[{"data":1,"prerenderedAt":1007},["ShallowReactive",2],{"questions:number-system":3},{"bank":4,"contentHash":998,"dependencyHashes":999,"releaseId":1006},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":46,"sourceIds":988,"reviewStatus":994,"authoring":995},1,"number-system","Number system: question bank","Practise every part of the number system: place value, number names, the Indian and International systems, expanded form, neighbours, comparing, forming numbers, rounding and estimation, and Roman numerals. Questions are graded **foundation → core → stretch → challenge**, and every one has a worked solution. Try each question before opening its solution.",[10,14,18,22,26,30,34,38,42],{"id":11,"title":12,"description":13},"pv","Place value and face value","Find what a digit is worth in its place, compare place and face value, and reason about digits in different places.",{"id":15,"title":16,"description":17},"names","Number names","Read and write large numbers in words in the Indian and International systems, and spot common naming mistakes.",{"id":19,"title":20,"description":21},"systems","Indian ↔ International","Place commas in both systems and convert between lakh, crore, million, billion and trillion.",{"id":23,"title":24,"description":25},"expanded","Expanded form","Write numbers as sums of place values, including with powers of ten and jumbled parts, and back again.",{"id":27,"title":28,"description":29},"neighbours","Predecessor and successor","Find the numbers just before and after, smallest and largest numbers of each size, and count whole numbers in a range.",{"id":31,"title":32,"description":33},"compare","Comparing and ordering","Compare numbers, arrange them in ascending and descending order, and count numbers that satisfy conditions.",{"id":35,"title":36,"description":37},"forming","Forming numbers from digits","Make the greatest and smallest numbers from given digits, with and without repetition, the zero rule and extra conditions.",{"id":39,"title":40,"description":41},"rounding","Rounding and estimation","Round to the nearest 10, 100, 1,000, 10,000 or lakh, estimate sums and products, and reason about rounding errors.",{"id":43,"title":44,"description":45},"roman","Roman numerals","Read and write Roman numerals using the addition, subtraction and repetition rules.",[47,60,69,78,88,98,107,132,150,160,173,182,191,200,212,230,240,258,268,278,297,315,324,334,344,362,371,380,397,412,430,439,448,457,466,475,485,495,505,515,524,533,542,551,559,575,591,610,628,646,664,673,682,692,701,711,720,730,739,749,758,767,776,785,794,803,812,821,831,840,849,858,868,884,894,903,913,923,932,942,960,969,979],{"id":48,"section":11,"level":49,"prompt":50,"check":51,"hints":55,"solution":57,"skills":58},"number-system.q001","foundation","What is the place value of 7 in **47,38,205**?",{"kind":52,"answer":53,"tolerance":54},"number",700000,0,[56],"Which place is the 7 in? The commas help.","Count places from the right: 5 ones, 0 tens, 2 hundreds, 8 thousands, 3 ten thousands, **7 lakhs**. Place value = 7 × 1,00,000 = **7,00,000**.",[59],"place value",{"id":61,"section":11,"level":49,"prompt":62,"check":63,"hints":65,"solution":66,"skills":67},"number-system.q002","What is the face value of 3 in **47,38,205**?",{"kind":52,"answer":64,"tolerance":54},3,[],"The face value of a digit is the digit itself, wherever it is: **3**. (Its place value would be 30,000.)",[68],"face value",{"id":70,"section":11,"level":49,"prompt":71,"check":72,"hints":74,"solution":76,"skills":77},"number-system.q003","In **92,615**, which digit is in the thousands place?",{"kind":52,"answer":73,"tolerance":54},2,[75],"Count places from the right.","From the right: 5 ones, 1 tens, 6 hundreds, **2 thousands**, 9 ten thousands. The thousands digit is **2**.",[59],{"id":79,"section":11,"level":80,"prompt":81,"check":82,"hints":84,"solution":86,"skills":87},"number-system.q004","core","In **5,85,358**, find the difference between the place values of the two 8s.",{"kind":52,"answer":83,"tolerance":54},79992,[85],"Find each 8’s place first.","The left 8 is in the ten thousands place: 80,000. The right 8 is in the ones place: 8. Difference: 80,000 − 8 = **79,992**.",[59],{"id":89,"section":11,"level":80,"prompt":90,"check":91,"hints":93,"solution":95,"skills":96},"number-system.q005","What is the sum of the place values of all the non-zero digits of **3,05,07,081**? What do you notice?",{"kind":52,"answer":92,"tolerance":54},30507081,[94],"Write each non-zero digit’s place value and add.","Place values: 3,00,00,000 + 5,00,000 + 7,000 + 80 + 1. Their sum is **3,05,07,081**, the number itself. A number always equals the sum of the place values of its digits; this is its expanded form.",[59,97],"expanded form",{"id":99,"section":11,"level":80,"prompt":100,"check":101,"hints":103,"solution":105,"skills":106},"number-system.q006","In **3,82,564**, what is the product of the place value and the face value of 8?",{"kind":52,"answer":102,"tolerance":54},640000,[104],"Place value first: which place is 8 in?","The 8 is in the ten thousands place: place value 80,000, face value 8. Product: 80,000 × 8 = **6,40,000**.",[59,68],{"id":108,"section":11,"level":109,"prompt":110,"check":111,"hints":127,"solution":129,"skills":130},"number-system.q007","stretch","Riya says: “In the 7-digit number I wrote, the digit 6 is in the crores place.” Which statement is