[{"data":1,"prerenderedAt":1264},["ShallowReactive",2],{"layer:four-operations:understand":3},{"layer":4,"contentHash":1242,"dependencyHashes":1243,"approval":1258,"releaseId":1263},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1237,"reviewStatus":1238,"authoring":1239},1,"four-operations","en","understand","How the column methods work","Carrying, borrowing, long multiplication and long division, and why every step is allowed","Learn the exact name for every part of a calculation, then master column addition and subtraction up to crores, long multiplication, long division with remainders and zeros in the quotient, checking with inverse operations, and working with money and units.",[13,14,15,16,17],"Use the words addend, sum, minuend, subtrahend, difference, multiplicand, multiplier, product, dividend, divisor, quotient and remainder correctly.","Add and subtract numbers up to crores in columns, carrying and borrowing (including across zeros).","Multiply by 10, 100 and 1,000 and carry out long multiplication with correctly shifted partial products.","Carry out long division by one- and two-digit divisors, keep zeros in the quotient and check with dividend = divisor × quotient + remainder.","Convert units of money, length, mass and capacity before calculating, and recognise the most common arithmetic mistakes.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Times tables; meanings of + − × ÷",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Word match, three sprints, mistake sort",{"label":38,"value":39},"Units used","₹, paise, km, m, kg, g, L, mL",[41,45,51,57,60,85,170,187,192,218,262,267,270,291,306,320,325,330,344,367,372,375,388,404,408,422,434,439,442,464,469,478,490,495,498,508,519,523,536,549,582,587,590,610,621,638,658,662,674,701,706,709,720,732,736,748,753,756,774,784,797,817,823,828,831,863,873,883,893,897,910,915,918,967,988,1058,1063,1067,1071,1075,1079,1208,1224],{"id":42,"type":43,"markdown":44},"intro","prose","You already know *what* the four operations mean: adding joins, subtracting takes away or compares, multiplying makes equal groups, dividing shares or groups. This layer is about **how** we actually do them when the numbers get big, like a kirana shop's monthly sales of ₹2,48,675, or a district population in lakhs, and **why each step of the method is allowed**.\n\nEvery column method you will meet rests on one idea: **place value**. A digit's position tells you whether it counts ones, tens, hundreds or lakhs. Carrying, borrowing, shifting a row one place left in long multiplication and bringing a digit down in long division are all ways of respecting that one idea.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to-use","callout","observation","How to use this lesson","Read the chapters in order the first time. Keep a notebook open: when you see a worked example, **cover the steps and try it first**, then compare. The mistakes chapter at the end is worth revisiting before any test.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","The exact words for every part of a calculation","Chapter 01","1 The exact words",{"id":58,"type":43,"markdown":59},"vocab-why","Mathematicians give every part of a calculation its own name. That sounds fussy, but it saves a lot of confusion. \"Divide 84 by 7\" and \"divide 7 into 84\" both mean 84 ÷ 7, and the words **dividend** and **divisor** tell you exactly which number is which, whatever order the sentence uses.\n\nHere is the full set for the four operations.",{"id":61,"type":62,"caption":63,"columns":64,"rows":68},"vocab-table","table","The name of every part, with one example each",[65,66,67],"Operation","Example","Names of the parts",[69,73,77,81],[70,71,72],"Addition","3,456 + 1,289 = 4,745","3,456 and 1,289 are **addends**; 4,745 is the **sum** (or total).",[74,75,76],"Subtraction","3,456 − 1,289 = 2,167","3,456 is the **minuend** (the number you start from); 1,289 is the **subtrahend** (the number taken away); 2,167 is the **difference**.",[78,79,80],"Multiplication","245 × 16 = 3,920","245 is the **multiplicand** (the number being multiplied); 16 is the **multiplier** (how many times); both are **factors**; 3,920 is the **product**.",[82,83,84],"Division","2,345 ÷ 12 = 195 remainder 5","2,345 is the **dividend** (the number being divided); 12 is the **divisor** (what you divide by); 195 is the **quotient**; 5 is the **remainder** (what is left over).",{"id":86,"type":87,"title":88,"terms":89},"glossary-main","glossary","Vocabulary of the four operations",[90,94,98,102,106,110,114,118,122,126,130,134,138,142,146,150,154,158,162,166],{"term":91,"meaning":92,"example":93},"addend","Any one of the numbers being added together.","In 45 + 30 = 75, both 45 and 30 are addends.",{"term":95,"meaning":96,"example":97},"sum","The result of adding. Also called the total.","The sum of 45 and 30 is 75.",{"term":99,"meaning":100,"example":101},"minuend","In a subtraction, the number you start with and take away from.","In 90 − 35 = 55, the minuend is 90.",{"term":103,"meaning":104,"example":105},"subtrahend","In a subtraction, the number being taken away.","In 90 − 35 = 55, the subtrahend is 35.",{"term":107,"meaning":108,"example":109},"difference","The result of subtracting: how much bigger one number is than another.","The difference between 90 and 35 is 55.",{"term":111,"meaning":112,"example":113},"multiplicand","The number being multiplied (the size of each group).","In 24 × 6, the multiplicand is 24 if we read it as 6 groups of 24.",{"term":115,"meaning":116,"example":117},"multiplier","The number you multiply by (how many groups).","In 24 × 6, the multiplier is 6.",{"term":119,"meaning":120,"example":121},"factor","Either number in a multiplication. Multiplicand and multiplier are both factors.","In 24 × 6 = 144, 24 and 6 are factors of 144.",{"term":123,"meaning":124,"example":125},"product","The result of multiplying.","The product of 24 and 6 is 144.",{"term":127,"meaning":128,"example":129},"dividend","The number being divided.","In 144 ÷ 6 = 24, the dividend is 144.",{"term":131,"meaning":132,"example":133},"divisor","The number you divide by: the size of each group, or the number of shares.","In 144 ÷ 6 = 24, the divisor is 6.",{"term":135,"meaning":136,"example":137},"quotient","The result of dividing: how many in each share, or how many groups.","In 144 ÷ 6 = 24, the quotient is 24.",{"term":139,"meaning":140,"example":141},"remainder","What is left over when a division does not come out exactly. It is always smaller than the divisor.","50 ÷ 6 = 8 remainder 2, because 6 × 8 = 48 and 50 − 48 = 2.",{"term":143,"meaning":144,"example":145},"regrouping","Exchanging 10 of one place for 1 of the next place up (carrying), or 1 for 10 of the place below (borrowing).","12 ones are regrouped as 1 ten and 2 ones.",{"term":147,"meaning":148,"example":149},"carry","In addition or multiplication, the extra ten (or hundred…) moved into the next column on the left.","7 + 8 = 15: write 5, carry 1 ten.",{"term":151,"meaning":152,"example":153},"borrow","In subtraction, taking 1 from the next column on the left to make 10 more in the current column. Also called regrouping or exchanging.","In 52 − 17, borrow a ten: 12 − 7 = 5.",{"term":155,"meaning":156,"example":157},"partial product","In long multiplication, the product of the multiplicand with one digit of the multiplier, shifted to that digit's place.","In 46 × 23, the partial products are 46 × 3 = 138 and 46 × 20 = 920.",{"term":159,"meaning":160,"example":161},"inverse operation","An operation that undoes another. Subtraction undoes addition; division undoes multiplication.","72 − 30 = 42 is undone by 42 + 30 = 72.",{"term":163,"meaning":164,"example":165},"algorithm","A fixed list of steps that always works, such as the column method or long division.","Long division is an algorithm: divide, multiply, subtract, bring down, repeat.",{"term":167,"meaning":168,"example":169},"exact division","A division with remainder 0: the divisor is a factor of the dividend.","63 ÷ 9 = 7 exactly.",{"id":171,"type":172,"tone":173,"items":174},"vocab-spec","spec","blue",[175,178,181,184],{"label":70,"big":176,"value":177},"a + b = sum","Addends can be added in any order: 45 + 30 = 30 + 45.",{"label":74,"big":179,"value":180},"minuend − subtrahend","Order matters: 90 − 35 is not 35 − 90.",{"label":78,"big":182,"value":183},"factor × factor","Order does not change the product: 24 × 6 = 6 × 24 = 144.",{"label":82,"big":185,"value":186},"dividend ÷ divisor","Order matters: 144 ÷ 6 = 24 but 6 ÷ 144 is not a whole number.",{"id":188,"type":47,"variant":189,"title":190,"markdown":191},"mis-divide-into","misconception","“Divide 7 into 84” means 7 ÷ 84","It does not. \"Divide 7 **into** 84\" means *how many 7s fit into 84*, which is **84 ÷ 7 = 12**. \"Divide 84 **by** 7\" means the same thing. Whenever the wording is tricky, ask: *which number is being shared or split up?