Class 9 Mathematics Chapter 3: A Peek Beyond the Point Important Questions (2024-25)

The CBSE Class 9 Mathematics paper allocates significant weightage to number systems and decimal operations. Chapter 3 forms the numeracy backbone—examiners expect you to manipulate decimals with fluency and convert between representations effortlessly. The 2024-25 rationalized syllabus emphasizes conceptual clarity over rote memorization, meaning questions test *why* place value works and *how* decimal operations relate to whole-number arithmetic. In recent board papers, you'll find 8–12 marks dedicated to this chapter across multiple question types: MCQs that check your immediate recall, short-answer questions requiring worked steps, and long-answer problems combining place value with real-world scenarios (pricing, measurements, data analysis). By practising these curated questions now, you develop pattern recognition—you'll spot instantly whether a question demands rounding, estimation, or exact computation. Moreover, decimal fluency directly supports Chapters 4–6 (exponents, rational numbers, algebraic expressions), where decimal coefficients and fractional equivalents appear regularly. Mastering these important questions ensures no chapter later feels disconnected.

Multiple-choice questions test your immediate understanding and decision-making speed. Each option is carefully designed to catch common misconceptions—selecting the wrong answer reveals a gap worth plugging before the board exam. **Question 1:** What is the place value of 7 in the decimal number 23.7456? (A) 7/10 (B) 7/100 (C) 7/1000 (D) 7/10000 **Answer:** (A) 7/10. The digit 7 occupies the first place after the decimal point, which is the tenths position. Its place value is 7 × (1/10) = 0.7. **Question 2:** Which of the following is the fractional form of 0.625? (A) 5/8 (B) 3/8 (C) 7/8 (D) 1/8 **Answer:** (A) 5/8. To convert: 0.625 = 625/1000. Dividing numerator and denominator by 125: 625÷125 = 5, 1000÷125 = 8. Thus, 0.625 = 5/8. **Question 3:** Add 12.35 + 8.47. Which statement is true? (A) The sum is 20.72 (B) The sum is 20.82 (C) The sum is 21.82 (D) The sum is 20.71 **Answer:** (B) 20.82. Align decimal points: 12.35 + 8.47 = 20.82. **Question 4:** Arrange in increasing order: 0.5, 0.05, 0.55, 0.505. (A) 0.05, 0.5, 0.505, 0.55 (B) 0.05, 0.505, 0.5, 0.55 (C) 0.05, 0.5, 0.55, 0.505 (D) 0.5, 0.05, 0.505, 0.55 **Answer:** (A) 0.05, 0.5, 0.505, 0.55. Compare place by place: 0.05 < 0.500 < 0.505 < 0.550. **Question 5:** Divide 45.6 ÷ 12. The quotient is: (A) 3.8 (B) 3.9 (C) 4.8 (D) 4.0 **Answer:** (A) 3.8. 45.6 ÷ 12 = 456 ÷ 120 = 3.8.

These questions require you to show one or two steps of working. Examiners award marks for method clarity and correct final answers. **Question 1:** Convert 7/8 into a decimal and identify whether it is a terminating or non-terminating decimal. **Solution:** Divide 7 by 8: 7 ÷ 8 = 0.875. Since the division terminates with a remainder of 0, 0.875 is a **terminating decimal**. (Alternatively, since 8 = 2³, and the denominator has only factors of 2, the fraction yields a terminating decimal.) **Question 2:** Find the difference: 25.08 − 13.245. Show your working. **Solution:** Align decimal points and subtract: 25.080 − 13.245 ________ 11.835 The difference is **11.835**. **Question 3:** Multiply 3.5 × 2.4 and verify your answer by converting to fractions. **Solution:** Direct multiplication: 3.5 × 2.4 = 8.4. Verification via fractions: 3.5 = 7/2, 2.4 = 12/5. Thus, (7/2) × (12/5) = 84/10 = 8.4. ✓ **Question 4:** Round 12.4567 to the nearest hundredth. State the digit you examined and explain your choice. **Solution:** To round to the hundredth, we look at the thousandth place (5). Since 5 ≥ 5, we round up. Thus, 12.4567 rounded to the nearest hundredth is **12.46**. **Question 5:** Estimate the product 4.9 × 3.1 by rounding each number to the nearest whole number. Then calculate the exact product. **Solution:** 4.9 ≈ 5 and 3.1 ≈ 3, so estimated product ≈ 5 × 3 = 15. Exact product: 4.9 × 3.1 = (49/10) × (31/10) = 1519/100 = **15.19**.