correct?",{"kind":112,"options":113,"correct":126},"choice",[114,117,120,123],{"id":115,"label":116},"a","That is impossible: a 7-digit number has no crores place",{"id":118,"label":119},"b","The place value of 6 is 6,00,00,000",{"id":121,"label":122},"c","The place value of 6 is 60,00,000",{"id":124,"label":125},"d","The face value of 6 is 6,00,00,000",[115],[128],"How many places does a 7-digit number have?","A 7-digit number reaches only the ten lakhs place (the largest is 99,99,999). The crores place is the 8th place from the right, so a 7-digit number cannot have a digit there. Statement **a** is correct.",[59,131],"reasoning",{"id":133,"section":11,"level":109,"prompt":134,"check":135,"hints":146,"solution":148,"skills":149},"number-system.q008","The place value of a digit in a number is **40,00,000**. What are the digit and its place (Indian system)?",{"kind":112,"options":136,"correct":145},[137,139,141,143],{"id":115,"label":138},"4 in the ten lakhs place",{"id":118,"label":140},"4 in the lakhs place",{"id":121,"label":142},"40 in the lakhs place",{"id":124,"label":144},"4 in the crores place",[115],[147],"A place value is always one digit times a power of ten.","40,00,000 = 4 × 10,00,000. The digit is 4 and the place is **ten lakhs** (four million in the International system).",[59],{"id":151,"section":11,"level":152,"prompt":153,"check":154,"hints":156,"solution":158,"skills":159},"number-system.q009","challenge","A number is made of place values 3,00,000; 50,000; 700 and 9. Its ten thousands digit and hundreds digit are swapped. By how much does the number increase?",{"kind":52,"answer":155,"tolerance":54},19800,[157],"Write the number first, then swap.","The number is 3,50,709. Ten thousands digit 5, hundreds digit 7. After swapping: 3,70,509. Increase: 3,70,509 − 3,50,709 = **19,800**. Shortcut: the 7 gains 70,000 − 700 = 69,300 and the 5 loses 50,000 − 500 = 49,500; 69,300 − 49,500 = 19,800.",[59,131],{"id":161,"section":15,"level":49,"prompt":162,"check":163,"hints":168,"solution":170,"skills":171},"number-system.q010","Write **45,060** in words (Indian system).",{"kind":164,"accept":165},"text",[166,167],"forty-five thousand sixty","forty five thousand sixty",[169],"Say the period name at each comma.","Read period by period at the commas: 45 | 060. In words: **forty-five thousand sixty**. Skip any period that is all zeros.",[172],"number names",{"id":174,"section":15,"level":80,"prompt":175,"check":176,"hints":179,"solution":180,"skills":181},"number-system.q011","Write **70,08,090** in words (Indian system).",{"kind":164,"accept":177},[178],"seventy lakh eight thousand ninety",[169],"Read period by period at the commas: 70 | 08 | 090. In words: **seventy lakh eight thousand ninety**. Skip any period that is all zeros.",[172],{"id":183,"section":15,"level":49,"prompt":184,"check":185,"hints":187,"solution":189,"skills":190},"number-system.q012","Write in digits: **three lakh four thousand twenty**.",{"kind":52,"answer":186,"tolerance":54},304020,[188],"Give every period its full number of digits.","Lakhs period: 3. Thousands period: 04. Ones period: 020. Joined: **3,04,020**.",[172],{"id":192,"section":15,"level":80,"prompt":193,"check":194,"hints":196,"solution":198,"skills":199},"number-system.q013","Write in digits: **five crore six lakh seven hundred**.",{"kind":52,"answer":195,"tolerance":54},50600700,[197],"Fill empty periods with zeros.","Crores 5 | lakhs 06 | thousands 00 | ones 700 → **5,06,00,700**.",[172],{"id":201,"section":15,"level":80,"prompt":202,"check":203,"hints":207,"solution":209,"skills":210},"number-system.q014","Write **72,305,008** in words (International system).",{"kind":164,"accept":204},[205,206],"seventy-two million three hundred five thousand eight","seventy two million three hundred five thousand eight",[208],"Read three-digit periods: millions, thousands, ones.","Periods: 72 | 305 | 008 → **seventy-two million three hundred five thousand eight**.",[172,211],"international",{"id":213,"section":15,"level":80,"prompt":214,"check":215,"hints":226,"solution":228,"skills":229},"number-system.q015","Which is the correct name of **60,06,006**?",{"kind":112,"options":216,"correct":225},[217,219,221,223],{"id":115,"label":218},"Sixty lakh six thousand six",{"id":118,"label":220},"Six lakh six thousand six",{"id":121,"label":222},"Sixty lakh sixty thousand six",{"id":124,"label":224},"Six million sixty thousand six",[115],[227],"Read the Indian commas.","Periods: 60 lakh | 06 thousand | 006, so **sixty lakh six thousand six**. In the International system the same number, 6,006,006, is six million six thousand six, so option d (sixty thousand) is wrong too.",[172],{"id":231,"section":15,"level":109,"prompt":232,"check":233,"hints":235,"solution":237,"skills":238},"number-system.q016","Spot the mistake. Arjun wrote *“nine crore nine lakh nine”* as **9,90,00,009**. What should it be?",{"kind":52,"answer":234,"tolerance":54},90900009,[236],"Each Indian period after the ones period has exactly two digits.","Crores 9 | lakhs **09** | thousands 00 | ones 009 → **9,09,00,009**. Arjun wrote 90 in the lakhs period (ninety lakh) instead of 09 (nine lakh).",[172,239],"spot the mistake",{"id":241,"section":15,"level":109,"prompt":242,"check":243,"hints":254,"solution":256,"skills":257},"number-system.q017","A cheque must say the amount in words. Which wording is correct for **₹12,50,075**?",{"kind":112,"options":244,"correct":253},[245,247,249,251],{"id":115,"label":246},"Rupees twelve lakh fifty thousand seventy-five only",{"id":118,"label":248},"Rupees one lakh twenty-five thousand seventy-five only",{"id":121,"label":250},"Rupees twelve lakhs fifty thousands seventy-five only",{"id":124,"label":252},"Rupees one million two hundred fifty thousand seventy-five only",[115],[255],"Read the periods at the commas.","Periods: 12 lakh | 50 thousand | 075 → **twelve lakh fifty thousand seventy-five**. Option c uses plurals, which number names avoid. Option d names the right number in the International system (1,250,075), but a rupee cheque is written in the Indian system, and a is exactly right.",[172],{"id":259,"section":15,"level":152,"prompt":260,"check":261,"hints":263,"solution":265,"skills":266},"number-system.q018","How many times does the word **“thousand”** appear when you write every number from 1,000 to 1,09,999 in words (Indian system)? Count numbers, not words: each number contains the word at most once.",{"kind":52,"answer":262,"tolerance":54},108000,[264],"Which numbers have a thousands period of 00?","A number contains “thousand” only if its thousands period (ten thousands and thousands digits) is not 00. From 1,000 to 99,999 every number has a non-zero thousands period: 99,000 numbers. From 1,00,000 to 1,09,999 the thousands period is 00 for 1,00,000–1,00,999 (1,000 numbers) and non-zero for the other 9,000. Total: 99,000 + 9,000 = **1,08,000**.",[172,267],"counting",{"id":269,"section":19,"level":49,"prompt":270,"check":271,"hints":273,"solution":275,"skills":276},"number-system.q019","How many lakh make one million?",{"kind":52,"answer":272,"tolerance":54},10,[274],"Count the zeros in each.","1 million = 10,00,000 = **10 lakh**.",[277],"conversion",{"id":279,"section":19,"level":49,"prompt":280,"check":281,"hints":292,"solution":294,"skills":295},"number-system.q020","Which is **4567890** written with **International** commas?",{"kind":112,"options":282,"correct":291},[283,285,287,289],{"id":115,"label":284},"45,67,890",{"id":118,"label":286},"4,567,890",{"id":121,"label":288},"456,7890",{"id":124,"label":290},"4,56,78,90",[118],[293],"Groups of three from the right.","Put a comma after every three digits from the right: **4,567,890** (four million five hundred sixty-seven thousand eight hundred ninety). Option a is the Indian way.",[296],"commas",{"id":298,"section":19,"level":49,"prompt":299,"check":300,"hints":311,"solution":313,"skills":314},"number-system.q021","Which is **12345678** written with **Indian** commas?",{"kind":112,"options":301,"correct":310},[302,304,306,308],{"id":115,"label":303},"12,345,678",{"id":118,"label":305},"1,23,45,678",{"id":121,"label":307},"12,34,56,78",{"id":124,"label":309},"1,234,5678",[118],[312],"3 digits, then groups of 2.","First comma after three digits from the right (678), then every two digits: **1,23,45,678**, one crore twenty-three lakh forty-five thousand six hundred seventy-eight.",[296],{"id":316,"section":19,"level":80,"prompt":317,"check":318,"hints":320,"solution":322,"skills":323},"number-system.q022","3 crore is how many million?",{"kind":52,"answer":319,"tolerance":54},30,[321],"1 crore = 10 million.","1 crore = 10 million, so 3 crore = **30 million**. In digits: 3,00,00,000 = 30,000,000.",[277],{"id":325,"section":19,"level":80,"prompt":326,"check":327,"hints":330,"solution":332,"skills":333},"number-system.q023","85 million is how many crore?",{"kind":52,"answer":328,"tolerance":329},8.5,0.001,[331],"Millions ÷ 10 = crores.","1 crore = 10 million, so divide by 10: 85 million = **8.5 crore** (8 crore 50 lakh).",[277],{"id":335,"section":19,"level":80,"prompt":336,"check":337,"hints":339,"solution":341,"skills":342},"number-system.q024","A city has **2.4 million** people. How many lakh is that?",{"kind":52,"answer":338,"tolerance":54},24,[340],"Millions × 10 = lakhs.","1 million = 10 lakh, so 2.4 million = **24 lakh** (24,00,000 = 2,400,000).",[277,343],"real life",{"id":345,"section":19,"level":80,"prompt":346,"check":347,"hints":358,"solution":360,"skills":361},"number-system.q025","Which pair shows the same number?",{"kind":112,"options":348,"correct":357},[349,351,353,355],{"id":115,"label":350},"5,00,00,000 and 5,000,000",{"id":118,"label":352},"50,00,000 and 5,000,000",{"id":121,"label":354},"5,00,000 and 5,000,000",{"id":124,"label":356},"50,00,00,000 and 50,000,000",[118],[359],"Remove commas and count digits.","Remove the commas and compare digits. 50,00,000 → 5000000 and 5,000,000 → 5000000: the same, **b** (fifty lakh = five million). The others differ by a factor of 10 or 100.",[277],{"id":363,"section":19,"level":109,"prompt":364,"check":365,"hints":367,"solution":369,"skills":370},"number-system.q026","India’s population is about **140 crore**. Express this in billions.",{"kind":52,"answer":366,"tolerance":329},1.4,[368],"100 crore = 1 billion.","100 crore = 1 billion, so 140 crore = 140 ÷ 100 = **1.4 billion**.",[277,343],{"id":372,"section":19,"level":109,"prompt":373,"check":374,"hints":376,"solution":378,"skills":379},"number-system.q027","A news site writes “₹24,000 crore”. A foreign site reports the same amount in billions of rupees. What number of billions should it show?",{"kind":52,"answer":375,"tolerance":54},240,[377],"Crores ÷ 100 = billions.","100 crore = 1 billion, so 24,000 crore = 24,000 ÷ 100 = **240 billion** rupees.",[277],{"id":381,"section":19,"level":109,"prompt":382,"check":383,"hints":392,"solution":394,"skills":395},"number-system.q028","True or false: *“The number 75,00,000 is more than 7 million.”