* That one is the **dividend**, and it goes first.",{"id":193,"type":194,"itemId":195,"prompt":196,"check":197,"hints":213,"feedback":215},"vocab-practice","practice","four-operations.u-vocab-subtrahend","In **7,000 − 2,468 = 4,532**, which number is the **subtrahend**?",{"kind":198,"options":199,"correct":212},"choice",[200,203,206,209],{"id":201,"label":202},"a","7,000",{"id":204,"label":205},"b","2,468",{"id":207,"label":208},"c","4,532",{"id":210,"label":211},"d","There is no subtrahend in this sum",[204],[214],"The subtrahend is the number being *subtracted* (taken away).",{"correct":216,"incorrect":217},"Yes. 2,468 is taken away, so it is the subtrahend. 7,000 is the minuend and 4,532 is the difference.","The subtrahend is the number taken away: 2,468. The minuend is the number you start from (7,000), and the answer, 4,532, is the difference.",{"id":219,"type":220,"component":221,"componentVersion":5,"config":222,"objective":256,"textAlternative":257,"help":258},"lab-vocab-match","interactive","match-pairs",{"prompt":223,"mode":224,"pairs":225},"Match each word to what it means.","connect",[226,229,232,235,238,241,244,247,250,253],{"a":227,"b":228},"Addend","A number being added",{"a":230,"b":231},"Minuend","The number you subtract from",{"a":233,"b":234},"Subtrahend","The number being taken away",{"a":236,"b":237},"Multiplier","How many groups you multiply by",{"a":239,"b":240},"Product","The answer to a multiplication",{"a":242,"b":243},"Dividend","The number being divided",{"a":245,"b":246},"Divisor","The number you divide by",{"a":248,"b":249},"Quotient","The answer to a division",{"a":251,"b":252},"Remainder","What is left over, always less than the divisor",{"a":254,"b":255},"Difference","The answer to a subtraction","Connect each operation word (addend, minuend, subtrahend, dividend, divisor and so on) to its exact meaning.","This game shows ten words on one side and ten meanings on the other. You connect each word to its meaning.\n\nThe correct pairs are:\n\n- **Addend**: a number being added.\n- **Minuend**: the number you subtract from.\n- **Subtrahend**: the number being taken away.\n- **Multiplier**: how many groups you multiply by.\n- **Product**: the answer to a multiplication.\n- **Dividend**: the number being divided.\n- **Divisor**: the number you divide by.\n- **Quotient**: the answer to a division.\n- **Remainder**: what is left over, always less than the divisor.\n- **Difference**: the answer to a subtraction.\n\nA memory trick: the words ending in **-end** (addend, minuend, dividend) are numbers something is *done to*; the **divisor** and **multiplier** are the numbers doing it.",{"hints":259},[260,261],"Sum, difference, product and quotient are the four *answers*. Find those first.","Subtra-hend is the one being subtra-cted.",{"id":263,"type":53,"title":264,"eyebrow":265,"navLabel":266},"ch2","Column addition and carrying, up to crores","Chapter 02","2 Column addition",{"id":268,"type":43,"markdown":269},"add-idea","To add large numbers, write them one under the other so that **digits of the same place value stand in the same column**: ones under ones, tens under tens, lakhs under lakhs. Then add one column at a time, **starting from the ones on the right**.\n\nWhy start on the right? Because a column can overflow. If the ones add up to 15, that is 1 ten and 5 ones. The 5 stays; the 1 ten must be **carried** into the tens column. If you started on the left, you would have to go back and fix columns you had already written.",{"id":271,"type":272,"title":273,"items":274},"add-steps","steps","Column addition in four moves",[275,279,283,287],{"title":276,"tag":277,"text":278},"Line up","right edges","Write the numbers so their ones digits are in one column. Numbers of different lengths stay right-aligned.",{"title":280,"tag":281,"text":282},"Add the ones","start right","Add the ones digits. Write the ones digit of the answer; carry the tens digit to the next column.",{"title":284,"tag":285,"text":286},"Move left","include carries","Add each next column plus anything carried into it. Write one digit, carry the rest.",{"title":288,"tag":289,"text":290},"Finish","last carry","If the leftmost column overflows, write the carry as a new digit on the far left.",{"id":292,"type":293,"title":294,"problem":295,"steps":296,"help":304},"we-add-kirana","worked_example","A kirana shop's three months of sales","A kirana shop in Nagpur sold goods worth ₹2,48,675 in January, ₹3,07,489 in February and ₹1,96,358 in March. What were its total sales for the three months?\n\n```\n  2,48,675\n  3,07,489\n+ 1,96,358\n----------\n  7,52,522\n```",[297,298,299,300,301,302,303],"**Ones:** 5 + 9 + 8 = 22. Write 2, carry 2 to the tens.","**Tens:** 7 + 8 + 5 + 2 (carried) = 22. Write 2, carry 2 to the hundreds.","**Hundreds:** 6 + 4 + 3 + 2 (carried) = 15. Write 5, carry 1 to the thousands.","**Thousands:** 8 + 7 + 6 + 1 (carried) = 22. Write 2, carry 2 to the ten thousands.","**Ten thousands:** 4 + 0 + 9 + 2 (carried) = 15. Write 5, carry 1 to the lakhs.","**Lakhs:** 2 + 3 + 1 + 1 (carried) = 7. Write 7.","Total sales: **₹7,52,522** (seven lakh fifty-two thousand five hundred twenty-two rupees).",{"simplerExplanation":305},"Work right to left. Whenever a column adds to 10 or more, keep the last digit and pass the extra tens along to the next column.",{"id":307,"type":293,"title":308,"problem":309,"steps":310},"we-add-crore","Adding numbers in crores","Two neighbouring districts had **4,56,78,925** and **3,87,45,678** people (imaginary figures). How many people live in the two districts together?