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

Chapter 1 of Class 9 Mathematics — Real Numbers — ends with its Review Exercise, and that is exactly what these notes complete. After the number skills of Exercise 1.1, the roots and powers of Exercise 1.2 and the word problems of Exercise 1.3, the review mixes all three toolboxes together, the way a board paper does.

These notes cover the Review Exercise completely, in handwritten style and in very simple words: the MCQ answer key with exam technique, both “verify the law” questions worked fraction by fraction, the small concept questions, the simplify trio, two story questions, and Salma’s two-round investment — every step shown, every answer checked.

Read This First: A Review Exercise Contains Nothing New

Let us be honest with each other. Students often fear the review exercise more than the exercises before it, because the questions arrive in a mixed order and nothing announces which chapter tool to use. That fear is backwards. A review exercise contains no new maths at all — it is a practice test built entirely from tools you already own, and its only real question is whether you can recognize them in disguise.

So these notes begin by laying out the three toolboxes of Chapter 1 side by side: number tools from Exercise 1.1 (rational or not, the three laws of real numbers, numbers between numbers), root and power tools from Exercise 1.2 (roots written as powers, rationalizing, the conjugate), and story tools from Exercise 1.3 (read, choose x, make the equation, solve, check). Each review question is then matched to its toolbox before any calculation starts. Learn to ask “which old tool is hiding inside this question?” and the review exercise — and the board paper after it — becomes a sorting game you already know how to win.

Notes Details

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

What Is Included in These Notes?

1. MCQ Answer Key and Exam Technique (Question 1)

The official key for all ten multiple-choice questions — (i) C, (ii) D, (iii) D, (iv) D, (v) A, (vi) B, (vii) A, (viii) B, (ix) D, (x) D — together with the elimination method: cross out wrong options first, then test the survivors with an easy number. The statements and options themselves are printed in your textbook.

2. Verify the Laws (Questions 2 and 3)

The distributive law verified with a = 3/2, b = 5/3, c = 7/5 (both sides 23/5 and 133/30), and the associative law for addition and multiplication with a = 4/3, b = 5/2, c = 7/4 (both sides 67/12 and 35/6). Every LCM is shown, and the three laws are contrasted in one memory box so MCQs cannot mix them up.

3. The Concept Trio (Questions 4, 5 and 6)

Why 0 is a rational number, the trichotomy property explained as “three doors”, and two rational numbers between 4 and 5 (17/4 and 9/2) found with the mean trick.

4. Simplify (Question 7)

A fifth root that becomes x³y⁷/z⁴, a cube root that simplifies to 9ˣ — with the famous 9^{2x} trap flagged in red — and a 3ⁿ expression where the common factor walks out and the answer lands on 27.

5. Stories and Money (Questions 8, 9 and 10)

Three consecutive odd integers (15, 17, 19), Abdullah’s two buckets (34 and 62 balls), and Salma’s investment growing to Rs 5,40,750 over seven years at two different rates. The notes close with a one-page revision sheet for the whole chapter and four try-yourself questions with answers.

Review Exercise Detailed Walkthrough: Every Question Explained

Question 1 — The MCQs. The review opens with ten multiple-choice questions, and the notes give the official answer key: (i) C, (ii) D, (iii) D, (iv) D, (v) A, (vi) B, (vii) A, (viii) B, (ix) D, (x) D. More valuable than the key is the method taught beside it: never wrestle all four options at once. Eliminate the options that are clearly wrong, and if two survive, substitute one easy number into both — the false one usually breaks immediately. Review MCQs recycle the definitions of Exercise 1.1, so a student who knows the three laws and the rational-number definition cold can bank these marks quickly.

Question 2 — Verify the distributive law. “Verify” asks only that both sides be calculated and shown equal. With a = 3/2, b = 5/3 and c = 7/5: b + c = 46/15 (LCM 15), so a(b + c) = (3/2)(46/15) = 23/5; separately, ab + ac = 5/2 + 21/10 = 23/5 (LCM 10). Both sides match. The second part repeats the pattern for (a + b)c = ac + bc, where both sides come to 133/30. The walkthrough shows every LCM step, because in verification questions the arithmetic — not the idea — is where marks leak.

Question 3 — Verify the associative law. Associative means the brackets may move without changing the answer. For addition with a = 4/3, b = 5/2, c = 7/4: b + c = 17/4, so a + (b + c) = 67/12; regrouping, a + b = 23/6, so (a + b) + c = 67/12. For multiplication: bc = 35/8 gives a(bc) = 35/6, and ab = 10/3 gives (ab)c = 35/6. The notes end the question with a family portrait — commutative swaps the order, associative moves the brackets, distributive shares one number over a bracket — because confusing these three is the most common MCQ error in the chapter.

