. In Q7, a table or timeline approach helps: if Reshma is x now, she will be x + 5 in five years and was x − 9 nine years ago. The condition gives x + 5 = 3(x − 9), so x = 16.

Population with percentage change (Q8): This is the most involved problem. Starting population is x; after adding 1200 and then reducing by 11%, the result is 32 fewer than the original x.

Common Mistakes to Avoid

• Misreading the problem — in age problems, clearly distinguish "present age," "age after n years," and "age n years ago" before writing any equation.
• Forgetting to find all unknowns — if the question asks for "the number and the two parts," solve for x but also calculate 5x and 3x separately. Leaving just x as the answer will lose marks.
• Two-digit number setup errors — a two-digit number with tens digit a and units digit b equals 10a + b, not a + b. This is a very common mistake in Q3-type problems.
• Percentage application errors (Q8) — a decrease of 11% means multiplying by (100 − 11)/100 = 89/100, not subtracting 11 directly from the population.

What This Exercise Prepares You For

The word-problem-to-equation skill built here is used throughout Class 9 and Class 10 mathematics. Age problems and number problems reappear in Class 9 linear equations, while ratio and percentage problems connect directly to Comparing Quantities using Proportion. Students appearing for CBSE, Telangana, or AP board exams should practise forming equations independently from word descriptions — this is a guaranteed 2–3 mark question in most papers. For more equation-solving practice, revisit Exercise 2.3 on equations with variables on both sides.