Exercise 2.2 — Applications

What Does Exercise 2.2 Cover?

Exercise 2.2 is the most important and varied exercise in Chapter 2 — Linear Equations in One Variable. It covers two main areas: using angle properties of triangles and parallel lines to set up and solve equations, and translating real-life word problems into linear equations. This exercise is heavily tested in Class 8 board exams for CBSE, Telangana, and Andhra Pradesh, and it builds the core problem-solving skill of forming an equation from a situation.

Geometry Background — Key Angle Properties Used

Before solving Questions 1(i) to 1(v), you need to know these three geometric facts:

• Angle sum of a triangle: The three interior angles of any triangle add up to 180°. Used in parts (ii), (iii), and (iv).
• Exterior angle theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles. Used in part (i): ∠ACD = ∠A + ∠B.
• Isosceles triangle: If two sides are equal, the angles opposite to them are also equal. Used in parts (iv) and (v).
• Vertically opposite angles: When two lines intersect, opposite angles are equal. Used in part (v) alongside isosceles triangle properties.

Solutions for Q1: (i) x = 67, (ii) x = 17, (iii) x = 125, (iv) x = 19°, (v) x = 20.

How to Approach Word Problems — The 5-Step Method

Every word problem in this exercise follows the same approach: read carefully, assign a variable, express all other unknowns in terms of that variable, form an equation from the given condition, and solve. Always state the answer in the context of the question — not just the value of x.

Word Problem Solutions — Q2 to Q17

• Q2 — Two numbers, difference 8: Let bigger = x, smaller = x − 8. Condition: x + 2 = 3(x − 8). Solving: x = 13. Numbers are 13 and 5.
• Q3 — Sum 58, difference 28: Let bigger = x, smaller = 58 − x. Condition: x − (58 − x) = 28. Solving: x = 43. Numbers are 43 and 15.
• Q4 — Two consecutive odd numbers, sum 56: Let numbers be 2x − 1 and 2x + 1. Equation: 4x = 56, x = 14. Numbers are 27 and 29.
• Q5 — Three consecutive multiples of 7, sum 777: Let numbers be x, x + 7, x + 14. Equation: 3x + 21 = 777, x = 252. Multiples are 252, 259, 266.
• Q6 — Journey by walk, train, bus totalling 70 km: Walk = 10 km, train = x km, bus = 2x km. Equation: 10 + 3x = 70, x = 20. Train distance = 20 km.
• Q7 — Cake cut into 3 pieces, total 300 g: First = x, second = x + 7, third = x − 4. Equation: 3x + 3 = 300, x = 99. Pieces weigh 99 g, 106 g, 95 g.
• Q8 — Rectangular field, perimeter 400 m, length 26 m more: Breadth = x, length = x + 26. Using l + b = 200: 2x + 26 = 200, x = 87. Breadth = 87 m, length = 113 m.
• Q9 — Perimeter 56 m, length 8 m less than twice breadth: Breadth = x, length = 2x − 8. Using l + b = 28: 3x − 8 = 28, x = 12. Breadth = 12 m, length = 16 m.
• Q10 — Isosceles triangle, perimeter 55 m: Third side = x, equal sides = 2x − 5 each. Equation: 5x − 10 = 55, x = 13. Sides are 21 m, 21 m, 13 m.
• Q11 — Two complementary angles, differ by 12°: Angles x and x + 12, sum = 90. Equation: 2x + 12 = 90, x = 39. Angles are 39° and 51°.
• Q12 — Ages in ratio 5:7, sum after 4 years = 56: Ages 5x and 7x. Equation: 12x + 8 = 56, x = 4. Present ages = 20 and 28 years.
• Q13 — 180 MCQs, +4 correct, −1 wrong, scored 450: Let correct = x, wrong = 180 − x. Equation: 4x − (180 − x) = 450, 5x = 630, x = 126. Correct answers = 126.
• Q14 — ₹500 in ₹5 and ₹10 notes, 90 notes total: ₹5 notes = x, ₹10 notes = 90 − x. Equation: 5x + 10(90 − x) = 500, −5x = −400, x = 80. Notes: 80 of ₹5 and 10 of ₹10.
• Q15 — ₹564 on pens (₹7) and pencils (₹3), 108 items: Pens = x, pencils = 108 − x. Equation: 7x + 3(108 − x) = 564, 4x = 240, x = 60. Bought 60 pens and 48 pencils.
• Q16 — Volleyball court, perimeter 177 ft, length = 2 × width: Width = x, length = 2x. Equation: 2(3x) = 177, x = 29.5. Width = 29.5 ft, length = 59 ft.
• Q17 — Facing page numbers sum to 373: Pages = x and x + 1. Equation: 2x + 1 = 373, x = 186. Pages are 186 and 187.

Common Mistakes to Avoid

• In angle problems, always check which property applies — exterior angle, angle sum, or isosceles triangle — before writing the equation.
• In ratio problems (like Q12), let the two quantities be kx and the second part times x — not just x and x + something.
• In problems involving two categories (notes, pens/pencils), always express the second quantity as (total − x) and expand brackets carefully before collecting terms.
• Complementary angles sum to 90°; supplementary angles sum to 180° — mixing these up is a common error.

What This Exercise Prepares You For

The word-problem skills from Exercise 2.2 appear throughout the rest of Class 8 and beyond — in comparing quantities, direct and inverse proportions, and geometry chapters. The same equation-formation technique is extended in Class 9 and 10 when solving quadratic equations and coordinate geometry problems. For the solving techniques behind this exercise, revisit Exercise 2.1.