Class 9 Maths Chapter 1 Real Numbers Notes – Exercise 1.3 Solved (Free PDF Download)

Real Numbers is the first chapter of Class 9 Mathematics, and Exercise 1.3 is where the chapter meets real life. After the number skills of Exercise 1.1 and the roots and powers of Exercise 1.2, this exercise hides the same maths inside small stories — three integers in a row, a father’s and son’s ages, a triangle, a rectangle, the Lahore heat, a shopkeeper’s profit, income tax and bank markup.

These notes cover Exercise 1.3 completely, in handwritten style and in very simple words. Every story is first translated into plain maths, then solved step by step, and every answer is checked back inside its story, so slow learners can follow the method without getting stuck.

Read This First: A Story Question Is Just an Equation in a Costume

Let us be honest with each other. Most students do not fear the maths in Exercise 1.3 — they fear the words. A paragraph about a father, a son and six years ago feels harder than a clean equation, even when the equation inside it is one line long. So students either skip word problems completely or guess at them, and lose easy marks.

Here is the secret these notes are built on: every story in this exercise undresses the same way. Read the story slowly, twice. Choose x for the unknown thing. Change the story words into an equation — the equals sign is the bridge between the story and the maths. Solve one small step at a time. Then check your answer back inside the story, where a wrong answer immediately looks wrong. Five steps, used nine times. The triangle and rectangle questions additionally reuse your conjugate tool from Exercise 1.2, and the money questions are each a single small formula. Learn the five steps and this whole exercise becomes reading practice, not a new chapter of maths.

Notes Details

Class	9
Subject	Mathematics
Chapter	1 – Real Numbers
Exercise	1.3 (complete)
Format	PDF (handwritten notes)
Pages	8
Language	Simple English
Price	Free

What Is Included in These Notes?

1. Number Stories (Questions 1 and 4)

Three consecutive integers that add to 42, and two numbers with a given sum and difference. Both are solved by the golden rule — choose x, write the equation, solve, and check inside the story.

2. The Age Story (Question 6)

A father and son whose ages add to 72, with a condition from six years ago. The notes slow down on the two traps: subtracting 6 from both ages, and remembering the question asks for the son’s age six years ago, not today.

3. Root Geometry (Questions 2 and 3)

A right-angled triangle with AC = (√3 + √5) cm and area (1 + √15) cm², where AB comes out as 4√3 − 2√5 cm, and a rectangle whose area simplifies to (11√2 − 2) m². Your rationalizing tool from Exercise 1.2 does the heavy lifting, and every multiplication is shown in small pieces.

4. Temperature (Question 5)

Changing Lahore’s 48°C into Fahrenheit with the given formula °F = (9/5)°C + 32 — worked slowly, with the memory line “times 9, divide by 5, add 32”.

5. Money Stories (Questions 7, 8 and 9)

Profit percentage on a toy, income tax after the exempt amount, and compound markup for one year — each reduced to one formula, with the numbers substituted step by step. The notes finish with a one-page revision sheet and four try-yourself questions with answers.

Exercise 1.3 Detailed Walkthrough: Every Question Explained

Question 1 — Three consecutive integers. Consecutive means one after another, so if the smallest integer is x, the three are x, x + 1 and x + 2. Their sum is 42, which gives the equation x + (x + 1) + (x + 2) = 42. Collecting like terms gives 3x + 3 = 42, so 3x = 39 and x = 13. The three integers are 13, 14 and 15, and the check is instant: 13 + 14 + 15 = 42. This single pattern — smallest = x, then keep adding 1 — answers every consecutive-number question in the book.

Question 2 — The right-angled triangle. The right angle is at A, so AB and AC are the two legs, and the area of a right triangle is half the product of its legs. With AC = (√3 + √5) cm and area (1 + √15) cm², we get AB = 2(1 + √15)/(√3 + √5). A root in the denominator calls for the conjugate from Exercise 1.2: multiplying top and bottom by (√3 − √5) makes the denominator 3 − 5 = −2. Expanding the numerator in four pieces gives √3 − √5 + √45 − √75, and since √45 = 3√5 and √75 = 5√3, the families collect to −4√3 + 2√5. Dividing by −2 flips both signs, and AB = 4√3 − 2√5 cm, so a = 4 and b = −2. A decimal sense-check (about 2.46 cm) confirms the length is sensible.

Question 3 — The rectangle. Clean each side before multiplying. The length 2 + √18 becomes 2 + 3√2 because √18 = √(9 × 2) = 3√2. The breadth 5 − 4/√2 becomes 5 − 2√2 because rationalizing 4/√2 gives 2√2. Now the area is (2 + 3√2)(5 − 2√2), multiplied in four pieces: 10, −4√2, +15√2 and −12. The plain numbers collect to −2 and the √2 terms to +11√2, so the area is (11√2 − 2) m², meaning a = −2 and b = 11. The notes also verify it decimally: roughly 6.24 m × 2.17 m ≈ 13.56 m², matching −2 + 11√2.

Question 4 — Two numbers. A sum and a difference give two equations: x + y = 68 and x − y = 22. From the second, x = 22 + y; substituting into the first gives 22 + 2y = 68, so y = 23 and x = 45. Both story sentences check out: 45 + 23 = 68 and 45 − 23 = 22.

Question 5 — Temperature. The formula is provided, so the only work is careful substitution: °F = (9/5) × 48 + 32. Now 9 × 48 = 432 and 432 ÷ 5 = 86.4, and adding 32 gives 118.4°F — a believable figure for a Lahore summer afternoon.