*",{"kind":112,"options":384,"correct":391},[385,388],{"id":386,"label":387},"t","True",{"id":389,"label":390},"f","False",[386],[393],"Convert 75 lakh to millions.","**True.** 75,00,000 = 75 lakh = 7.5 million = 7,500,000, which is more than 7,000,000.",[277,396],"comparing",{"id":398,"section":19,"level":152,"prompt":399,"check":400,"hints":407,"solution":409,"skills":410},"number-system.q029","The Union Budget is **₹50 lakh crore**. Write this amount in the International system using the largest suitable name (a whole number of that unit).",{"kind":164,"accept":401},[402,403,404,405,406],"50 trillion","fifty trillion","₹50 trillion","rs 50 trillion","rupees 50 trillion",[408],"Write lakh and crore as powers of ten.","1 lakh crore = 10⁵ × 10⁷ = 10¹² = 1 trillion. So 50 lakh crore = **50 trillion** rupees = 50,000,000,000,000.",[277,411],"powers of ten",{"id":413,"section":23,"level":49,"prompt":414,"check":415,"hints":426,"solution":428,"skills":429},"number-system.q030","Write **6,02,315** in expanded form. Which is correct?",{"kind":112,"options":416,"correct":425},[417,419,421,423],{"id":115,"label":418},"6,00,000 + 2,000 + 300 + 10 + 5",{"id":118,"label":420},"6,00,000 + 20,000 + 300 + 10 + 5",{"id":121,"label":422},"6 + 0 + 2 + 3 + 1 + 5",{"id":124,"label":424},"60,000 + 2,000 + 300 + 15",[115],[427],"Find the place of each digit.","Place values: 6 lakhs, 0 ten thousands, 2 thousands, 3 hundreds, 1 ten, 5 ones → **6,00,000 + 2,000 + 300 + 10 + 5**.",[97],{"id":431,"section":23,"level":49,"prompt":432,"check":433,"hints":435,"solution":437,"skills":438},"number-system.q031","Write as a number: 50,000 + 3,000 + 40 + 7.",{"kind":52,"answer":434,"tolerance":54},53047,[436],"The hundreds place is empty.","Ten thousands 5, thousands 3, hundreds 0, tens 4, ones 7 → **53,047**.",[97],{"id":440,"section":23,"level":80,"prompt":441,"check":442,"hints":444,"solution":446,"skills":447},"number-system.q032","Write as a number: 8 × 1,00,000 + 5 × 1,000 + 6 × 10.",{"kind":52,"answer":443,"tolerance":54},805060,[445],"Make a slot for every place.","Lakhs 8, ten thousands 0, thousands 5, hundreds 0, tens 6, ones 0 → **8,05,060**.",[97],{"id":449,"section":23,"level":80,"prompt":450,"check":451,"hints":453,"solution":455,"skills":456},"number-system.q033","Write as a number: 700 + 4 + 90,00,000 + 20,000 (the parts are jumbled).",{"kind":52,"answer":452,"tolerance":54},9020704,[454],"Sort the parts from biggest to smallest first.","Sort by place: 90,00,000 (ten lakhs 9) + 20,000 (ten thousands 2) + 700 + 4 → **90,20,704**.",[97],{"id":458,"section":23,"level":109,"prompt":459,"check":460,"hints":462,"solution":464,"skills":465},"number-system.q034","Write as a number: 3 × 10⁷ + 4 × 10⁵ + 5 × 10² + 6 × 10⁰.",{"kind":52,"answer":461,"tolerance":54},30400506,[463],"10ⁿ is 1 followed by n zeros.","10⁷ is crores, 10⁵ lakhs, 10² hundreds, 10⁰ ones. Fill the places: 3 | 0 | 4 | 0 | 0 | 5 | 0 | 6 → **3,04,00,506**.",[97,411],{"id":467,"section":23,"level":109,"prompt":468,"check":469,"hints":471,"solution":473,"skills":474},"number-system.q035","Spot the mistake: Neha wrote 40,05,020 = 4,00,000 + 5,000 + 20. What is the value of the number Neha’s expansion actually gives?",{"kind":52,"answer":470,"tolerance":54},405020,[472],"Check the place of the 4.","Neha used 4,00,000 (four lakh) for the 4, but in 40,05,020 the 4 is in the ten lakhs place: 40,00,000. Her sum is 4,00,000 + 5,000 + 20 = **4,05,020**, ten times too small in the first term. Correct: 40,05,020 = 40,00,000 + 5,000 + 20.",[97,239],{"id":476,"section":23,"level":152,"prompt":477,"check":478,"hints":480,"solution":482,"skills":483},"number-system.q036","A number has 1 crore, 12 lakh, 15 thousand and 25 ones (some of these are more than 9 of a unit). Write the number in standard form.",{"kind":52,"answer":479,"tolerance":54},11215025,[481],"Add them as numbers; do not just write the parts side by side.","Add the parts: 1,00,00,000 + 12,00,000 + 15,000 + 25 = **1,12,15,025**. Regrouping: 12 lakh is 1 ten lakh and 2 lakh; 15 thousand is 1 ten thousand and 5 thousand; 25 is 2 tens and 5 ones.",[97,484],"regrouping",{"id":486,"section":27,"level":49,"prompt":487,"check":488,"hints":490,"solution":492,"skills":493},"number-system.q037","What is the successor of **69,999**?",{"kind":52,"answer":489,"tolerance":54},70000,[491],"Add 1.","69,999 + 1 = **70,000**. All the 9s roll over to 0 and the ten thousands digit goes from 6 to 7.",[494],"successor",{"id":496,"section":27,"level":49,"prompt":497,"check":498,"hints":500,"solution":502,"skills":503},"number-system.q038","What is the predecessor of **5,00,000**?",{"kind":52,"answer":499,"tolerance":54},499999,[501],"Subtract 1; the zeros become 9s.","5,00,000 − 1 = **4,99,999**.",[504],"predecessor",{"id":506,"section":27,"level":80,"prompt":507,"check":508,"hints":510,"solution":512,"skills":513},"number-system.q039","What is the smallest 6-digit number minus the largest 4-digit number?",{"kind":52,"answer":509,"tolerance":54},90001,[511],"Smallest 6-digit number = 1,00,000.","1,00,000 − 9,999 = **90,001**.",[514],"smallest and largest",{"id":516,"section":27,"level":80,"prompt":517,"check":518,"hints":520,"solution":522,"skills":523},"number-system.q040","What