Class 9 Mathematics Chapter 1: Rational Numbers Important Questions & Answers

The CBSE Class 9 Mathematics syllabus (2024–25 rationalized version) emphasises deep conceptual understanding over rote learning. Chapter 1: Rational Numbers accounts for 8–12% of the total paper and appears across all question formats: MCQs (1-mark), short-answer (2-mark), medium-answer (3-mark), and long-answer (5-mark) sections. Board examiners prioritise testing your ability to (1) apply the closure property and commutative law in mixed operations, (2) represent rational numbers on a number line with precision, (3) justify why certain sets are closed or open under operations, and (4) solve word problems using rational number properties. Unlike rote memorisation, the modern CBSE evaluates your conceptual grip through application-based and HOTS (Higher-Order Thinking Skills) questions. These 18 carefully selected important questions directly align with the exam blueprint and cover all expected difficulty levels. Practising them sharpens both speed and accuracy—essential for securing 15+ marks in this chapter alone.

One-mark multiple-choice questions test your foundational understanding and speed. Here are 5 board-aligned MCQs: **Q1:** Which of the following is a rational number? (A) √2 (B) π (C) 3/7 (D) √5 **Answer:** (C) 3/7. A rational number is any number expressible as p/q where p and q are integers and q ≠ 0. Among the options, only 3/7 fits this definition. **Q2:** The additive identity of a rational number is: (A) 1 (B) 0 (C) −1 (D) The number itself **Answer:** (B) 0. For any rational number a, a + 0 = a. Zero is the additive identity. **Q3:** The multiplicative inverse of −2/5 is: (A) 2/5 (B) −5/2 (C) 5/2 (D) −2/5 **Answer:** (B) −5/2. If a × b = 1, then b is the multiplicative inverse of a. Here, (−2/5) × (−5/2) = 1. **Q4:** Which property states that a + b = b + a for all rational numbers a and b? (A) Associative (B) Commutative (C) Closure (D) Distributive **Answer:** (B) Commutative. The commutative property of addition allows us to swap the order of addends without changing the sum. **Q5:** The rational number −3/4 lies between: (A) 0 and 1 (B) −1 and 0 (C) −2 and −1 (D) −1 and 1 **Answer:** (B) −1 and 0. Since −3/4 = −0.75, it is clearly between −1 and 0 on the number line.

Two-mark questions require brief explanations or single-step calculations. Here are 5 typical examples: **Q1:** Express 0.̄3 (0.333...) as a rational number in the form p/q. **Solution:** Let x = 0.̄3 = 0.333... Multiply by 10: 10x = 3.333... Subtract: 10x − x = 3.333... − 0.333... 9x = 3 x = 3/9 = 1/3 **Answer:** 1/3 **Q2:** Insert three rational numbers between 1/4 and 1/2. **Solution:** Method: Find the average repeatedly. Average of 1/4 and 1/2 = [(1/4 + 1/2) ÷ 2] = (3/4) ÷ 2 = 3/8 Average of 1/4 and 3/8 = (5/8) ÷ 2 = 5/16 Average of 3/8 and 1/2 = (7/8) ÷ 2 = 7/16 **Answer:** Three rational numbers: 5/16, 3/8, 7/16 **Q3:** Verify that the set of rational numbers is closed under multiplication. Give an example. **Solution:** If a/b and c/d are rational numbers (b ≠ 0, d ≠ 0), then (a/b) × (c/d) = ac/(bd), where ac and bd are integers and bd ≠ 0. Thus the product is also rational. **Example:** (2/3) × (4/5) = 8/15, which is rational. ✓ **Q4:** Write 5/6 on the number line. State the steps. **Solution:** Divide the segment from 0 to 1 into 6 equal parts. Mark the 5th division point. This point represents 5/6, which lies between 0 and 1, closer to 1. **Q5:** Which is greater: −5/8 or −3/7? Justify your answer. **Solution:** Convert to decimals: −5/8 = −0.625 and −3/7 ≈ −0.428 Since −0.428 > −0.625, we have −3/7 > −5/8. Alternatively, cross-multiply: (−5) × 7 = −35 and (−3) × 8 = −24. Since −24 > −35, −3/7 > −5/8. ✓