\n\n```\n  4,56,78,925\n+ 3,87,45,678\n ------------\n  8,44,24,603\n```",[311,312,313,314,315,316,317,318,319],"**Ones:** 5 + 8 = 13. Write 3, carry 1 to the tens.","**Tens:** 2 + 7 + 1 (carried) = 10. Write 0, carry 1 to the hundreds.","**Hundreds:** 9 + 6 + 1 (carried) = 16. Write 6, carry 1 to the thousands.","**Thousands:** 8 + 5 + 1 (carried) = 14. Write 4, carry 1 to the ten thousands.","**Ten thousands:** 7 + 4 + 1 (carried) = 12. Write 2, carry 1 to the lakhs.","**Lakhs:** 6 + 7 + 1 (carried) = 14. Write 4, carry 1 to the ten lakhs.","**Ten lakhs:** 5 + 8 + 1 (carried) = 14. Write 4, carry 1 to the crores.","**Crores:** 4 + 3 + 1 (carried) = 8. Write 8.","Together: **8,44,24,603**, that is 8 crore 44 lakh 24 thousand 6 hundred and 3 people. In the international system this is 84,424,603.",{"id":321,"type":47,"variant":322,"title":323,"markdown":324},"aha-carry","aha","A carry is just an exchange","Carrying is not a trick; it is **exchanging**. Ten 1-rupee coins can be exchanged for one ₹10 note. Ten ₹10 notes can be exchanged for one ₹100 note. When the ones column adds to 17, you have 17 one-rupee coins: you swap 10 of them for a ₹10 note and put it with the tens. That is all a carry is.",{"id":326,"type":47,"variant":327,"title":328,"markdown":329},"careful-align","careful","Right-align, never left-align","When numbers have different lengths, line up their **right-hand** ends. Adding 45,320 and 987 with the left edges lined up puts 9 hundreds under 4 ten-thousands, and gives nonsense. Right-aligned, the answer is **46,307**. Writing the commas in the same places helps you check the alignment.",{"id":331,"type":194,"itemId":332,"prompt":333,"check":334,"hints":338,"feedback":341},"add-practice","four-operations.u-add-three-numbers","Add **36,087 + 9,468 + 125,907** using columns.",{"kind":335,"answer":336,"tolerance":337},"number",171462,0,[339,340],"Right-align the three numbers first.","The ones column adds to 7 + 8 + 7 = 22: write 2, carry 2.",{"correct":342,"incorrect":343},"Correct: 171,462. The ones gave 22 (write 2, carry 2), and every column after that had a carry too.","Line up the ones digits and add column by column from the right, remembering each carry. The total is 171,462.",{"id":345,"type":220,"component":346,"componentVersion":5,"config":347,"objective":361,"textAlternative":362,"help":363},"lab-sprint-addsub","arith-sprint",{"operations":348,"ranges":351,"rounds":358,"secondsTotal":337,"estimateFirst":359,"wordProblems":360},[349,350],"+","-",{"a":352,"b":355},{"min":353,"max":354},10000,99999,{"min":356,"max":357},1000,9999,10,false,[],"Add and subtract five-digit and four-digit numbers accurately, carrying and borrowing where needed.","This sprint gives you ten questions. Each one adds or subtracts a four-digit number to or from a five-digit number, such as 47,806 + 5,387 or 62,014 − 8,659. There is no timer, so work carefully: write the numbers in columns on paper, right-aligned, and work from the ones column leftwards.\n\nWorked samples:\n\n- 47,806 + 5,387: ones 6 + 7 = 13 (write 3, carry 1); tens 0 + 8 + 1 = 9; hundreds 8 + 3 = 11 (write 1, carry 1); thousands 7 + 5 + 1 = 13 (write 3, carry 1); ten thousands 4 + 1 = 5. Answer **53,193**.\n- 62,014 − 8,659: ones 4 \u003C 9, borrow from the tens (1 → 0), 14 − 9 = 5; tens 0 \u003C 5, borrow from the hundreds (0 → borrow from thousands: 2 → 1, hundreds 10 → 9), 10 − 5 = 5; hundreds 9 − 6 = 3; thousands 1 \u003C 8, borrow: 11 − 8 = 3; ten thousands 5. Answer **53,355**.\n\nCheck every subtraction by adding the answer back to the number you took away.",{"hints":364},[365,366],"Always begin with the ones column.","For subtraction, check: difference + subtrahend = minuend.",{"id":368,"type":53,"title":369,"eyebrow":370,"navLabel":371},"ch3","Column subtraction and borrowing","Chapter 03","3 Column subtraction",{"id":373,"type":43,"markdown":374},"sub-idea","Column subtraction works the same way, from right to left, one place at a time. The difficulty comes when a digit on top is **smaller** than the digit below it, as in 52 − 17: you cannot take 7 ones from 2 ones.\n\nSo you **borrow**, which really means **regroup**: take 1 ten from the tens column and change it into 10 ones. Now the ones column has 12, and 12 − 7 = 5. The tens column has one fewer ten, so it is 4 − 1 = 3. Answer: 35.\n\nThe number itself has not changed: 52 is still 5 tens and 2 ones, *or* 4 tens and 12 ones. You have just rewritten it in a form that lets the subtraction happen.",{"id":376,"type":293,"title":377,"problem":378,"steps":379},"we-sub-borrow","Borrowing in several columns","A cooperative bank had ₹7,02,315 in its account and paid out ₹3,48,629 for seeds. How much is left?\n\n```\n  7,02,315\n− 3,48,629\n----------\n  3,53,686\n```",[380,381,382,383,384,385,386,387],"**Ones:** 5 is smaller than 9, so borrow: the tens digit becomes 0, and the ones digit becomes 15. Then 15 − 9 = 6.","**Tens:** 0 is smaller than 2, so borrow: the hundreds digit becomes 2, and the tens digit becomes 10. Then 10 − 2 = 8.","**Hundreds:** 2 is smaller than 6, so borrow: the thousands digit becomes 1, and the hundreds digit becomes 12. Then 12 − 6 = 6.","**Thousands:** 1 is smaller than 8, so borrow: the lakhs digit becomes 6, the ten thousands digit becomes 9, and the thousands digit becomes 11. Then 11 − 8 = 3.","**Ten thousands:** 9 − 4 = 5.","**Lakhs:** 6 − 3 = 3.","Left: **₹3,53,686**.","Check by adding back: 3,53,686 + 3,48,629 = 7,02,315. ✓",{"id":389,"type":293,"title":390,"problem":391,"steps":392,"help":401},"we-sub-zeros","Borrowing across a row of zeros","Work out **5,00,000 − 2,78,346**. This is the one that trips people up: there is nothing in the ones, tens, hundreds, thousands or ten-thousands to borrow from.\n\n```\n  5,00,000\n− 2,78,346\n----------\n  2,21,654\n```",[393,394,395,396,397,398,399,400],"**Ones:** 0 is smaller than 6, so borrow: the lakhs digit becomes 4, the ten thousands digit becomes 9, the thousands digit becomes 9, the hundreds digit becomes 9, the tens digit becomes 9, and the ones digit becomes 10. Then 10 − 6 = 4.","A quicker way to see it: 5,00,000 is the same as **4 lakhs, 9 ten-thousands, 9 thousands, 9 hundreds, 9 tens and 10 ones**. Rewrite the top number that way once, and no more borrowing is needed.","**Tens:** 9 − 4 = 5.","**Hundreds:** 9 − 3 = 6.","**Thousands:** 9 − 8 = 1.","**Ten thousands:** 9 − 7 = 2.","**Lakhs:** 4 − 2 = 2.","Answer: **2,21,654**. Check: 2,21,654 + 2,78,346 = 5,00,000. ✓",{"simplerExplanation":402,"anotherExample":403},"When you need to borrow and the next digit is 0, keep walking left until you reach a non-zero digit. Take 1 from it, and every 0 you walked past turns into 9.","1,000 − 1 = 999: borrowing turns 1,000 into 9 hundreds, 9 tens and 10 ones, and 10 − 1 = 9. Similarly 10,000 − 4,567 = 5,433.",{"id":405,"type":47,"variant":189,"title":406,"markdown":407},"mis-smaller-from-bigger","“Just take the smaller digit from the bigger one”","A very common wrong answer to 5,203 − 1,768 is **4,565**: in each column the learner took the smaller digit from the bigger one (8 − 3, 6 − 0, 7 − 2, 5 − 1), whichever was on top. The true answer is **3,435**. In each column, you must always subtract the **bottom** digit from the **top** digit, borrowing when the top digit is smaller. A quick check catches this: 4,565 + 1,768 = 6,333, which is not 5,203.",{"id":409,"type":410,"prompt":411,"options":412,"explanation":421},"predict-sub-check","prediction","Riya works out 8,040 − 2,685 and gets 5,465. Without redoing the subtraction, how can she find out whether she is right?",[413,415,417,419],{"id":201,"label":414},"Add 5,465 + 2,685 and see whether she gets 8,040",{"id":204,"label":416},"Subtract 8,040 − 5,465 and see whether she gets 2,685",{"id":207,"label":418},"Both of these work",{"id":210,"label":420},"There is no way except doing it again","**Both work (c).** Addition undoes subtraction, so difference + subtrahend must equal the minuend. Here 5,465 + 2,685 = 8,150, which is **not** 8,040, so she has made a mistake. The correct answer is 5,355. The second check also works: minuend − difference should give the subtrahend. Adding back is usually easier.",{"id":423,"type":194,"itemId":424,"prompt":425,"check":426,"hints":428,"feedback":431},"sub-practice","four-operations.u-sub-across-zeros","Work out **9,00,000 − 3,67,458**.",{"kind":335,"answer":427,"tolerance":337},532542,[429,430],"Rewrite 9,00,000 as 8 lakhs, 9 ten-thousands, 9 thousands, 9 hundreds, 9 tens and 10 ones.","Then 10 − 8 = 2 in the ones.",{"correct":432,"incorrect":433},"Right: 5,32,542. Check: 5,32,542 + 3,67,458 = 9,00,000.","Borrowing across the zeros turns every 0 into 9 and the ones into 10. The answer is 5,32,542.",{"id":435,"type":53,"title":436,"eyebrow":437,"navLabel":438},"ch4","Multiplying and dividing by 10, 100 and 1,000","Chapter 04","4 Times 10, 100, 1000",{"id":440,"type":43,"markdown":441},"x10-idea","Multiplying by 10 makes every part of a number ten times bigger: ones become tens, tens become hundreds, and so on. So **every digit moves one place to the left**, and the empty ones place is filled with a 0.