is the successor of the largest 7-digit number?",{"kind":52,"answer":519,"tolerance":54},10000000,[521],"Largest 7-digit number = seven 9s.","The largest 7-digit number is 99,99,999. Its successor is **1,00,00,000**: one crore, or ten million (10,000,000).",[494],{"id":525,"section":27,"level":80,"prompt":526,"check":527,"hints":529,"solution":531,"skills":532},"number-system.q041","How many whole numbers lie **strictly between** 1,000 and 10,000?",{"kind":52,"answer":528,"tolerance":54},8999,[530],"Count from 1,001 to 9,999.","Numbers from 1,001 to 9,999: 9,999 − 1,001 + 1 = **8,999**. (Careful: “between” here does not include 1,000 and 10,000.)",[267],{"id":534,"section":27,"level":109,"prompt":535,"check":536,"hints":538,"solution":540,"skills":541},"number-system.q042","The successor of a number is **10,00,000**. What is the predecessor of the same number?",{"kind":52,"answer":537,"tolerance":54},999998,[539],"First find the number itself.","The number is 10,00,000 − 1 = 9,99,999. Its predecessor is 9,99,999 − 1 = **9,99,998**.",[504,494],{"id":543,"section":27,"level":109,"prompt":544,"check":545,"hints":547,"solution":549,"skills":550},"number-system.q043","How many 5-digit numbers are there?",{"kind":52,"answer":546,"tolerance":54},90000,[548],"Count from the smallest to the largest 5-digit number.","From 10,000 to 99,999: 99,999 − 10,000 + 1 = **90,000**. Or: 9 choices for the first digit × 10 × 10 × 10 × 10 = 90,000.",[267],{"id":552,"section":27,"level":152,"prompt":553,"check":554,"hints":555,"solution":557,"skills":558},"number-system.q044","How many whole numbers have exactly 7 digits in the Indian system **and** a successor with 8 digits? And how many 7-digit numbers have a predecessor with 6 digits? Give the total of the two counts.",{"kind":52,"answer":73,"tolerance":54},[556],"When does adding 1 increase the number of digits?","A successor gains a digit only when the number is all 9s: only 99,99,999 works (1 number). A predecessor loses a digit only when the number is 1 followed by zeros: only 10,00,000 works (1 number). Total **2**.",[131],{"id":560,"section":31,"level":49,"prompt":561,"check":562,"hints":571,"solution":573,"skills":574},"number-system.q045","Which is greater: **4,56,789** or **4,65,789**?",{"kind":112,"options":563,"correct":570},[564,566,568],{"id":115,"label":565},"4,56,789",{"id":118,"label":567},"4,65,789",{"id":121,"label":569},"They are equal",[118],[572],"Compare from the left.","Both have 6 digits. Lakhs: 4 = 4. Ten thousands: 5 against 6. So **4,65,789** is greater.",[396],{"id":576,"section":31,"level":49,"prompt":577,"check":578,"hints":587,"solution":589,"skills":590},"number-system.q046","Which symbol makes this true: 99,999 ___ 1,00,001?",{"kind":112,"options":579,"correct":586},[580,582,584],{"id":115,"label":581},">",{"id":118,"label":583},"\u003C",{"id":121,"label":585},"=",[118],[588],"Count the digits.","99,999 has 5 digits and 1,00,001 has 6, so **99,999 \u003C 1,00,001**.",[396],{"id":592,"section":31,"level":80,"prompt":593,"check":594,"hints":605,"solution":607,"skills":608},"number-system.q047","Arrange in ascending order: 4,308; 43,080; 4,380; 40,380. Which list is correct?",{"kind":112,"options":595,"correct":604},[596,598,600,602],{"id":115,"label":597},"4,308; 4,380; 40,380; 43,080",{"id":118,"label":599},"4,380; 4,308; 43,080; 40,380",{"id":121,"label":601},"4,308; 40,380; 4,380; 43,080",{"id":124,"label":603},"43,080; 40,380; 4,380; 4,308",[115],[606],"Group by number of digits first.","4-digit numbers first: 4,308 \u003C 4,380 (hundreds 3 against 3, tens 0 against 8). Then 5-digit: 40,380 \u003C 43,080. Ascending: **4,308; 4,380; 40,380; 43,080**. (d is the descending order.)",[609],"ordering",{"id":611,"section":31,"level":80,"prompt":612,"check":613,"hints":624,"solution":626,"skills":627},"number-system.q048","Approximate river lengths: Godavari 1,465 km, Yamuna 1,376 km, Krishna 1,400 km, Narmada 1,312 km. Which river is the **second longest** of these four?",{"kind":112,"options":614,"correct":623},[615,617,619,621],{"id":115,"label":616},"Godavari",{"id":118,"label":618},"Yamuna",{"id":121,"label":620},"Krishna",{"id":124,"label":622},"Narmada",[121],[625],"Compare the hundreds digits, then the tens.","Descending order: 1,465 (Godavari) > 1,400 (Krishna) > 1,376 (Yamuna) > 1,312 (Narmada). The second longest is **Krishna**.",[609,343],{"id":629,"section":31,"level":80,"prompt":630,"check":631,"hints":642,"solution":644,"skills":645},"number-system.q049","Which number is the smallest?",{"kind":112,"options":632,"correct":641},[633,635,637,639],{"id":115,"label":634},"10,01,000",{"id":118,"label":636},"10,00,100",{"id":121,"label":638},"10,10,000",{"id":124,"label":640},"10,00,010",[124],[643],"Find the first place where they differ.","All have 7 digits and start 10. Next compare the ten thousands, thousands, hundreds…: 10,00,010 has zeros for longest. So **10,00,010** is the smallest (ten lakh ten).",[396],{"id":647,"section":31,"level":109,"prompt":648,"check":649,"hints":660,"solution":662,"skills":663},"number-system.q050","Which is the greatest?",{"kind":112,"options":650,"correct":659},[651,653,655,657],{"id":115,"label":652},"7 million",{"id":118,"label":654},"65 lakh",{"id":121,"label":656},"0.6 crore",{"id":124,"label":658},"6,999,999",[115],[661],"Convert everything to