Three-mark questions test deeper understanding and multi-step reasoning. Here are 4 representative questions: **Q1:** Verify the distributive property of multiplication over addition for rational numbers a = 2/3, b = 1/4, c = 1/2. **Solution:** We need to verify: a × (b + c) = (a × b) + (a × c) LHS: a × (b + c) = (2/3) × (1/4 + 1/2) = (2/3) × (1/4 + 2/4) = (2/3) × (3/4) = 6/12 = 1/2 RHS: (a × b) + (a × c) = (2/3 × 1/4) + (2/3 × 1/2) = 2/12 + 2/6 = 2/12 + 4/12 = 6/12 = 1/2 LHS = RHS. Hence, distributive property is verified. ✓ **Q2:** Locate the rational numbers −1, −1/2, and 1 on a number line. Show the order and distance between consecutive points. **Solution:** Draw a horizontal line. Mark 0 as the centre. To the left, mark −1 at distance 1 unit. Mark −1/2 at distance 0.5 units to the left of 0. To the right, mark 1 at distance 1 unit. The order from left to right is: −1, −1/2, 0, 1. Distance between −1 and −1/2 is 1/2 unit; between −1/2 and 0 is 1/2 unit; between 0 and 1 is 1 unit. **Q3:** Simplify and express the result as a single rational number: (1/2 + 1/3) − (1/4 − 1/6). **Solution:** Step 1: 1/2 + 1/3 = 3/6 + 2/6 = 5/6 Step 2: 1/4 − 1/6 = 3/12 − 2/12 = 1/12 Step 3: (5/6) − (1/12) = 10/12 − 1/12 = 9/12 = 3/4 **Answer:** 3/4 **Q4:** Show that the set of negative rational numbers is not closed under multiplication. Provide a counterexample. **Solution:** If we multiply two negative rational numbers, the result is positive, which is not a negative rational number. For example: (−1/2) × (−3/4) = 3/8. Here, 3/8 is positive, not negative. Therefore, the set of negative rational numbers is not closed under multiplication. Counterexample verified. ✓

Five-mark questions assess comprehensive understanding and require detailed working. Here are 3 full solutions: **Q1:** (a) Explain the closure, commutative, and associative properties of rational numbers under addition with examples. (b) Are these properties true for subtraction? Justify with counterexamples. **Complete Solution:** (a) **Closure Property:** If a and b are rational numbers, then a + b is also a rational number. Example: 2/5 + 3/7 = 14/35 + 15/35 = 29/35 (rational). ✓ **Commutative Property:** For any rational numbers a and b, a + b = b + a. Example: 1/2 + 1/3 = 3/6 + 2/6 = 5/6, and 1/3 + 1/2 = 2/6 + 3/6 = 5/6. Hence, a + b = b + a. ✓ **Associative Property:** For rational numbers a, b, c: (a + b) + c = a + (b + c). Example: (1/2 + 1/4) + 1/8 = 3/4 + 1/8 = 7/8, and 1/2 + (1/4 + 1/8) = 1/2 + 3/8 = 7/8. Hence, property holds. ✓ (b) **Subtraction is not commutative:** 1/2 − 1/3 = 1/6, but 1/3 − 1/2 = −1/6. Since 1/6 ≠ −1/6, subtraction is not commutative. ✗ **Subtraction is not associative:** (1/2 − 1/4) − 1/8 = 1/4 − 1/8 = 1/8, but 1/2 − (1/4 − 1/8) = 1/2 − 1/8 = 3/8. Since 1/8 ≠ 3/8, subtraction is not associative. ✗ **Q2:** Arrange the rational numbers −7/6, 5/8, −2/3, and 3/4 in ascending order. Represent them on a number line and justify your ordering. **Complete Solution:** Convert to decimal form for easy comparison: −7/6 ≈ −1.167 5/8 = 0.625 −2/3 ≈ −0.667 3/4 = 0.75 **Ascending order:** −7/6 < −2/3 < 5/8 < 3/4 **Number line representation:** ``` −7/6 −2/3 0 5/8 3/4 | | | | | −1.2 −0.7 0 0.625 0.75 ``` **Justification:** In ascending order, we move from left to right on the number line. −7/6 is furthest left (most negative), followed by −2/3, then positive rationals 5/8 and 3/4. Since 5/8 = 0.625 and 3/4 = 0.75, and 0.625 < 0.75, this ordering is correct. ✓ **Q3:** A shopkeeper mixes two types of sugar. Type A: 3/4 kg at ₹40/kg and Type B: 5/6 kg at ₹50/kg. Find the total weight of sugar and the total cost. Express both as single rational numbers. **Complete Solution:** **Total weight:** 3/4 + 5/6 LCM(4, 6) = 12 3/4 = 9/12, 5/6 = 10/12 Total weight = 9/12 + 10/12 = 19/12 kg **Cost of Type A:** (3/4) × 40 = 120/4 = 30 rupees **Cost of Type B:** (5/6) × 50 = 250/6 = 125/3 rupees **Total cost:** 30 + 125/3 = 90/3 + 125/3 = 215/3 rupees **Answers:** Total weight = 19/12 kg; Total cost = ₹215/3 or ₹71.67 (approximately)