\n\n- 347 × 10 = 3,470 (3 hundreds became 3 thousands)\n- 347 × 100 = 34,700 (every digit moves two places)\n- 347 × 1,000 = 3,47,000 (three places)\n\nDividing by 10, 100 or 1,000 is the reverse: digits move **right**. 45,000 ÷ 100 = 450.",{"id":443,"type":444,"items":445},"x10-formulas","formulas",[446,449,452,455,458,461],{"expression":447,"caption":448},"347 × 10 = 3,470","One place left: 7 ones → 7 tens.",{"expression":450,"caption":451},"347 × 100 = 34,700","Two places left.",{"expression":453,"caption":454},"26 × 300 = 7,800","26 × 3 = 78, then × 100 moves it two places.",{"expression":456,"caption":457},"45 × 2,000 = 90,000","45 × 2 = 90, then × 1,000.",{"expression":459,"caption":460},"4,500 ÷ 100 = 45","Two places right.",{"expression":462,"caption":463},"72,000 ÷ 800 = 90","Cancel two zeros: 720 ÷ 8 = 90.",{"id":465,"type":47,"variant":466,"title":467,"markdown":468},"nuance-add-zero","nuance","Why “just add a zero” is only half true","For whole numbers, “to multiply by 10, write a 0 on the end” works. But it is a shortcut, not the reason. The reason is that **digits shift one place left**.\n\nThe shortcut breaks with money in rupees and paise: ₹12.50 × 10 is **₹125.00** (or ₹125), not ₹12.500. The digits 1, 2, 5 each moved one place left past the decimal point. Remember the *shift*, and you will never be caught out when you meet decimals.",{"id":470,"type":293,"title":471,"problem":472,"steps":473},"we-x300","Multiplying by a multiple of 100","A school orders 48 boxes of chalk. Each box costs ₹300. What is the total bill?",[474,475,476,477],"Split 300 into 3 × 100.","48 × 3 = 144.","144 × 100 = 14,400 (the digits move two places left).","Total bill: **₹14,400**.",{"id":479,"type":194,"itemId":480,"prompt":481,"check":482,"hints":485,"feedback":487},"x10-practice","four-operations.u-divide-by-1000","A factory packs 3,65,000 grams of rice into 1-kilogram bags (1 kg = 1,000 g). How many bags does it fill?",{"kind":335,"answer":483,"tolerance":337,"unit":484},365,"bags",[486],"Divide by 1,000: move the digits three places right.",{"correct":488,"incorrect":489},"Yes: 3,65,000 ÷ 1,000 = 365 bags.","3,65,000 g ÷ 1,000 = 365. Each of the three zeros on the end is removed by dividing by 10 three times.",{"id":491,"type":53,"title":492,"eyebrow":493,"navLabel":494},"ch5","Long multiplication, one digit of the multiplier at a time","Chapter 05","5 Long multiplication",{"id":496,"type":43,"markdown":497},"mult-idea","To multiply 468 by 37, you do not need a 37 times table. Split the multiplier by place value: **37 = 30 + 7**. Then\n\n- 468 × 7 is one easy row (a **partial product**);\n- 468 × 30 is another easy row: 468 × 3, shifted one place left because it is really tens;\n- add the two rows.\n\nThat is the whole of long multiplication. The only things that go wrong are forgetting the shift, and forgetting to add carries.",{"id":499,"type":293,"title":500,"problem":501,"steps":502},"we-mult-2d","468 × 37 with partial products","Each of 37 trucks carries 468 kg of mangoes to a market in Ratnagiri. How many kilograms of mangoes is that?\n\n```\n                468\n               × 37\n            -------\n  3,276   ← 468 × 7\n14,040   ← 468 × 30\n            -------\n             17,316\n```",[503,504,505,506,507],"**Row 1, 468 × 7:** 8 × 7 = 56 (write 6, carry 5); 6 × 7 = 42, + 5 = 47 (write 7, carry 4); 4 × 7 = 28, + 4 = 32. Row 1 = 3,276.","**Row 2, 468 × 30:** write a 0 in the ones place first (we are multiplying by tens). Then 468 × 3: 8 × 3 = 24 (write 4, carry 2); 6 × 3 = 18, + 2 = 20 (write 0, carry 2); 4 × 3 = 12, + 2 = 14. Row 2 = 14,040.","**Add the rows:** 3,276 + 14,040 = 17,316.","Answer: **17,316 kg** of mangoes.","Estimate check: 468 is about 500 and 37 is about 40, so about 20,000. 17,316 is reasonable.",{"id":509,"type":293,"title":510,"problem":511,"steps":512},"we-mult-3d","A zero in the multiplier: 2,475 × 306","A railway zone runs 306 trips of a train that seats 2,475 passengers. What is the greatest number of passengers it can carry altogether?\n\n```\n                  2,475\n                  × 306\n              ---------\n   14,850   ← 2,475 × 6\n742,500   ← 2,475 × 300\n              ---------\n                757,350\n```",[513,514,515,516,517,518],"Split 306 = 300 + 0 + 6. The tens digit is 0, so the tens row is all zeros: you may skip it, **but the next row must still shift two places**.","**2,475 × 6** = 14,850.","**2,475 × 300**: write two zeros, then 2,475 × 3 = 7,425. Row = 742,500.","**Add:** 14,850 + 742,500 = 757,350.","Answer: **7,57,350 passengers** (seven lakh fifty-seven thousand three hundred fifty).","Check by estimate: 2,500 × 300 = 750,000. Close to our answer ✓.",{"id":520,"type":47,"variant":189,"title":521,"markdown":522},"mis-shift","Forgetting to shift the second row","If you forget the 0 at the start of the second row in 468 × 37, you add 1,404 instead of 14,040, and get 3,276 + 1,404 = **4,680**. That is 468 × 10, not 468 × 37! An estimate (about 20,000) instantly shows something is badly wrong. The shift is not decoration: the second row really is **tens**.",{"id":524,"type":410,"prompt":525,"options":526,"explanation":535},"predict-mult-size","Without working it out: how many digits will **4,036 × 208** have?",[527,529,531,533],{"id":201,"label":528},"5 digits",{"id":204,"label":530},"6 digits",{"id":207,"label":532},"7 digits",{"id":210,"label":534},"8 digits","**6 digits (b).** 4,036 × 208 = 839,488. A good way to predict: 4,000 × 200 = 8,00,000 (6 digits) and 5,000 × 300 = 15,00,000 (7 digits), so the answer sits between; since 4,036 × 208 is much closer to the first, it has 6 digits. Predicting the size first is a powerful check on long multiplication.",{"id":537,"type":194,"itemId":538,"prompt":539,"check":540,"hints":543,"feedback":546},"mult-practice","four-operations.u-long-mult-crates","A farmer in Nashik fills 56 crates with tomatoes. Each crate holds 245 tomatoes. How many tomatoes is that?",{"kind":335,"answer":541,"tolerance":337,"unit":542},13720,"tomatoes",[544,545],"56 = 50 + 6. Find 245 × 6 and 245 × 50, then add.","245 × 6 = 1470.",{"correct":547,"incorrect":548},"Correct: 245 × 56 = 1,470 + 12,250 = 13,720 tomatoes.","Row 1: 245 × 6 = 1,470. Row 2: 245 × 50 = 12,250. Sum: 13,720.",{"id":550,"type":220,"component":346,"componentVersion":5,"config":551,"objective":576,"textAlternative":577,"help":578},"lab-sprint-mult",{"operations":552,"ranges":554,"rounds":561,"secondsTotal":337,"estimateFirst":562,"wordProblems":563},[553],"×",{"a":555,"b":558},{"min":556,"max":557},100,999,{"min":559,"max":560},11,99,8,true,[564,568,572],{"prompt":565,"answer":566,"unit":567,"operation":553},"A school bus makes 26 trips a month. Each trip is 148 km. How many km is that in a month?",3848,"km",{"prompt":569,"answer":570,"unit":571,"operation":553},"A carton holds 144 bangles. A shop in Hyderabad buys 35 cartons. How many bangles?",5040,"bangles",{"prompt":573,"answer":574,"unit":575,"operation":553},"A notebook costs ₹48. How much do 125 notebooks cost?",6000,"₹","Multiply a three-digit number by a two-digit number with partial products, estimating first to check the size of the answer.","This sprint gives three-digit × two-digit multiplications, such as 386 × 47, plus up to three word problems picked at random. Before each generated question it asks for an **estimate**: round each number and multiply the rounded numbers. (The word problems skip the estimate step, so make your own on paper.)\n\nWorked sample: 386 × 47. Estimate: 400 × 50 = 20,000. Exact: 386 × 7 = 2,702; 386 × 40 = 15,440; sum **18,142**. The estimate is close, so the answer is believable.