digits in one system.","Convert all to Indian digits: 7 million = 70,00,000; 65 lakh = 65,00,000; 0.6 crore = 60,00,000; 6,999,999 = 69,99,999. The greatest is **7 million** (70 lakh), just 1 more than 6,999,999.",[396,277],{"id":665,"section":31,"level":109,"prompt":666,"check":667,"hints":669,"solution":671,"skills":672},"number-system.q051","How many whole numbers are greater than 4,99,995 and less than 5,00,003?",{"kind":52,"answer":668,"tolerance":54},7,[670],"List the smallest and the largest such number.","The numbers are 4,99,996 up to 5,00,002. Count: 5,00,002 − 4,99,996 + 1 = **7**.",[396,267],{"id":674,"section":31,"level":152,"prompt":675,"check":676,"hints":678,"solution":680,"skills":681},"number-system.q052","Using each of the digits 1, 2, 3, 4, 5 exactly once, how many 5-digit numbers are **greater than 43,215**?",{"kind":52,"answer":677,"tolerance":54},33,[679],"Split by the first two digits.","Count by the first place where a number beats 43,215. **Starts with 5:** the other four digits in any order, 4 × 3 × 2 × 1 = 24. **Starts with 45:** 3 × 2 × 1 = 6. **Starts with 43:** the rest must beat 215 using 1, 2, 5: 251, 512, 521 → 3 (125, 152 and 215 itself do not). Total 24 + 6 + 3 = **33**.",[396,267],{"id":683,"section":35,"level":49,"prompt":684,"check":685,"hints":687,"solution":689,"skills":690},"number-system.q053","Using each of the digits 3, 9, 1, 6 exactly once, what is the **greatest** 4-digit number?",{"kind":52,"answer":686,"tolerance":54},9631,[688],"Biggest digit in the biggest place.","Arrange the digits in descending order: **9,631**.",[691],"forming numbers",{"id":693,"section":35,"level":49,"prompt":694,"check":695,"hints":697,"solution":699,"skills":700},"number-system.q054","Using each of the digits 3, 9, 1, 6 exactly once, what is the **smallest** 4-digit number?",{"kind":52,"answer":696,"tolerance":54},1369,[698],"Smallest digit in the biggest place.","Arrange the digits in ascending order: **1,369**.",[691],{"id":702,"section":35,"level":80,"prompt":703,"check":704,"hints":706,"solution":708,"skills":709},"number-system.q055","Using each of 0, 7, 2, 5, 8 exactly once, what is the smallest 5-digit number?",{"kind":52,"answer":705,"tolerance":54},20578,[707],"Which digit cannot go first?","Ascending order is 0, 2, 5, 7, 8, but a number cannot start with 0. Put the smallest non-zero digit, 2, first, then 0: **20,578**.",[691,710],"zero",{"id":712,"section":35,"level":80,"prompt":713,"check":714,"hints":716,"solution":718,"skills":719},"number-system.q056","Using each of 0, 7, 2, 5, 8 exactly once, find the difference between the greatest and the smallest 5-digit numbers.",{"kind":52,"answer":715,"tolerance":54},66942,[717],"Find both numbers first.","Greatest: 87,520. Smallest: 20,578. Difference: 87,520 − 20,578 = **66,942**.",[691],{"id":721,"section":35,"level":80,"prompt":722,"check":723,"hints":725,"solution":727,"skills":728},"number-system.q057","Using the digit 4 and the digit 0, with repetition allowed, and using **both** digits, what is the greatest 6-digit number?",{"kind":52,"answer":724,"tolerance":54},444440,[726],"Where should the single 0 go?","Fill as many high places as possible with 4, but a 0 must appear, so put it in the ones place: **4,44,440**.",[691,729],"repetition",{"id":731,"section":35,"level":80,"prompt":732,"check":733,"hints":735,"solution":737,"skills":738},"number-system.q058","Using the digits 4 and 0 (repetition allowed, both used), what is the smallest 6-digit number?",{"kind":52,"answer":734,"tolerance":54},400000,[736],"The first digit cannot be 0.","The first digit must be 4. Then all other places 0: **4,00,000**. Both digits are used.",[691,729],{"id":740,"section":35,"level":109,"prompt":741,"check":742,"hints":744,"solution":746,"skills":747},"number-system.q059","Using each of 3, 9, 1, 4, 6 exactly once, what is the greatest **even** 5-digit number?",{"kind":52,"answer":743,"tolerance":54},96314,[745],"Which digits can go last?","The last digit must be even: 4 or 6. Ending in 4: 9, 6, 3, 1 in front → 96,314. Ending in 6: 9, 4, 3, 1 in front → 94,316. The greater is **96,314**: use the smaller even digit at the end so the larger one can go higher up.",[691,748],"conditions",{"id":750,"section":35,"level":109,"prompt":751,"check":752,"hints":754,"solution":756,"skills":757},"number-system.q060","Using each of 5, 0, 8, 2, 7 exactly once, what is the smallest **odd** 5-digit number?",{"kind":52,"answer":753,"tolerance":54},20587,[755],"Fix the last digit first.","It must end in 5 or 7. The first digit should be the smallest non-zero digit, 2, then 0. Ending in 7: 2, 0, 5, 8, 7 → 20,587. Ending in 5: 2, 0, 7, 8, 5 → 20,785. Smallest: **20,587**.",[691,748],{"id":759,"section":35,"level":109,"prompt":760,"check":761,"hints":763,"solution":765,"skills":766},"number-system.q061","How many different 3-digit numbers can be made from the digits 0, 3, 6 if each digit is used exactly once?",{"kind":52,"answer":762,"tolerance":54},4,[764],"List them all.","There are 3 × 2 × 1 = 6 arrangements, but 036 and 063 start with 0 and are not 3-digit numbers. That leaves **4**: 306, 360, 603, 630.",[691,267],{"id":768,"section":35,"level":152,"prompt":769,"check":770,"hints":772,"solution":774,"skills":775},"number-system.q062","Using each of 1, 5, 7, 9 exactly once, form the 4-digit number **closest to 6,000**.",{"kind":52,"answer":771,"tolerance":54},5971,[773],"Try