Class 9 Mathematics Chapter 3 Number Play: Previous Year Questions with Complete Solutions

Reading your NCERT textbook twice won't improve your marks as much as solving 10 carefully chosen past papers once. Here's why: (1) **Pattern Recognition** — Examiners repeat certain question types across years. Once you've solved a place-value question from 2022 and 2023, you recognise the structure instantly. (2) **Speed Training** — In a real exam, you have 3 hours for 80 marks. Theory reading doesn't teach you to work fast; solving PYQ under timed conditions does. (3) **Confidence Gaps** — When you solve a 3-mark question correctly, you *feel* ready. Passive reading creates illusion of readiness. (4) **NCERT Alignment** — CBSE Class 9 questions are drawn directly from the rationalized 2024–25 syllabus. PYQ show you *exactly* which topics appear, and at what difficulty. For Number Play, this means: place value to 8 digits (not 10), Indian and International systems, estimation/rounding, operations on large numbers, Roman numerals, and number puzzles. (5) **Error Learning** — When you get a question wrong, the solution sticks harder than any explanation. PYQ let you fail in practice, not in the exam.

**Question 1:** Write 45,678,901 in words using the Indian numbering system. **Answer:** Forty-five crore, sixty-seven lakh, eighty-nine thousand, nine hundred and one. **Explanation:** In the Indian system, we group digits in pairs from right to left: ones, tens, hundreds, thousands, tens of thousands, lakhs, tens of lakhs, crores. This number has 8 digits: 4 (crores) 5 (ten-millions) 6 (lakhs) 7 (ten-thousands) 8 (thousands) 9 (hundreds) 0 (tens) 1 (ones). **Question 2:** Express 89,234,567 in the International numbering system. **Answer:** Eighty-nine million, two hundred thirty-four thousand, five hundred sixty-seven. **Explanation:** The International system groups in threes: ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions. So 89,234,567 = 89 million + 234 thousand + 567 ones. **Question 3:** Round 7,654,389 to the nearest lakh. **Answer:** 76,54,400 (or 76.54 lakh). **Explanation:** The digit in the ten-thousands place is 5. When rounding and the digit is ≥5, round up. So 76,54,389 → 76,54,400. **Question 4:** What is the place value of 8 in 12,345,678? **Answer:** 80 or eight tens. **Explanation:** The 8 is in the ones place from the right: position 2 (tens). So place value = 8 × 10 = 80. **Question 5:** Convert MMXXIV to Hindu-Arabic numerals. **Answer:** 2024. **Explanation:** M = 1000, M = 1000, X = 10, X = 10, IV = 4. Total = 1000 + 1000 + 10 + 10 + 4 = 2024.

**Question 1:** A factory produced 1,234,567 items in 2023 and 1,876,543 items in 2024. Find the total production and express it in both Indian and International numbering systems. **Solution:** Total = 1,234,567 + 1,876,543 = 3,111,110 **Indian system:** 31 lakh, 11 thousand, 110 (or three lakh, eleven thousand, one hundred and ten). **International system:** Three million, one hundred eleven thousand, one hundred ten. *Mark allocation:* Addition (1 mark) + Indian form (1 mark) + International form (1 mark). **Question 2:** Estimate the product of 8,765 and 1,234 by rounding each to the nearest thousand. Then calculate the actual product and find the difference. **Solution:** Rounding to nearest thousand: 8,765 ≈ 9,000; 1,234 ≈ 1,000. Estimated product = 9,000 × 1,000 = 9,000,000. Actual product = 8,765 × 1,234 = 10,819,610. Difference = 10,819,610 − 9,000,000 = 1,819,610. *Teaching note:* Estimation first builds intuition; students often catch calculation errors by comparing. **Question 3:** In a number puzzle, a 6-digit number has the following properties: (i) the ten-thousands digit is twice the units digit, (ii) the sum of all digits is 27. If the number is 5A3B1C (where A, B, C are unknowns), find A, B, and C. **Solution:** The number is 5A3B1C. Ten-thousands digit = B, units digit = C. Condition 1: B = 2C. Condition 2: 5 + A + 3 + B + 1 + C = 27 → A + B + C = 18. Substitute B = 2C: A + 2C + C = 18 → A + 3C = 18. If C = 3, then A = 9, B = 6. Number = 593,613. Check: 5+9+3+6+1+3 = 27 ✓; 6 = 2×3 ✓. **Answer:** A = 9, B = 6, C = 3. **Question 4:** Convert the following to Roman numerals: 1949, 2015, 3008. (Sub-marks for each.) **Solution:** 1949 = 1000 + 900 + 40 + 9 = M + CM + XL + IX = **MCMXLIX**. 2015 = 2000 + 15 = MM + XV = **MMXV**. 3008 = 3000 + 8 = MMM + VIII = **MMMVIII**. *Rule reminder:* Subtractive notation (like CM = 900) works only for: IV (4), IX (9), XL (40), XC (90), CD (400), XM (900). **Question 5:** A number has a digit sum of 36 and lies between 5,000,000 and 6,000,000. If the first digit is 5 and all other digits are the same, what is the number? **Solution:** Let the number be 5ddddd (5 followed by six d's, making it 7 digits). Digit sum: 5 + 6d = 36 → 6d = 31 → d = 31/6 ≈ 5.17 (not an integer; adjust problem). Alternatively, if the number is 5dddd (5 followed by four d's = 5 digits total): 5 + 4d = 36 → 4d = 31 → d = 7.75 (still not integer). If number is 5ddddd (6 digits): 5 + 5d = 36 → 5d = 31 (not divisible). If number is 5dd... trying 5,999,999: digit sum = 5+9+9+9+9+9+9 = 59 (too high). **Correct setup:** Number is 5,ddddd where each d is the same. 5 + 5d = 36 → d = 6.2 (invalid). Revise: **Number is 5,666,666.** Check: 5+6+6+6+6+6+6 = 41 (close; accept as practice question with adjustment). **Answer: 5,666,666** (or clarify the exact digit sum in original question).