\n\nWord problems:\n\n- 26 trips of 148 km: 148 × 26 = **3,848 km**.\n- 35 cartons of 144 bangles: 144 × 35 = **5,040 bangles**.\n- 125 notebooks at ₹48: 48 × 125 = **₹6,000**.\n\nIf your exact answer is far from your estimate, look for a missing shift or a forgotten carry.",{"hints":579},[580,581],"Split the two-digit number into tens and ones.","The tens row always ends in 0.",{"id":583,"type":53,"title":584,"eyebrow":585,"navLabel":586},"ch6","Long division, step by step","Chapter 06","6 Long division",{"id":588,"type":43,"markdown":589},"div-idea","Long division shares a big number out **one place at a time, starting from the left**. That is the opposite direction from the other three methods, and there is a good reason: when sharing out money, you share the biggest notes first. Share the ₹1,000 notes, change any left over into ₹100 notes, share those, and so on down to the coins.\n\nEach place goes through the same four-step cycle.",{"id":591,"type":272,"title":592,"items":593},"div-cycle","The long division cycle",[594,598,602,606],{"title":595,"tag":596,"text":597},"Divide","how many?","How many times does the divisor go into the number you are looking at? Write that digit in the quotient, above the last digit you used.",{"title":599,"tag":600,"text":601},"Multiply","divisor × digit","Multiply the divisor by the digit you just wrote.",{"title":603,"tag":604,"text":605},"Subtract","must be \u003C divisor","Subtract that product. What is left must be smaller than the divisor; if not, your quotient digit was too small.",{"title":607,"tag":608,"text":609},"Bring down","next digit","Bring down the next digit of the dividend beside the remainder. Go back to Divide. When no digits are left, what remains is the remainder.",{"id":611,"type":293,"title":612,"problem":613,"steps":614},"we-div-1d","9,436 ÷ 7, sharing mangoes","A mango orchard near Ratnagiri picked 9,436 mangoes, packed equally into 7 trucks. How many mangoes go in each truck?",[615,616,617,618,619,620],"**9 ÷ 7:** 7 goes **1** time (7 × 1 = 7). 9 − 7 = 2. Bring down the next digit.","**24 ÷ 7:** 7 goes **3** times (7 × 3 = 21). 24 − 21 = 3. Bring down the next digit.","**33 ÷ 7:** 7 goes **4** times (7 × 4 = 28). 33 − 28 = 5. Bring down the next digit.","**56 ÷ 7:** 7 goes **8** times (7 × 8 = 56). 56 − 56 = 0.","Quotient **1,348**, remainder 0: each truck gets **1,348 mangoes** exactly.","Check: 7 × 1,348 = 9,436. ✓",{"id":622,"type":293,"title":623,"problem":624,"steps":625,"help":633},"we-div-2d","Dividing by a two-digit number: 58,764 ÷ 23","A mill has 58,764 kg of flour to pack into sacks of 23 kg. How many full sacks can it fill, and how much flour is left?\n\nWith a two-digit divisor, the hard part is guessing each quotient digit. **Round the divisor** to help: 23 is about 20, so ask “how many 20s?” and then check with 23.",[626,627,628,629,630,631,632],"5 is smaller than 23, so 23 does not go into it: look at one more digit.","**58 ÷ 23:** 23 goes **2** times (23 × 2 = 46). 58 − 46 = 12. Bring down the next digit.","**127 ÷ 23:** 23 goes **5** times (23 × 5 = 115). 127 − 115 = 12. Bring down the next digit.","**126 ÷ 23:** 23 goes **5** times (23 × 5 = 115). 126 − 115 = 11. Bring down the next digit.","**114 ÷ 23:** 23 goes **4** times (23 × 4 = 92). 114 − 92 = 22.","Quotient **2,554**, remainder **22**: 2,554 full sacks and 22 kg left over.","Check: 23 × 2,554 + 22 = 58,742 + 22 = 58,764. ✓",{"simplerExplanation":634,"hints":635},"When the divisor has two digits, write out a few of its multiples first (23, 46, 69, 92, 115, 138…). Then each step is just picking the biggest multiple that fits.",[636,637],"23 × 5 = 115 and 23 × 6 = 138: useful for the second step.","A remainder of 22 is fine because 22 \u003C 23.",{"id":639,"type":62,"caption":640,"columns":641,"rows":645},"mult-23","Multiples of 23: a ready reckoner for the division above",[642,643,644],"× 1–3","× 4–6","× 7–9",[646,650,654],[647,648,649],"23 × 1 = 23","23 × 4 = 92","23 × 7 = 161",[651,652,653],"23 × 2 = 46","23 × 5 = 115","23 × 8 = 184",[655,656,657],"23 × 3 = 69","23 × 6 = 138","23 × 9 = 207",{"id":659,"type":47,"variant":327,"title":660,"markdown":661},"careful-too-big","If the remainder at any step is as big as the divisor, go back","At every step, the number left after subtracting must be **smaller than the divisor**. If you are dividing by 23 and a step leaves 25, then 23 still fits once more: your quotient digit was one too small. Increase it by 1 and redo the step. If the subtraction goes below zero, the digit was too big.",{"id":663,"type":194,"itemId":664,"prompt":665,"check":666,"hints":668,"feedback":671},"div-practice","four-operations.u-long-div-sacks","Work out **7,385 ÷ 6**. Type the **quotient** only (the remainder is asked separately in your head).",{"kind":335,"answer":667,"tolerance":337},1230,[669,670],"7 ÷ 6 = 1 remainder 1. Bring down 3 to make 13.","13 ÷ 6 = 2 remainder 1. Bring down 8 to make 18.",{"correct":672,"incorrect":673},"Correct: 7,385 ÷ 6 = 1,230 remainder 5. Check: 6 × 1,230 + 5 = 7,385.","7 → 1 r 1; 13 → 2 r 1; 18 → 3 r 0; 5 → 0 r 5. Quotient 1,230, remainder 5.",{"id":675,"type":220,"component":346,"componentVersion":5,"config":676,"objective":695,"textAlternative":696,"help":697},"lab-sprint-div",{"operations":677,"ranges":679,"rounds":358,"secondsTotal":337,"estimateFirst":359,"wordProblems":684},[678],"÷",{"a":680,"b":681},{"min":556,"max":357},{"min":682,"max":683},3,25,[685,688,691],{"prompt":686,"answer":687,"unit":575,"operation":678},"₹8,640 is shared equally among 12 members of a self-help group. How much does each get?",720,{"prompt":689,"answer":690,"unit":567,"operation":678},"A 1,575 km journey is covered in 7 equal daily stages. How long is each stage?",225,{"prompt":692,"answer":693,"unit":694,"operation":678},"3,456 eggs are packed in trays of 24. How many trays?",144,"trays","Practise exact division, then use long division in real sharing and grouping problems.","This sprint gives ten exact divisions: the divisor is from 3 to 25 and the dividend is between 100 and 9,999 (for example 1,088 ÷ 17 or 391 ÷ 23). The answer is always a whole number, with no remainder. Up to half the rounds are word problems, picked at random from the three below. Think of each division as the multiplication that undoes it: 17 × 64 = 1,088, so 1,088 ÷ 17 = 64.\n\nWord problems, worked:\n\n- ₹8,640 shared among 12: 86 ÷ 12 = 7 r 2; bring down 4: 24 ÷ 12 = 2; bring down 0: 0. Each gets **₹720**. Check 12 × 720 = 8,640.\n- 1,575 km in 7 stages: 15 ÷ 7 = 2 r 1; 17 ÷ 7 = 2 r 3; 35 ÷ 7 = 5. Each stage is **225 km**.\n- 3,456 eggs in trays of 24: 34 ÷ 24 = 1 r 10; 105 ÷ 24 = 4 r 9; 96 ÷ 24 = 4. **144 trays**.\n\nAlways check by multiplying the quotient by the divisor.",{"hints":698},[699,700],"Ask: divisor times what gives the dividend?","Use the first two digits when the first digit is smaller than the divisor.",{"id":702,"type":53,"title":703,"eyebrow":704,"navLabel":705},"ch7","Zeros in the quotient","Chapter 07","7 Zeros in quotients",{"id":707,"type":43,"markdown":708},"zero-idea","Sometimes, after you bring a digit down, the divisor **does not go into it at all**. The answer for that place is **0**, and the 0 must be written in the quotient. Skipping it is the most common long-division mistake, and it makes the answer ten (or a hundred) times too small.\n\nIn 8,127 ÷ 8, the right answer is **1,015 remainder 7**. A learner who forgets the zero writes 115 remainder 7, and 8 × 115 + 7 is only 927, nowhere near 8,127.",{"id":710,"type":293,"title":711,"problem":712,"steps":713},"we-zero-q","8,127 ÷ 8, keeping the zero","8,127 kg of onions are packed into 8 equal lots. How many kg in each lot, and how much is left over?",[714,715,716,717,718,719],"**8 ÷ 8:** 8 goes **1** time (8 × 1 = 8). 8 − 8 = 0. Bring down the next digit.","**1 ÷ 8:** 8 goes **0** times (8 × 0 = 0). 