the first digits either side of 6.","The best candidates start with 5 (just below 6,000) or 7 (just above). Starting with 5, make the rest as large as possible: 5,971, which is 29 below. Starting with 7, make the rest as small as possible: 7,159, which is 1,159 above. Closest: **5,971**.",[691,131],{"id":777,"section":35,"level":152,"prompt":778,"check":779,"hints":781,"solution":783,"skills":784},"number-system.q063","Using only the digits 2 and 5 (repetition allowed; both need not appear), how many 4-digit numbers can be formed that are **greater than 5,000**?",{"kind":52,"answer":780,"tolerance":54},8,[782],"What must the first digit be?","The first digit must be 5 (a 2 in front gives less than 5,000). The other three places each have 2 choices: 2 × 2 × 2 = 8. All of them, from 5,222 to 5,555, are greater than 5,000, and none equals it. Answer: **8**.",[691,267],{"id":786,"section":39,"level":49,"prompt":787,"check":788,"hints":790,"solution":792,"skills":793},"number-system.q064","Round **3,468** to the nearest hundred.",{"kind":52,"answer":789,"tolerance":54},3500,[791],"Look at the tens digit.","The deciding digit (tens) is 6, which is 5 or more, so round up: **3,500**.",[39],{"id":795,"section":39,"level":49,"prompt":796,"check":797,"hints":799,"solution":801,"skills":802},"number-system.q065","Round **85** to the nearest ten.",{"kind":52,"answer":798,"tolerance":54},90,[800],"5 rounds up.","85 is exactly halfway between 80 and 90. By the usual rule, halfway rounds up: **90**.",[39],{"id":804,"section":39,"level":80,"prompt":805,"check":806,"hints":808,"solution":810,"skills":811},"number-system.q066","Round **7,54,321** to the nearest thousand.",{"kind":52,"answer":807,"tolerance":54},754000,[809],"Look only at the hundreds digit.","The deciding digit (hundreds) is 3, so round down: **7,54,000**.",[39],{"id":813,"section":39,"level":80,"prompt":814,"check":815,"hints":817,"solution":819,"skills":820},"number-system.q067","Round **23,47,856** to the nearest lakh.",{"kind":52,"answer":816,"tolerance":54},2300000,[818],"The deciding digit is one place to the right of the lakhs.","Rounding to lakhs: the deciding digit is the ten thousands digit, 4. Round down: **23,00,000**.",[39],{"id":822,"section":39,"level":80,"prompt":823,"check":824,"hints":826,"solution":828,"skills":829},"number-system.q068","Estimate **4,896 + 2,134** by rounding each number to the nearest thousand.",{"kind":52,"answer":825,"tolerance":54},7000,[827],"Round each number first.","4,896 → 5,000 and 2,134 → 2,000. Estimate: 5,000 + 2,000 = **7,000**. (Exact: 7,030.)",[830],"estimation",{"id":832,"section":39,"level":80,"prompt":833,"check":834,"hints":836,"solution":838,"skills":839},"number-system.q069","Estimate **58 × 312** by rounding each number to its greatest place.",{"kind":52,"answer":835,"tolerance":54},18000,[837],"Round to one non-zero digit each.","58 → 60 and 312 → 300. Estimate: 60 × 300 = **18,000**. (Exact: 18,096.)",[830],{"id":841,"section":39,"level":109,"prompt":842,"check":843,"hints":845,"solution":847,"skills":848},"number-system.q070","What is the **smallest** whole number that rounds to 7,000 when rounded to the nearest thousand?",{"kind":52,"answer":844,"tolerance":54},6500,[846],"What is halfway between 6,000 and 7,000?","Numbers from 6,500 up to 7,499 round to 7,000. The smallest is **6,500** (it is exactly halfway and rounds up).",[39],{"id":850,"section":39,"level":109,"prompt":851,"check":852,"hints":854,"solution":856,"skills":857},"number-system.q071","Round **9,97,500** to the nearest ten thousand.",{"kind":52,"answer":853,"tolerance":54},1000000,[855],"Carry, just like in addition.","The ten thousands digit is 9 and the deciding digit (thousands) is 7, so round up. 99 ten thousands + 1 = 100 ten thousands: **10,00,000**. The carry makes a 7-digit number.",[39],{"id":859,"section":39,"level":109,"prompt":860,"check":861,"hints":863,"solution":865,"skills":866},"number-system.q072","Sana rounds 3,448 to the nearest ten, then rounds her answer to the nearest hundred. Ravi rounds 3,448 directly to the nearest hundred. What is **Sana’s answer minus Ravi’s answer**?",{"kind":52,"answer":862,"tolerance":54},100,[864],"Do both carefully.","Sana: 3,448 → 3,450 → 3,500. Ravi: 3,448 → 3,400 (tens digit 4). Difference: 3,500 − 3,400 = **100**. Ravi is right: rounding twice can push a number past the halfway point.",[39,867],"double rounding",{"id":869,"section":39,"level":109,"prompt":870,"check":871,"hints":880,"solution":882,"skills":883},"number-system.q073","A school has 1,238 students. Each needs 18 notebooks. The headteacher estimates by rounding 1,238 to the nearest hundred and 18 to the nearest ten. Is her estimate more or less than the exact number of notebooks needed?",{"kind":112,"options":872,"correct":879},[873,875,877],{"id":115,"label":874},"More",{"id":118,"label":876},"Less",{"id":121,"label":878},"Exactly equal",[115],[881],"Compare how much each number changed, as a fraction of its size.","Estimate: 1,200 × 20 = 24,000. Exact: 1,238 × 18 = 22,284. The estimate is **more**, by 1,716. One number was rounded down (1,238 → 1,200, about 3% smaller) and one up (18 → 20, about 11% bigger); the bigger relative change wins, so the estimate is too high.",[830,131],{"id":885,"section":39,"level":152,"prompt":886,"check":887,"hints":889,"solution":891,"skills":892},"number-system.q074","Six