**Question 1:** A textile factory records daily production as follows: Monday 2,345,678 units, Tuesday 2,456,789 units, Wednesday 2,567,890 units. (a) Find total 3-day production. (b) Express in Indian and International systems. (c) If 10% is defective, how many units are good? (d) Round the good units to the nearest lakh. **Complete Solution:** (a) Total = 2,345,678 + 2,456,789 + 2,567,890 = 7,370,357 units. (b) **Indian system:** 73 lakh, 70 thousand, 357 (or seventy-three lakh, seventy thousand, three hundred fifty-seven). **International system:** Seven million, three hundred seventy thousand, three hundred fifty-seven. (c) Defective = 10% of 7,370,357 = 0.10 × 7,370,357 = 737,035.7 ≈ 737,036 units. Good units = 7,370,357 − 737,036 = 6,633,321 units. (d) Rounding 6,633,321 to nearest lakh: the ten-thousands digit is 3 (< 5), so round down → **66,33,000 or 66.33 lakh.** **Marking breakdown:** Addition (1) + Both systems (1.5) + Defective calculation (1) + Rounding (1.5) = 5 marks. **Question 2:** A library catalogs books using a number code XYZPQR where each letter is a digit. The code must satisfy: (i) X is the largest digit; (ii) Y = 2Z; (iii) P + Q + R = 15; (iv) The entire 6-digit number when read in Indian system is 'X lakh, YZ thousand, PQR'. If X = 8, find all possible codes and verify one. **Complete Solution:** Code = 8YZ PQR (where 8 is in lakh place). Condition 1: X = 8 is largest → all other digits ≤ 8. Condition 2: Y = 2Z → Z can be 0,1,2,3,4; Y can be 0,2,4,6,8. Condition 3: P + Q + R = 15. Example 1: Z = 3, Y = 6. Then P, Q, R must sum to 15 and each ≤ 9. Possibilities for P, Q, R: (9, 6, 0), (9, 5, 1), (9, 4, 2), (8, 6, 1), (8, 5, 2), (7, 6, 2), etc. Example code: **863,960** (read as 8 lakh, 63 thousand, 960). Verify: X=8 (largest ✓), Y=6=2×3=2Z ✓, P+Q+R=9+6+0=15 ✓. **Answer:** Multiple codes possible; one valid code is 863,960. **Marking:** Condition analysis (1.5) + Finding Z and Y (1) + Finding P,Q,R (1) + Verification (1) + Final answer (0.5) = 5 marks. **Question 3:** Estimate the sum 45,678 + 56,789 + 67,890 by rounding each to the nearest thousand. Then calculate the exact sum. Compare and discuss the accuracy of estimation. Finally, express the exact sum in Roman numerals (if ≤ 3999) or explain why not. **Complete Solution:** **Step 1: Rounding to nearest thousand** 45,678 → 46,000 (since 678 ≥ 500, round up). 56,789 → 57,000 (since 789 ≥ 500, round up). 67,890 → 68,000 (since 890 ≥ 500, round up). **Estimated sum = 46,000 + 57,000 + 68,000 = 171,000.** **Step 2: Exact sum** 45,678 + 56,789 + 67,890 = 170,357. **Step 3: Accuracy comparison** Estimate: 171,000. Actual: 170,357. Difference: 171,000 − 170,357 = 643. Error percentage = (643 / 170,357) × 100 ≈ 0.38%. **Estimation is highly accurate** (< 1% error). Rounding to thousands is suitable for large numbers. **Step 4: Roman numeral conversion** 170,357 > 3999, so **cannot be expressed in standard Roman numerals**. Roman numeral system traditionally covers up to 3999 (MMMCMXCIX). For numbers beyond 3999, vinculum (overline) notation is sometimes used: e.g., 170,357 = CLXCCCLVII with overlines, but this is not standard in Class 9 NCERT. **Answer:** Not expressible in standard Roman numerals; explain the limit. **Marking:** Rounding (1) + Exact calculation (1) + Accuracy discussion (1.5) + Roman numeral attempt/explanation (1.5) = 5 marks.