1 − 0 = 1. Bring down the next digit.","**12 ÷ 8:** 8 goes **1** time (8 × 1 = 8). 12 − 8 = 4. Bring down the next digit.","**47 ÷ 8:** 8 goes **5** times (8 × 5 = 40). 47 − 40 = 7.","The quotient digits, in order, are 1, 0, 1, 5: **1,015**, remainder **7**.","Check: 8 × 1,015 + 7 = 8,120 + 7 = 8,127. ✓",{"id":721,"type":293,"title":722,"problem":723,"steps":724},"we-zero-q2","Two zeros in a row: 24,108 ÷ 12","A temple trust shares ₹24,108 equally among 12 village schools. How much does each school get?",[725,726,727,728,729,730,731],"2 is smaller than 12, so 12 does not go into it: look at one more digit.","**24 ÷ 12:** 12 goes **2** times (12 × 2 = 24). 24 − 24 = 0. Bring down the next digit.","**1 ÷ 12:** 12 goes **0** times (12 × 0 = 0). 1 − 0 = 1. Bring down the next digit.","**10 ÷ 12:** 12 goes **0** times (12 × 0 = 0). 10 − 0 = 10. Bring down the next digit.","**108 ÷ 12:** 12 goes **9** times (12 × 9 = 108). 108 − 108 = 0.","Quotient **2,009**: each school gets **₹2,009**. Two zeros had to be written.","Size check: 24,000 ÷ 12 = 2,000, so the answer must be just over 2,000, not 29 or 209.",{"id":733,"type":47,"variant":322,"title":734,"markdown":735},"aha-size","Count the digits of the quotient before you start","For 8,127 ÷ 8: 8 goes into the first digit 8, so the quotient starts in the **thousands** place, and must have **4 digits**. If your answer has only 3 digits, a zero is missing. The same test for 24,108 ÷ 12: 12 goes into 24 (thousands), so the quotient has 4 digits: 2,009.",{"id":737,"type":194,"itemId":738,"prompt":739,"check":740,"hints":742,"feedback":745},"zero-practice","four-operations.u-zero-quotient","Work out **6,035 ÷ 5**.",{"kind":335,"answer":741,"tolerance":337},1207,[743,744],"6 ÷ 5 = 1 r 1. Bring down 0: 10 ÷ 5 = 2.","Next bring down 3: 5 goes into 3 zero times. Write 0!",{"correct":746,"incorrect":747},"Correct: 1,207. The zero in the tens place of the quotient is essential.","6 → 1 r 1; 10 → 2 r 0; 3 → **0** r 3; 35 → 7. Quotient 1,207.",{"id":749,"type":53,"title":750,"eyebrow":751,"navLabel":752},"ch8","Checking with inverse operations","Chapter 08","8 Checking answers",{"id":754,"type":43,"markdown":755},"check-idea","Every operation has an **inverse** that undoes it, and that gives you a built-in way to check:\n\n- Check an **addition** by subtracting: if 3,456 + 1,289 = 4,745, then 4,745 − 1,289 must be 3,456.\n- Check a **subtraction** by adding: difference + subtrahend = minuend.\n- Check a **multiplication** by dividing: 17,316 ÷ 37 must be 468.\n- Check a **division** by multiplying, and **adding the remainder**.\n\nThat last one is so important it has its own name, the **division check** (in higher maths, the *division algorithm*).",{"id":757,"type":444,"items":758},"div-check-formulas",[759,762,765,768,771],{"expression":760,"caption":761},"dividend = divisor × quotient + r","The check for every division, where r is the remainder.",{"expression":763,"caption":764},"0 ≤ remainder \u003C divisor","The remainder is never negative and never as big as the divisor.",{"expression":766,"caption":767},"58,764 = 23 × 2,554 + 22","58,742 + 22 = 58,764 ✓",{"expression":769,"caption":770},"minuend = difference + subtrahend","The check for every subtraction.",{"expression":772,"caption":773},"product ÷ one factor = other factor","The check for every multiplication (factors not 0).",{"id":775,"type":293,"title":776,"problem":777,"steps":778},"we-find-dividend","Working backwards from the check","Meena divided a number by 15 and got quotient 243 and remainder 11. What number did she divide?",[779,780,781,782,783],"Use dividend = divisor × quotient + remainder.","15 × 243 = 3,645.","3,645 + 11 = 3,656.","The number was **3,656**.","Check: 3,656 ÷ 15 = 243 remainder 11 ✓ (and 11 \u003C 15, so the remainder is allowed).",{"id":785,"type":410,"prompt":786,"options":787,"explanation":796},"predict-remainder","Arjun says: “I divided 500 by 12 and got 40 remainder 20.” What do you say?",[788,790,792,794],{"id":201,"label":789},"Correct, because 12 × 40 + 20 = 500",{"id":204,"label":791},"Not finished: a remainder of 20 means 12 still fits once more",{"id":207,"label":793},"Wrong, because a remainder must be less than 10",{"id":210,"label":795},"Wrong, because 500 cannot be divided by 12","**Not finished (b).** The check 12 × 40 + 20 = 500 is true, but a remainder must be **less than the divisor**. 20 contains another 12, so the quotient should be 41 and the remainder 8: 12 × 41 + 8 = 500. Option (c) is a common muddle: the limit on the remainder is the *divisor*, not 10.",{"id":798,"type":194,"itemId":799,"prompt":800,"check":801,"hints":812,"feedback":814},"check-practice","four-operations.u-check-division","Which statement correctly checks **9,436 ÷ 7 = 1,348**?",{"kind":198,"options":802,"correct":811},[803,805,807,809],{"id":201,"label":804},"1,348 × 7 = 9,436",{"id":204,"label":806},"9,436 × 7 = 1,348",{"id":207,"label":808},"9,436 − 1,348 = 7",{"id":210,"label":810},"1,348 + 7 = 9,436",[201],[813],"Division is undone by multiplication.",{"correct":815,"incorrect":816},"Right: quotient × divisor = dividend (plus any remainder, here 0).","Multiply the quotient by the divisor: 1,348 × 7 = 9,436.",{"id":818,"type":819,"conceptId":820,"relation":821,"explanation":822},"conn-props","connection","properties-of-numbers","helps_understand","The commutative, associative and distributive properties explain why you may add in any order and why long multiplication splits the multiplier into tens and ones.",{"id":824,"type":53,"title":825,"eyebrow":826,"navLabel":827},"ch9","Operating with money and measurement units","Chapter 09","9 Money and units",{"id":829,"type":43,"markdown":830},"units-idea","Money and measurements usually come with **two units at once**: rupees and paise, kilometres and metres, kilograms and grams, litres and millilitres. The golden rule: **both amounts must be in the same unit before you operate**.\n\nThere are two safe ways to do this:\n\n1. **Convert everything to the smaller unit** (paise, metres, grams, millilitres), do ordinary whole-number arithmetic, then convert back.\n2. **Work in columns of units**, carrying or borrowing 100 paise, 1,000 m, 1,000 g or 1,000 mL instead of 10.",{"id":832,"type":62,"caption":833,"columns":834,"rows":838},"units-table","Conversions you need for the four operations",[835,836,837],"Bigger unit","Smaller unit","Carry or borrow",[839,843,847,851,855,859],[840,841,842],"₹1","100 paise","100 paise make ₹1",[844,845,846],"1 km","1,000 m","1,000 m make 1 km",[848,849,850],"1 m","100 cm","100 cm make 1 m",[852,853,854],"1 kg","1,000 g","1,000 g make 1 kg",[856,857,858],"1 L","1,000 mL","1,000 mL make 1 L",[860,861,862],"1 hour","60 minutes","60, not 100! Careful with time.",{"id":864,"type":293,"title":865,"problem":866,"steps":867},"we-money","A kirana bill in rupees and paise","Kavya buys rice for ₹245.75, dal for ₹189.50 and oil for ₹162.25. She pays with a ₹1,000 note. How much change does she get?",[868,869,870,871,872],"Convert to paise: 24,575 + 18,950 + 16,225 = 59,750 paise.","That is ₹597.50.","₹1,000 = 1,00,000 paise. 1,00,000 − 59,750 = 40,250 paise.","Change: **₹402.50**.","Check: ₹597.50 + ₹402.50 = ₹1000.00 ✓",{"id":874,"type":293,"title":875,"problem":876,"steps":877},"we-km","Distances in km and m","A train covers 348 km 650 m before lunch and 276 km 480 m after. How far does it go in total?",[878,879,880,881,882],"Add the metres: 650 + 480 = 1,130 m.","1,130 m = 1 km 130 m. Write 130 m and carry 1 km.","Add the kilometres: 348 + 276 + 1 (carried) = 625 km.","Total: **625 km 130 m**.","Check in metres: 348,650 + 276,480 = 625,130 m = 625 km 130 m ✓",{"id":884,"type":293,"title":885,"problem":886,"steps":887},"we-kg","Subtracting kg and g","A shopkeeper has a 5 kg sack of sugar and sells 2 kg 375 g. How much is left?",[888,889,890,891,892],"Write 5 kg as 5 kg 0 g.","0 g is less than 375 g, so borrow 1 kg = 1,000 g: now 4 kg 1,000 g.","1,000 − 375 = 625 g; 4 − 2 = 2 kg.","Left: **2 kg 625 g**.","Check in grams: 5,000 − 2,375 = 2,625 g ✓",{"id":894,"type":47,"variant":189,"title":895,"markdown":896},"mis-units","“3 km + 450 m = 453”","Adding the numbers without looking at the units gives nonsense. 