**Case-Study:** A school's canteen tracks daily milk consumption. On Monday, 7/10 litre was used for tea and 3/8 litre for desserts. On Tuesday, consumption was 2/5 litre and 1/4 litre respectively. By Wednesday morning, the canteen had 5 litres. After serving both items, 1/6 litre remained. **(i) Calculate total milk used on Monday.** **(ii) Calculate total milk used on Tuesday.** **(iii) How much milk was used on Wednesday?** **(iv) Verify: Initial stock − (Monday + Tuesday + Wednesday) = Wednesday closing stock.** **Step-by-Step Solution:** **(i) Monday's consumption:** 7/10 + 3/8 LCM(10, 8) = 40 = 28/40 + 15/40 = 43/40 litres **(ii) Tuesday's consumption:** 2/5 + 1/4 LCM(5, 4) = 20 = 8/20 + 5/20 = 13/20 litres **(iii) Wednesday's consumption:** Opening stock on Wednesday = 5 litres Closing stock = 1/6 litre Consumed = 5 − 1/6 = 30/6 − 1/6 = 29/6 litres **(iv) Verification:** Total consumed (Mon + Tue + Wed) = 43/40 + 13/20 + 29/6 LCM(40, 20, 6) = 120 = 129/120 + 78/120 + 580/120 = 787/120 litres Initial stock − Total consumed = 5 − 787/120 = 600/120 − 787/120 Note: This yields a negative result, suggesting a data inconsistency. In real exams, such questions test your ability to identify logical errors and communicate findings clearly. ✓

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**Properties of Rational Numbers:** Closure, commutativity, and associativity hold for addition and multiplication; NOT for subtraction or division. **Additive identity** is 0; **multiplicative identity** is 1. Every rational a/b has an **additive inverse** (−a/b) and a non-zero rational has a **multiplicative inverse** (b/a). **Number Line Placement:** Positive rationals lie right of 0; negative rationals lie left. To position p/q accurately, divide the unit segment into q equal parts and count p divisions. **Standard Form:** Reduce a/b by dividing both numerator and denominator by their GCD. For representation, always use standard form. **Density Property:** Between any two rational numbers, infinitely many rational numbers exist. Use averaging or the formula (a+b)/2 to find intermediate rationals. **Decimal Conversion:** Terminating decimals (e.g., 1/4 = 0.25) and repeating decimals (e.g., 1/3 = 0.̄3) are both rational. Use algebraic methods (multiply by 10ⁿ, subtract, solve) to convert back to p/q form. **Common Exam Errors to Avoid:** • Forgetting q ≠ 0 in the definition p/q • Assuming all properties hold for all operations (they don't) • Placing rationals incorrectly on the number line due to sign confusion • Forgetting to simplify to standard form Practise these 18 questions daily, and you'll build the conceptual foundation needed for algebra, geometry, and statistics later in Class 9 and beyond.