Question 6 — Father and son. Let the son be x and the father y today, so x + y = 72. Six years ago both were six years younger: the father was y − 6 and the son x − 6, and the condition says y − 6 = 2(x − 6), which tidies to y = 2x − 6. Substituting gives 3x − 6 = 72, so x = 26 — the son’s age today. The question, however, asks for his age six years ago: 26 − 6 = 20 years. The check closes the story: the father is 46 today, so six years ago they were 40 and 20, and 40 is exactly twice 20.

Question 7 — Profit percentage. Mirha’s cost price is Rs 1500 and her selling price Rs 1520, so the profit is Rs 20. Percentage profit is always calculated on the cost price: (20 ÷ 1500) × 100 = 1.33%. The notes stress the base — dividing by the selling price instead is the classic error here.

Question 8 — Income tax. Tax is charged only on what remains after the exempt amount, so the taxable income is 9,60,000 − 1,30,000 = 8,30,000 rupees. At 0.75%, the tax is (0.75 ÷ 100) × 8,30,000 = Rs 6,225. The notes also show the gentle route: 1% of 8,30,000 is 8,300, and three-quarters of that is 6,225.

Question 9 — Compound markup. For one year, markup = principal × rate × time ÷ 100 = 3,75,000 × 14 × 1 ÷ 100 = Rs 52,500. The notes point out why the simple formula suffices here: compounding only begins to differ from the second year onward, and after this first year the total amount is 3,75,000 + 52,500 = Rs 4,27,500.

Why These Notes Help Slow Learners

Word problems punish students who read quickly and reward students who read slowly — which means this exercise should belong to slow learners, yet it is usually the one they fear most. These notes reverse that. The story is always translated first, in plain words, before any symbol appears. One golden rule of five steps is repeated for all nine questions until it becomes a habit. Traps are named before you meet them: the “six years ago” that must be subtracted from both ages, the final line that asks for a different age than the one you solved for, the cost price that must sit under the profit percentage. And every solution ends by checking the answer inside the story, a habit that catches mistakes in the exam hall too, where nobody is standing by to mark your working.

How to Use These Notes

Read the golden rule on the first page until you can repeat it. Then do the questions in order, covering the solution and trying each story yourself first — even five honest minutes of trying teaches more than reading three solutions. For Questions 2 and 3, revise the conjugate steps from Exercise 1.2 if they feel rusty. For the money questions, write the formula down before touching the numbers. The night before a test, redo the one-page revision sheet and the four try-yourself questions without looking.

Download Class 9 Maths Real Numbers Notes PDF

The complete Exercise 1.3 notes are available as a free PDF — the same handwritten pages described above, ready to read on your phone, print for your notebook, or share with your class fellows. No sign-up, no payment, just the download.

5 Common Mistakes Students Make in Exercise 1.3

1. Answering the age you found, not the age asked. In Question 6, x = 26 is the son’s age today, but the question asks for six years ago — 20. Underlining the final line of the question before starting prevents this expensive slip.

2. Subtracting six years from only one person. Six years ago, the father was also six years younger. Writing (x − 6) for the son but plain y for the father breaks the story and the equation.

3. Forgetting the half in the triangle’s area. The area of a right triangle is (leg × leg) ÷ 2. Students who multiply the legs and stop there double the area and lose AB completely.

4. Dividing profit by the selling price. Profit percentage is always on the cost price — what was paid, not what was received. In Question 7 the correct base is 1500, not 1520.

5. Taxing the whole income. The exempt amount exists precisely to be subtracted first. In Question 8, taxing all 9,60,000 instead of the taxable 8,30,000 inflates the answer and misses the point of the question.

Frequently Asked Questions

Are these Class 9 Maths notes free?

Yes. The Exercise 1.3 PDF is completely free to download, print and share. Everything on All Free Assets is free.

Which exercise is covered in these notes?

Exercise 1.3 of Chapter 1 (Real Numbers) from the Class 9 Mathematics textbook — all nine questions: consecutive integers, the triangle, the rectangle, two numbers, temperature conversion, the age problem, profit percentage, income tax and compound markup.

Are these notes suitable for slow learners?

Yes, they are written especially for slow learners: very simple English, one golden rule repeated for every story, traps named before you meet them, an answer-check inside every story, and a one-page revision sheet at the end.

Which syllabus do these notes follow?

They follow the PCTB (Punjab Curriculum and Textbook Board) Class 9 Mathematics book, Chapter 1 — Real Numbers, and they are equally useful for other BISE boards teaching the same chapter.

I find word problems very hard. Where should I start?

Start with Question 1 and learn the golden rule on the first page of the notes: read, choose x, make the equation, solve, check. Questions 4 and 5 use the same rule in shorter stories. Leave the triangle and rectangle until you have revised rationalizing from Exercise 1.2.

Can I print these notes or share them with my class fellows?

Yes. The PDF prints cleanly on A4 paper, and you may share it freely with your class fellows and study groups.

Conclusion

Exercise 1.3 is the friendliest exercise in Chapter 1 once you see its trick: there is no new maths in it at all, only stories. Translate each story with the same five steps, reuse the conjugate tool where roots appear, put money numbers into their one small formula each, and check every answer back inside its story. Download the free PDF, work through it question by question, and the word problems in your board exam will start looking like the easiest marks on the paper. For the full chapter, see the other exercises on our Free Class 9 Notes and Free Math Notes pages — and everything here on AllFreeAssets.com is free, always.