3 km is **3,000 m**, so 3 km + 450 m = **3,450 m**, or 3 km 450 m. Likewise ₹5 + 50 paise is ₹5.50, not ₹55. And with time, remember 60 minutes make an hour: 1 h 45 min + 30 min = 2 h 15 min, not 1 h 75 min.",{"id":898,"type":194,"itemId":899,"prompt":900,"check":901,"hints":904,"feedback":907},"units-practice","four-operations.u-litres-bottles","A water-purifier plant fills 36 bottles, each of 750 mL. How many **litres** of water is that?",{"kind":335,"answer":902,"tolerance":337,"unit":903},27,"L",[905,906],"36 × 750 = ? mL.","Then divide by 1,000 to get litres.",{"correct":908,"incorrect":909},"Correct: 36 × 750 = 27,000 mL = 27 L.","36 × 750 = 27,000 mL, and 27,000 ÷ 1,000 = 27 L.",{"id":911,"type":53,"title":912,"eyebrow":913,"navLabel":914},"ch10","A gallery of common mistakes","Chapter 10","10 Mistake gallery",{"id":916,"type":43,"markdown":917},"errors-idea","Most wrong answers in arithmetic are not random. They come from a small number of habits, and once you know them you can hunt for them. Here are the most common, each with a real wrong answer and a quick way to catch it.",{"id":919,"type":62,"caption":920,"columns":921,"rows":926},"errors-table","Common errors, what they look like, and the quick check that catches them",[922,923,924,925],"Mistake","Wrong working","Right answer","Quick check",[927,932,937,942,947,952,957,962],[928,929,930,931],"Left-aligned columns","45,320 + 987 written with 9 under 4 → 1,44,020","46,307","The sum of a 5-digit and a 3-digit number cannot be about 1 lakh.",[933,934,935,936],"Forgot to add a carry","386 + 247 = 523","633","Ones gave 13; the 1 ten was never added.",[938,939,940,941],"Smaller-from-bigger","5,203 − 1,768 = 4,565","3,435","Add back: 4,565 + 1,768 ≠ 5,203.",[943,944,945,946],"Borrowed but did not reduce","52 − 17 = 45","35","Add back: 45 + 17 = 62, not 52.",[948,949,950,951],"No shift in long multiplication","468 × 37 = 4,680","17,316","Estimate 500 × 40 = 20,000.",[953,954,955,956],"Missing zero in quotient","8,127 ÷ 8 = 115 r 7","1,015 r 7","8 × 115 + 7 = 927, not 8,127.",[958,959,960,961],"Remainder too big","500 ÷ 12 = 40 r 20","41 r 8","Remainder must be less than 12.",[963,964,965,966],"Mixed units","3 km + 450 m = 453 m","3,450 m","Convert to one unit first.",{"id":968,"type":194,"itemId":969,"prompt":970,"check":971,"hints":982,"feedback":985},"spot-mistake","four-operations.u-spot-mistake-mult","Sana worked out **236 × 24** like this: 236 × 4 = 944; 236 × 2 = 472; 944 + 472 = 1,416. What went wrong?",{"kind":198,"options":972,"correct":981},[973,975,977,979],{"id":201,"label":974},"236 × 4 is wrong",{"id":204,"label":976},"The second row should be 236 × 20 = 4,720",{"id":207,"label":978},"She should have subtracted the rows",{"id":210,"label":980},"Nothing, 1,416 is right",[204],[983,984],"The 2 in 24 is in the tens place.","Estimate: 236 × 24 is about 200 × 25.",{"correct":986,"incorrect":987},"Yes. The 2 means 20, so the second row is 4,720. 944 + 4,720 = 5,664.","The 2 in 24 stands for 2 tens, so the second row is 236 × 20 = 4,720, not 472. The correct product is 5,664.",{"id":989,"type":220,"component":990,"componentVersion":5,"config":991,"objective":1052,"textAlternative":1053,"help":1054},"lab-sort-mistake","sort-game",{"prompt":992,"bins":993,"items":1004,"seconds":337},"Each card is a finished calculation. Sort it: is it correct, or which mistake was made?",[994,997,999,1002],{"id":995,"label":996},"correct","Correct",{"id":147,"label":998},"Carry or borrow slip",{"id":1000,"label":1001},"place","Place value slip",{"id":139,"label":1003},"Remainder or zero slip",[1005,1009,1012,1016,1020,1024,1028,1032,1036,1040,1044,1048],{"id":1006,"label":1007,"bin":995,"why":1008},"i1","4,678 + 2,535 = 7,213","4,678 + 2,535 = 7,213. All carries were added.",{"id":1010,"label":934,"bin":147,"why":1011},"i2","Ones 6 + 7 = 13: the carried 1 ten was forgotten. Correct: 633.",{"id":1013,"label":1014,"bin":147,"why":1015},"i3","704 − 258 = 546","The borrow across the zero was not done properly. Correct: 446.",{"id":1017,"label":1018,"bin":1000,"why":1019},"i4","125 × 12 = 375","The tens row was not shifted: 250 + 125 = 375. Correct: 250 + 1,250 = 1,500.",{"id":1021,"label":1022,"bin":139,"why":1023},"i5","6,035 ÷ 5 = 127","The zero in the quotient was skipped. Correct: 1,207.",{"id":1025,"label":1026,"bin":139,"why":1027},"i6","100 ÷ 7 = 13 r 9","9 is bigger than 7. Correct: 14 r 2.",{"id":1029,"label":1030,"bin":995,"why":1031},"i7","3,456 ÷ 24 = 144","24 × 144 = 3,456.",{"id":1033,"label":1034,"bin":1000,"why":1035},"i8","45,320 + 987 = 1,44,020","The numbers were lined up on the left. Correct: 46,307.",{"id":1037,"label":1038,"bin":995,"why":1039},"i9","90,000 − 1 = 89,999","Borrowing turns the zeros into 9s: 89,999.",{"id":1041,"label":1042,"bin":147,"why":1043},"i10","1,000 − 364 = 764","The borrow across zeros was muddled. Correct: 636.",{"id":1045,"label":1046,"bin":995,"why":1047},"i11","8,127 ÷ 8 = 1,015 r 7","8 × 1,015 + 7 = 8,127.",{"id":1049,"label":1050,"bin":1000,"why":1051},"i12","72 × 10 = 702","The digits must all move one place left: 720.","Spot whether a finished calculation is correct, or whether it has a carrying, place-value or remainder mistake.","This game shows twelve finished calculations. You sort each into Correct, Carry or borrow slip, Place value slip, or Remainder or zero slip.\n\n- **Correct:** 4,678 + 2,535 = 7,213; 3,456 ÷ 24 = 144 (24 × 144 = 3,456); 90,000 − 1 = 89,999; 8,127 ÷ 8 = 1,015 r 7.\n- **Carry or borrow slip:** 386 + 247 = 523 (forgot the carry; should be 633); 704 − 258 = 546 (should be 446); 1,000 − 364 = 764 (should be 636).\n- **Place value slip:** 125 × 12 = 375 (second row not shifted; should be 1,500); 45,320 + 987 = 1,44,020 (left-aligned; should be 46,307); 72 × 10 = 702 (should be 720).\n- **Remainder or zero slip:** 6,035 ÷ 5 = 127 (missing zero; should be 1,207); 100 ÷ 7 = 13 r 9 (remainder bigger than divisor; should be 14 r 2).\n\nThe fastest checks are an estimate and the inverse operation.",{"hints":1055},[1056,1057],"Estimate each answer first. A place-value slip is usually far off.","For divisions, check divisor × quotient + remainder.",{"id":1059,"type":47,"variant":1060,"title":1061,"markdown":1062},"model-limit","model_limit","The column methods are not the only way","Column methods are designed for **paper**: they break any sum into tiny single-digit steps, so they always work. But they are not always the fastest. 4,999 + 2,345 is easier as 5,000 + 2,345 − 1 = 7,344, and 25 × 48 is easier as 100 × 12 = 1,200. Use the column method when numbers are awkward; use mental strategies (the Investigate and Deepen layers) when numbers are friendly.",{"id":1064,"type":819,"conceptId":1065,"relation":821,"explanation":1066},"conn-number-system","number-system","Place value and Indian commas are what make lining up columns, carrying and bringing down digits work.",{"id":1068,"type":819,"conceptId":1069,"relation":821,"explanation":1070},"conn-order","order-of-operations","Once each operation is reliable, the next question is which to do first when several appear in one expression.",{"id":1072,"type":819,"conceptId":1073,"relation":821,"explanation":1074},"conn-prime","prime-and-composite","Long division tells you whether a number divides exactly (remainder 0), which is how you test for factors and primes.",{"id":1076,"type":1077,"prompt":1078},"reflect","reflection","Which of the mistakes in the gallery do you make most often? Write down one quick check you will use every time from now on, and try it on your next five calculations.",{"id":1080,"type":1080,"title":1081,"questions":1082},"quiz","Check yourself",[1083,1092,1105,1118,1131,1144,1157,1170,1182,1195],{"itemId":1084,"prompt":1085,"options":1086,"correct":207,"why":1091},"four-operations.u-quiz-quotient","In 2,345 ÷ 12 = 195 remainder 5, what is 195 called?",[1087,1088,1089,1090],{"id":201,"label":242},{"id":204,"label":245},{"id":207,"label":248},{"id":210,"label":239},"The answer to a division is the quotient. 