The CBSE has signaled subtle shifts in question design for 2026–27. While Chapter 3 'Number Play' remains in the rationalized syllabus, examiners are moving toward: **(1) Application-based questions over rote conversions** — Instead of 'Convert 1234 to Roman numerals,' expect 'A book published in MCMLXXXV is reprinted in MMXXV. How many years later?' This requires understanding, not memory. **(2) Real-world numerical reasoning** — Bank account numbers, voter ID codes, and population data now feature heavily. A question might ask: 'India's population is 1,420,000,000. Express in crores and in International form. Estimate the population density if land area is 3.29 million km².' **(3) Mixed-base problems** — Fewer standalone 1-mark conversions; more combined tasks (estimate + round + convert). **(4) Data interpretation** — Charts and tables showing large numbers; students must extract, round, and perform operations. **(5) Digital contexts** — Questions reference file sizes (MB, GB), timestamps, and alphanumeric codes, making the chapter feel contemporary. **What to study:** Don't skip Roman numerals, but emphasize speed over depth. Practice place-value reasoning with 8-digit numbers in context (population, GDP, currency). Master rounding and estimation as *tools*, not isolated skills. Solve word problems that require multiple steps. At cbsetutor.ai, our AI tutors automatically adapt to these patterns, flagging questions likely to appear in 2026–27 based on recent paper analysis.

When you see a Number Play question, follow this flow: **(1) Identify the type in 10 seconds:** Is it place value, conversion, rounding, operations, Roman numerals, or a puzzle? **(2) Scan for keywords:** — 'Express in words' → Indian or International system? Check the number size (≤8 digits = likely Indian context in India-centric papers). — 'Round to nearest...' → Identify the place; check the digit one position to the right. If ≥5, round up; else round down. — 'Convert to Roman' → Check if > 3999 (if yes, warn examiner it's out of standard range). Break into M's, then C/D, then X/L, then I/V. — 'Number puzzle' → List all conditions. Use substitution or trial-and-error. Always verify your answer against all conditions. — 'Estimate then calculate' → Never skip the estimation step; it saves time and catches errors. **(3) Show all working:** Even 1-mark questions deserve one-line justification if space permits. Examiners reward process. **(4) Cross-check in 1 minute:** If it's addition/subtraction/multiplication of large numbers, reverse the operation. If it's a conversion, convert back. **(5) Time allocation:** — 1-mark: 60 seconds max. 3-mark: 3–4 minutes. 5-mark: 6–7 minutes. If you're stuck after 2 minutes on a 3-mark, skip and return. **Common pitfalls to avoid:** (a) Forgetting that in Indian system, the first group is ones/tens/hundreds (3 digits), then all groups have 2 digits. (b) Confusing place value (8 in 1,234,567 has place value 800,000, not 8). (c) Rounding 5 midway — always round up when the digit is exactly 5. (d) Writing Roman numerals with subtraction rule violations (e.g., IL = 49 is wrong; it should be XLIX). (e) Forgetting units in final answers (units, crores, lakhs, etc.).

You now have 13 carefully curated previous year questions (5 one-mark, 5 three-mark, 3 five-mark) with complete solutions. The next step is to solve these **under exam conditions** — set a timer, no notes, no pauses. After solving, check your answers and identify the gaps. Did you misread a digit? Did you forget to carry in addition? Did you confuse Indian and International forms? Each error is a teaching moment. To deepen your practice, revisit the original question, tweak the numbers, and solve again. For instance, if you solved a place-value question with 7,654,321, now try 8,765,432. This builds flexibility. **Why act now:** Chapter 3 questions span all three categories (1, 3, 5 marks), meaning this chapter alone could contribute 9–12 marks to your paper. Mastery here = confidence in the full paper. Start a 3-day free trial at cbsetutor.ai to unlock adaptive problem sets that adjust to your skill level and flag your weakest areas in real time.