numbers are each rounded to the nearest hundred and then added. What is the **largest possible** difference between this estimated total and the exact total?",{"kind":52,"answer":888,"tolerance":54},300,[890],"What is the most one number can change?","Rounding to the nearest hundred changes a number by at most 50 (for example 350 → 400). With six numbers, the errors can add up to at most 6 × 50 = **300**, if every number is pushed the same way by the full 50. (Strictly, the most a number can be rounded **down** is 49, as in 449 → 400, but rounding up can be a full 50, as in 350 → 400.)",[830,893],"error bounds",{"id":895,"section":39,"level":152,"prompt":896,"check":897,"hints":899,"solution":901,"skills":902},"number-system.q075","A number rounds to **40,000** to the nearest ten thousand and to **35,000** to the nearest thousand. What is the smallest whole number it could be?",{"kind":52,"answer":898,"tolerance":54},35000,[900],"Write the range of numbers for each condition, then overlap them.","To the nearest ten thousand = 40,000 means 35,000 ≤ n ≤ 44,999. To the nearest thousand = 35,000 means 34,500 ≤ n ≤ 35,499. Both together: 35,000 ≤ n ≤ 35,499. Smallest: **35,000**.",[39,131],{"id":904,"section":43,"level":49,"prompt":905,"check":906,"hints":908,"solution":910,"skills":911},"number-system.q076","What number is **XVII**?",{"kind":52,"answer":907,"tolerance":54},17,[909],"Add from left to right.","X = 10, V = 5, II = 2. All added (biggest first): 10 + 5 + 2 = **17**.",[912],"roman numerals",{"id":914,"section":43,"level":49,"prompt":915,"check":916,"hints":919,"solution":921,"skills":922},"number-system.q077","Write **9** in Roman numerals.",{"kind":164,"accept":917},[918],"IX",[920],"One less than ten.","9 = 10 − 1: put I before X → **IX**.",[912],{"id":924,"section":43,"level":80,"prompt":925,"check":926,"hints":928,"solution":930,"skills":931},"number-system.q078","What number is **XLIV**?",{"kind":52,"answer":927,"tolerance":54},44,[929],"Look for subtraction pairs.","XL = 50 − 10 = 40 and IV = 5 − 1 = 4. Total **44**.",[912],{"id":933,"section":43,"level":80,"prompt":934,"check":935,"hints":938,"solution":940,"skills":941},"number-system.q079","Write **2026** in Roman numerals.",{"kind":164,"accept":936},[937],"MMXXVI",[939],"Split by place value.","2,026 = 2,000 + 20 + 6 = MM + XX + VI = **MMXXVI**.",[912],{"id":943,"section":43,"level":80,"prompt":944,"check":945,"hints":956,"solution":958,"skills":959},"number-system.q080","Which Roman numeral is **not** written correctly?",{"kind":112,"options":946,"correct":955},[947,949,951,953],{"id":115,"label":948},"XC",{"id":118,"label":950},"CM",{"id":121,"label":952},"VX",{"id":124,"label":954},"XIX",[121],[957],"Which symbols may be subtracted?","V is never subtracted, so **VX** is wrong (5 is just V). XC = 90, CM = 900 and XIX = 19 are all correct.",[912],{"id":961,"section":43,"level":109,"prompt":962,"check":963,"hints":965,"solution":967,"skills":968},"number-system.q081","What is **MCMXLVII + XXIX**? Give the answer as an ordinary number.",{"kind":52,"answer":964,"tolerance":54},1976,[966],"Convert both to ordinary numbers first.","MCMXLVII = 1,000 + 900 + 40 + 7 = 1,947 (the year India became independent). XXIX = 29. Sum: **1,976**, which is MCMLXXVI.",[912],{"id":970,"section":43,"level":109,"prompt":971,"check":972,"hints":975,"solution":977,"skills":978},"number-system.q082","Spot the mistake: a student wrote 99 as **IC**. What is the correct Roman numeral for 99?",{"kind":164,"accept":973},[974],"XCIX",[976],"Split 99 into tens and ones.","I may only be subtracted from V and X. Split 99 by place value: 90 + 9 = XC + IX = **XCIX**.",[912,239],{"id":980,"section":43,"level":152,"prompt":981,"check":982,"hints":984,"solution":986,"skills":987},"number-system.q083","Using the standard rules, which whole number from 1 to 3,999 has the **longest** Roman numeral? (Enter the number.)",{"kind":52,"answer":983,"tolerance":54},3888,[985],"Which digit needs the most symbols?","Each place is written separately, and the digit 8 needs the most symbols in any place (VIII, LXXX, DCCC = 4 symbols each). The thousands place can hold at most 3 (MMM). So the longest is 3,888 = MMM + DCCC + LXXX + VIII = **MMMDCCCLXXXVIII**, with 15 symbols.",[912,131],[989,990,991,992,993],"number-system-ncert-class6-knowing-numbers","number-system-ncert-class7-large-numbers","number-system-khan-place-value","number-system-mathsisfun-roman","number-system-wiki-indian-numbering","needs_review",{"generatedBy":996,"notes":997},"claude-code","Draft; every numeric answer computed and asserted in Python. Pending owner review.","d156a0258cf7967c6a8120bc88ee043f52ab99f57e9b8a92f049fab032d61926",{"logic:questions":1000,"source:number-system-khan-place-value":1001,"source:number-system-mathsisfun-roman":1002,"source:number-system-ncert-class6-knowing-numbers":1003,"source:number-system-ncert-class7-large-numbers":1004,"source:number-system-wiki-indian-numbering":1005},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","8d110718a4c12de7f84b3e0653db1fd8b1147917a7dc31c48861b97c25731e0d","ef0f2abd2b3d7e393f1878aa51359c7a18194adf57596671bc0dff39cd1aa2f5","4e646d96c29f6b98469c9d11ee30da4e543f849b673fd067975aa685d8fef8a3","9dc5f1741ed9fad68db69882c1cc09c8d4b3cf5902d5d0273891bad1da1d10c0","045032c28b7bfe33f8bc73c5c09ee8dd8076293c90dbe922d8a461e059edd6e3","preview-7e1cbbcc4f",1789899599873]