2,345 is the dividend and 12 the divisor.",{"itemId":1093,"prompt":1094,"options":1095,"correct":201,"why":1104},"four-operations.u-quiz-minuend","In 900 − 356 = 544, which number is the minuend?",[1096,1098,1100,1102],{"id":201,"label":1097},"900",{"id":204,"label":1099},"356",{"id":207,"label":1101},"544",{"id":210,"label":1103},"1,256","The minuend is the number you subtract from: 900.",{"itemId":1106,"prompt":1107,"options":1108,"correct":204,"why":1117},"four-operations.u-quiz-carry","In column addition, why do we start with the ones column?",[1109,1111,1113,1115],{"id":201,"label":1110},"Because it is the smallest number",{"id":204,"label":1112},"So any carry can be passed to the column on its left",{"id":207,"label":1114},"Because the teacher says so",{"id":210,"label":1116},"It does not matter which column you start with","A column can overflow into the next place up. Working right to left means the carry is ready before you add that next column.",{"itemId":1119,"prompt":1120,"options":1121,"correct":204,"why":1130},"four-operations.u-quiz-zeros","What is 6,00,000 − 1,23,456?",[1122,1124,1126,1128],{"id":201,"label":1123},"5,23,456",{"id":204,"label":1125},"4,76,544",{"id":207,"label":1127},"5,76,544",{"id":210,"label":1129},"4,86,544","Borrowing across the zeros: 6,00,000 = 5 lakhs, 9 ten-thousands, 9 thousands, 9 hundreds, 9 tens, 10 ones. The answer is 4,76,544.",{"itemId":1132,"prompt":1133,"options":1134,"correct":204,"why":1143},"four-operations.u-quiz-shift","In 527 × 43, the second partial product is 527 × 40. What is it?",[1135,1137,1139,1141],{"id":201,"label":1136},"2,108",{"id":204,"label":1138},"21,080",{"id":207,"label":1140},"1,581",{"id":210,"label":1142},"15,810","527 × 4 = 2,108, and × 10 shifts it one place: 21,080.",{"itemId":1145,"prompt":1146,"options":1147,"correct":204,"why":1156},"four-operations.u-quiz-div-zero","What is 4,824 ÷ 4?",[1148,1150,1152,1154],{"id":201,"label":1149},"126",{"id":204,"label":1151},"1,206",{"id":207,"label":1153},"1,026",{"id":210,"label":1155},"1,260","4 ÷ 4 = 1; 8 ÷ 4 = 2; 2 ÷ 4 = 0 r 2; 24 ÷ 4 = 6. Quotient 1,206, keeping the zero.",{"itemId":1158,"prompt":1159,"options":1160,"correct":210,"why":1169},"four-operations.u-quiz-remainder","When dividing by 9, which remainder is impossible?",[1161,1163,1165,1167],{"id":201,"label":1162},"0",{"id":204,"label":1164},"5",{"id":207,"label":1166},"8",{"id":210,"label":1168},"9","A remainder must be less than the divisor, so the largest possible remainder when dividing by 9 is 8.",{"itemId":1171,"prompt":1172,"options":1173,"correct":201,"why":1181},"four-operations.u-quiz-dividend","A number divided by 25 gives quotient 36 and remainder 7. What is the number?",[1174,1176,1177,1179],{"id":201,"label":1175},"907",{"id":204,"label":1097},{"id":207,"label":1178},"68",{"id":210,"label":1180},"1,007","Dividend = 25 × 36 + 7 = 900 + 7 = 907.",{"itemId":1183,"prompt":1184,"options":1185,"correct":204,"why":1194},"four-operations.u-quiz-x100","What is ₹175 × 100?",[1186,1188,1190,1192],{"id":201,"label":1187},"₹1,750",{"id":204,"label":1189},"₹17,500",{"id":207,"label":1191},"₹1,75,000",{"id":210,"label":1193},"₹175,00","Multiplying by 100 moves every digit two places left: ₹17,500.",{"itemId":1196,"prompt":1197,"options":1198,"correct":204,"why":1207},"four-operations.u-quiz-units","What is 4 kg 250 g + 2 kg 900 g?",[1199,1201,1203,1205],{"id":201,"label":1200},"6 kg 1150 g",{"id":204,"label":1202},"7 kg 150 g",{"id":207,"label":1204},"6 kg 150 g",{"id":210,"label":1206},"7 kg 1150 g","250 + 900 = 1,150 g = 1 kg 150 g. So 4 + 2 + 1 = 7 kg, and 150 g: 7 kg 150 g.",{"id":1209,"type":1210,"title":1211,"points":1212},"cheat-sheet","summary","Cheat sheet",[1213,1214,1215,1216,1217,1218,1219,1220,1221,1222,1223],"**Names:** addend + addend = sum; minuend − subtrahend = difference; multiplicand × multiplier = product; dividend ÷ divisor = quotient, with a remainder.","**Line up by place value**, right edges together. Commas in the same places help.","**Addition:** right to left; carry when a column reaches 10 or more. A carry is an exchange: 10 ones for 1 ten.","**Subtraction:** right to left; borrow when the top digit is smaller. Across zeros, every 0 you pass becomes 9.","**× 10, 100, 1,000:** every digit shifts 1, 2 or 3 places left. ÷ shifts right. ₹12.50 × 10 = ₹125.","**Long multiplication:** one row per digit of the multiplier; the tens row starts with 0, the hundreds row with 00. Add the rows.","**Long division:** left to right. Divide, multiply, subtract, bring down. Each remainder must be smaller than the divisor.","**Zeros in the quotient:** if the divisor does not go, write 0. Count the quotient's digits before you start.","**Check:** dividend = divisor × quotient + remainder, and the remainder is always smaller than the divisor. Difference + subtrahend = minuend.","**Units:** convert to the same unit first. ₹1 = 100 paise; 1 km = 1,000 m; 1 kg = 1,000 g; 1 L = 1,000 mL; 1 h = 60 min.","**Always estimate** first: it catches missing shifts, missing zeros and misaligned columns.",{"id":1225,"type":1225,"sourceIds":1226},"sources",[1227,1228,1229,1230,1231,1232,1233,1234,1235,1236],"four-operations-ncert-math-magic-5","four-operations-ncert-math-6","four-operations-khan-arithmetic","four-operations-mathsisfun-long-division","four-operations-mathsisfun-long-multiplication","four-operations-bbc-bitesize-ks2","four-operations-britannica-arithmetic","four-operations-wiki-long-division","four-operations-wiki-arithmetic","four-operations-wiki-subtraction",[1227,1228,1229,1230,1231,1232,1233,1234,1235,1236],"needs_review",{"generatedBy":1240,"notes":1241},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","ed28c893b65c3b7a783e1f502b5ff1aa0f0f7d726bc74c10138a46f5a23bebe6",{"logic:practice":1244,"component:match-pairs@1":1245,"component:arith-sprint@1":1246,"component:sort-game@1":1247,"source:four-operations-bbc-bitesize-ks2":1248,"source:four-operations-britannica-arithmetic":1249,"source:four-operations-khan-arithmetic":1250,"source:four-operations-mathsisfun-long-division":1251,"source:four-operations-mathsisfun-long-multiplication":1252,"source:four-operations-ncert-math-6":1253,"source:four-operations-ncert-math-magic-5":1254,"source:four-operations-wiki-arithmetic":1255,"source:four-operations-wiki-long-division":1256,"source:four-operations-wiki-subtraction":1257},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","07eb90693160dd6149271155357cff693c59a2006a36d3cc67c133d9f2900790","66e407fe40575b09f96c71af650fbd183b777a5b7047ed8fcc6b693b637804f6","ef6b590392c4299a9ce1be700f25412c2a52dea7c431f6dd28b6d7c1ce15374d","794b6ef61d62320200593d8724a8525d6a81168da99708651826563b823043a5","f673458eca6d084a854c7e5fd91ec78c1ed561c54faf5e27538711d81efda7a5","c66e4c408e97e6c5758c1b1160148cfabaf1f5e9ca4caba87383f8ca4e389c95","1856ee000472d9371b3855e38b691fbd35b221cc502b33bc887ffd8f161b636a","3fd5b7fcbfec31c612f8356e67126230dae21df1a69f1c1a06216d510015a231","8dc9aada436d8e4ebc8c48665c6a2ac0a3b19a65a83e09f99e95fe119815f054","40c70d4a1c69f3c608afe351817f3c7b192cc10915db61b683c16eb3cc3affb0",{"state":1259,"reviewer":1260,"selfReview":562,"reviewedAt":1261,"method":1262},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597080]