Three-mark questions demand multi-step reasoning, often combining two concepts or requiring justification. **Question 1:** The cost of 1 litre of milk is ₹45.50. How much will 6.5 litres cost? Show all steps. **Solution:** Cost per litre = ₹45.50 Quantity = 6.5 litres Total cost = 45.50 × 6.5 = (4550/100) × (65/10) = (4550 × 65)/(100 × 10) = 295,750/1000 = **₹295.75** **Question 2:** A student has three decimal numbers: 5.246, 5.26, and 5.262. Arrange them in descending order and explain your method. **Solution:** Compare from left: all three start with 5.2__. Second decimal place: all are 4, 6, 6 → first two differ only after. Compare 5.246 vs 5.26: 5.260 vs 5.246 → 5.260 > 5.246 Compare 5.26 vs 5.262: 5.260 vs 5.262 → 5.262 > 5.260 **Descending order: 5.262 > 5.26 > 5.246** **Question 3:** Convert 0.8̄ (where 8 repeats) into a fraction. Verify your answer. **Solution:** Let x = 0.8̄ = 0.8888... Multiply by 10: 10x = 8.8888... Subtract: 10x − x = 8.8888... − 0.8888... 9x = 8 x = **8/9** Verification: 8 ÷ 9 = 0.8̄ ✓ **Question 4:** A tailor has cloth measuring 15.75 metres. She cuts it into pieces of 1.5 metres each. How many complete pieces can she cut, and what length remains? **Solution:** Divide total length by piece length: 15.75 ÷ 1.5 = (1575/100) ÷ (15/10) = (1575/100) × (10/15) = 15750/1500 = 10.5 So she gets **10 complete pieces** of 1.5 m each (total 15 m) and a **remaining 0.75 m**.

These questions integrate multiple concepts, often presented as word problems requiring interpretation, calculation, and explanation. **Question 1:** A shopkeeper recorded the daily sales (in ₹) for a week: Monday 245.50, Tuesday 320.75, Wednesday 198.25, Thursday 267.30, Friday 342.80, Saturday 289.45. (a) Find the total sales for the week. (b) Calculate the average daily sales. (c) On which days did sales exceed the average? **Full Solution:** (a) Total sales = 245.50 + 320.75 + 198.25 + 267.30 + 342.80 + 289.45 = **₹1664.05** (b) Average = 1664.05 ÷ 6 = **₹277.34** (rounded to nearest paisa) (c) Days exceeding ₹277.34: - Tuesday: 320.75 ✓ - Friday: 342.80 ✓ So, **Tuesday and Friday exceeded the average.** **Question 2:** A student's marks in five subjects are represented in decimal form: English 78.5, Math 85.25, Science 79.75, Social Studies 82.5, Hindi 88.25. (a) Express each mark as a fraction in lowest terms. (b) Identify which subject has the highest fractional equivalent. (c) If the total marks needed for distinction is 410, will the student achieve it? **Full Solution:** (a) Fractions: - English: 78.5 = 785/10 = **157/2** - Math: 85.25 = 8525/100 = **341/4** - Science: 79.75 = 7975/100 = **319/4** - Social Studies: 82.5 = 825/10 = **165/2** - Hindi: 88.25 = 8825/100 = **353/4** (b) Convert to common denominator (4): 157/2 = 314/4, 165/2 = 330/4. Highest is 353/4 = 88.25 (Hindi) (c) Total marks = 78.5 + 85.25 + 79.75 + 82.5 + 88.25 = **414.25** Since 414.25 > 410, **yes, the student achieves distinction.** **Question 3:** A rectangular garden has length 12.4 metres and width 8.6 metres. (a) Calculate the perimeter and area. (b) If fencing costs ₹35.50 per metre, find the total fencing cost. (c) If the gardener wants to plant flowers in a square plot of side 3.5 metres within the garden, what area remains for grass? **Full Solution:** (a) Perimeter = 2(length + width) = 2(12.4 + 8.6) = 2(21) = **42 metres** Area = 12.4 × 8.6 = **106.64 square metres** (b) Fencing cost = 42 × 35.50 = **₹1491** (c) Square plot area = 3.5² = **12.25 square metres** Area for grass = 106.64 − 12.25 = **94.39 square metres**