Questions 4 to 6 — The concept trio. Zero is rational because it fits the definition: 0 = 0/2 = 0/(−9); only a zero denominator is forbidden. The trichotomy property says that for any two real numbers a and b, exactly one of a = b, a > b or a < b holds — three doors, exactly one open. And two rationals between 4 and 5 come from the mean trick: the midpoint of 4 and 5 is 9/2, and the midpoint of 4 and 9/2 is 17/4, giving 4 < 17/4 < 9/2 < 5 — a trick that works between any two rationals, endlessly. Question 7 — Simplify. Part (i) turns the fifth root into the power 1/5 and shares it out: x¹⁵ → x³, y³⁵ → y⁷, z²⁰ → z⁴, giving x³y⁷/z⁴. Part (ii) is the trap of the exercise: the cube root of (27)^{2x} becomes (3³)^{2x/3} = 3^{2x} = 9ˣ — and emphatically not 9^{2x}, which would mean 3^{4x}. Part (iii) factors 3ⁿ out of the denominator, 3^{n+1} − 3ⁿ = 3ⁿ(3 − 1), the 3ⁿ cancels against the numerator’s 54 × 3ⁿ, and the answer is a clean 27.

Questions 8 and 9 — Two stories. Consecutive odd integers step by 2, so x, x + 2, x + 4 sum to 51 through 3x + 6 = 51, giving 15, 17 and 19. Abdullah’s buckets follow the “one has K more” pattern: x + (x + 28) = 96, so the buckets hold 34 and 62 balls. Both solutions end with the Exercise 1.3 habit — checking the answer back inside the story.

Question 10 — Salma’s investment. Simple profit is calculated in two rounds because the rate changes. Round one: Rs 3,50,000 at 7¼% for 2 years earns 3,50,000 × 7.25 × 2 ÷ 100 = Rs 50,750. Round two: the same original principal at 8% for 5 years earns Rs 1,40,000. The final amount is 3,50,000 + 50,750 + 1,40,000 = Rs 5,40,750. The notes stress why the principal does not grow between rounds: in simple profit, only the original amount earns — the idea that separates this question from compound markup in Exercise 1.3.

Why These Notes Help Slow Learners

A mixed exercise is hardest on students who learned each topic separately and never practised switching between them. These notes teach the switch itself. Every question is first sorted into its toolbox in plain words, so the student practises recognition — the actual exam skill — before touching a calculation. Verification questions, which slow learners often abandon as “too long”, are broken into two small independent sums with the LCM shown at each step. The chapter’s two most expensive traps are named in red before they can bite: the 9^{2x} slip in Question 7 and the +1 step wrongly used for odd integers in Question 8. And the closing revision page compresses the entire chapter into seven lines, so the night before a test the student reviews everything at once instead of hunting through three separate exercises.

How to Use These Notes

Use this review the way it was designed: as a test. Cover the solutions and attempt each question under a timer — ten minutes per question is generous — then compare with the notes and mark honestly which toolbox each mistake came from. If a whole toolbox is weak, go back to that exercise (1.1, 1.2 or 1.3) before re-attempting the review. The night before your chapter test, read only the one-page revision sheet and redo the four try-yourself questions from memory.

Download Class 9 Maths Real Numbers Notes PDF

The complete Review Exercise 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 the Review Exercise

1. Writing 9^{2x} in Question 7(ii). The cube root of (27)^{2x} simplifies to 3^{2x}, which is (3²)^x = 9ˣ. Doubling the exponent as well — 9^{2x} = 3^{4x} — is the single most repeated error in this exercise.

2. Stepping by 1 for odd integers. Consecutive odd (or even) integers differ by 2: x, x + 2, x + 4. Using x, x + 1, x + 2 in Question 8 produces even-spaced answers that fail the story’s check.

3. Mixing up the three laws. Commutative swaps order, associative moves brackets, distributive shares over a bracket. In MCQs the names are the question — learn each law with its picture, not just its formula.

4. Growing the principal in Question 10. Under simple profit, round two is still calculated on the original Rs 3,50,000. Adding the first profit to the principal is compound thinking in a simple-profit question.

5. Believing 0 is not rational. Zero fits the p/q definition perfectly (0 = 0/2). What is banned is a zero in the denominator, not a zero on top — a distinction examiners love to test.

Frequently Asked Questions

Are these Class 9 Maths notes free?

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

Which exercise is covered in these notes?

The Review Exercise of Chapter 1 (Real Numbers) from the Class 9 Mathematics textbook — all ten questions, from the MCQ key and the two verification questions to the simplify trio, both story questions and the investment question.

Do the notes include the MCQ questions themselves?

The notes include the official answer key for all ten MCQs and the elimination method for solving them. The statements and options are printed in your textbook — use the two together: statement from the book, method and key from these notes.

Are these notes suitable for slow learners?

Yes, they are written especially for slow learners: every question is first matched to its toolbox from Exercises 1.1 to 1.3, LCM steps are never skipped, traps are flagged in red, and the whole chapter is compressed into a one-page revision sheet.

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.

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

With this review, Chapter 1 is complete: rational and irrational numbers, the three laws, roots and powers, and the story problems that use them all. If you can sort each review question into its toolbox and finish it with a check, you are ready for the chapter test. Download the free PDF, sit the review like a real test, and then revisit whichever of Exercise 1.1, Exercise 1.2 or Exercise 1.3 your mistakes point to. The full chapter — and much more — lives on our Free Class 9 Notes and Free Math Notes pages, and everything on AllFreeAssets.com is free, always.