**Case Study:** A pharmacy supply company receives an order for medicine bottles. Each bottle costs ₹12.75 wholesale and sells for ₹18.50 retail. The company has ₹5000 available for purchasing stock. They also offer a 5% discount to bulk buyers. A hospital orders 200 bottles and qualifies for the bulk discount. (i) How much profit (in ₹) does the company make per bottle sold at retail? (ii) How many bottles can the company purchase with ₹5000? (iii) What is the discounted retail price per bottle for the hospital? (iv) What is the total revenue from selling 200 bottles to the hospital at the discounted price? (v) If the company wants a minimum profit of ₹2000, is this hospital order sufficient? **Step-by-Step Solution:** (i) **Profit per bottle:** Retail price − Wholesale price = 18.50 − 12.75 = **₹5.75 per bottle** (ii) **Number of bottles purchasable:** 5000 ÷ 12.75 = (5000 × 100) ÷ 1275 = 500000 ÷ 1275 ≈ **392 bottles** (maximum; exact: 392.15, so 392 complete bottles) (iii) **Discounted retail price:** 5% discount on 18.50 = 18.50 × 0.05 = 0.925 Discounted price = 18.50 − 0.925 = **₹17.575 per bottle** (or ₹17.58 rounded) (iv) **Total revenue from hospital order:** 200 bottles × 17.575 = **₹3515** (v) **Profit analysis:** Total cost for 200 bottles = 200 × 12.75 = ₹2550 Total revenue = ₹3515 Profit = 3515 − 2550 = **₹965** Since ₹965 < ₹2000, **no, the hospital order alone is insufficient to meet the ₹2000 profit target.** The company would need additional orders. This question integrates decimal arithmetic (multiplication, subtraction, percentages), real-world logic, and decision-making—exactly what examiners test at the higher difficulty level.

Solving questions once is not mastery—repetition under varied conditions is. CBSETUTOR.ai's AI tutor personalizes your Chapter 3 practice by analysing which decimal operations you struggle with (place value identification, fraction conversion, multi-step word problems) and drilling exactly those patterns daily. Here's how it works: **Adaptive Question Selection:** The AI assesses your answers to 1-mark MCQs and instantly flags misconceptions (e.g., confusing tenths with hundredths, forgetting decimal alignment in subtraction). It then queues targeted 2-mark and 3-mark drills to reinforce the weak concept before moving to harder problems. **Worked Explanations:** After each answer, you receive step-by-step solutions with reasoning—not just the final number. For example, when you struggle with 0.8̄ conversion, the tutor shows you the repeating decimal method and lets you practise similar patterns (0.3̄, 0.27̄) until fluency clicks. **Real-Time Feedback:** If you round 12.4567 incorrectly, the AI explains *why* 5 means round-up and suggests practising borderline cases (12.4549, 12.4551) to strengthen your judgment. **Mock Board Papers:** Every week, the tutor generates a mini Chapter 3 test in authentic CBSE format—5 MCQs, 5 short-answer, 2 long-answer, 1 HOTS question—and tracks your time, accuracy, and confidence. Over 4–6 weeks of consistent drills, you'll see your Chapter 3 score climb from 12/20 to 19/20. Start a 3-day free trial at cbsetutor.ai and experience adaptive learning that saves you hours of guesswork.