[{"data":1,"prerenderedAt":1144},["ShallowReactive",2],{"layer:four-operations:deepen":3},{"layer":4,"contentHash":1114,"dependencyHashes":1115,"approval":1138,"releaseId":1143},{"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":1109,"reviewStatus":1110,"authoring":1111},1,"four-operations","en","deepen","Why the methods work","Regrouping, the distributive property, the division algorithm, checks, proportion and the history behind them","Prove why carrying, borrowing, long multiplication and long division work, meet the division algorithm and why dividing by zero is impossible, check with casting out nines, use the unitary method wisely, and solve India-sized multi-step problems.",[13,14,15,16,17],"Explain carrying and borrowing as regrouping using expanded form, and use equal additions as an alternative.","Show that long multiplication is the distributive property and long division is repeated subtraction of chunks.","State the division algorithm, explain why the remainder is unique and why division by zero is undefined.","Check calculations with casting out nines, explain why it works and name errors it cannot catch.","Use the unitary method, bar models and working backwards on multi-step and large-number problems, recognising when proportion fails.",55,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Go deeper",{"label":26,"value":27},"Reading time","≈ 55 minutes",{"label":29,"value":30},"Prior knowledge","Column methods and checking (Understand)",{"label":32,"value":33},"Chapters","11",{"label":35,"value":36},"Labs","÷ sprint, match-pairs, rounding race, big-number sprint",{"label":38,"value":39},"Big idea","Place value + distributive property",[41,45,51,57,60,102,107,125,139,144,149,152,155,183,195,200,205,208,236,239,243,257,268,281,286,289,313,316,320,331,343,366,371,374,390,393,422,426,429,433,445,456,488,493,496,499,504,514,535,541,546,549,559,563,566,578,598,603,606,615,626,668,681,685,690,693,703,706,710,721,779,784,789,792,829,833,836,841,846,929,1060,1064,1080,1084,1088],{"id":42,"type":43,"markdown":44},"intro","prose","You can already add, subtract, multiply and divide numbers with lakhs and crores in them. You carry, you borrow, you shift rows, you bring digits down. This layer asks a harder question: **why are those moves allowed?**\n\nThat matters for three reasons. First, a method you understand is a method you can **repair**: if you forget a step, you can rebuild it. Second, once you see *why* it works, you can invent faster methods of your own, and spot when a \"trick\" is really a trap. Third, the reasons are beautiful. Every column method in this layer rests on just two ideas: **place value** (each place is worth ten of the place to its right) and a handful of **properties** of the operations, especially the distributive property.\n\nAt the end we meet the people who worked these methods out, many of them in India, more than a thousand years ago.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-how","callout","observation","How to read this layer","Keep a pencil handy. Most chapters make a claim (\"carrying is just regrouping\", \"long division is repeated subtraction\") and then **prove** it on a real example written out in full. Try to follow every line, then invent your own example and check that the argument still works. That is what mathematicians mean by understanding a proof.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","Carrying is regrouping: why column addition works","Chapter 01","1 Why carrying works",{"id":58,"type":43,"markdown":59},"carry-expanded","Take 2,47,368 + 1,85,974. Write each number in **expanded form**, place by place:\n\n- 2,47,368 = 2 lakh + 4 ten-thousands + 7 thousands + 3 hundreds + 6 tens + 8 ones\n- 1,85,974 = 1 lakh + 8 ten-thousands + 5 thousands + 9 hundreds + 7 tens + 4 ones\n\nBecause addition can be done in any order and in any grouping (the **commutative** and **associative** properties), we are allowed to add like places together:\n\n- ones: 8 + 4 = 12 ones\n- tens: 6 + 7 = 13 tens\n- hundreds: 3 + 9 = 12 hundreds\n- thousands: 7 + 5 = 12 thousands\n- ten-thousands: 4 + 8 = 12 ten-thousands\n- lakhs: 2 + 1 = 3 lakhs\n\nThat is already the correct answer, but it is written untidily: no place is allowed to hold more than 9. **Carrying** is simply tidying up. 12 ones = 1 ten + 2 ones, so 1 ten moves up to the tens place. Then the tens hold 13 + 1 = 14 tens = 1 hundred + 4 tens, and so on.",{"id":61,"type":62,"caption":63,"columns":64,"rows":70},"tab-carry","table","Tidying 2,47,368 + 1,85,974 from the right: each carry is 10 of one place becoming 1 of the next",[65,66,67,68,69],"Place","Raw total","Plus carry in","Write","Carry out",[71,77,83,89,93,97],[72,73,74,75,76],"Ones","8 + 4 = 12","12","2","1 ten",[78,79,80,81,82],"Tens","6 + 7 = 13","13 + 1 = 14","4","1 hundred",[84,85,86,87,88],"Hundreds","3 + 9 = 12","12 + 1 = 13","3","1 thousand",[90,91,86,87,92],"Thousands","7 + 5 = 12","1 ten-thousand",[94,95,86,87,96],"Ten-thousands","4 + 8 = 12","1 lakh",[98,99,100,81,101],"Lakhs","2 + 1 = 3","3 + 1 = 4","none",{"id":103,"type":47,"variant":104,"title":105,"markdown":106},"aha-carry","aha","A carry is a bundle, not a magic 1","When you write a small 1 above the tens column, you are not creating a mysterious extra 1. You are taking **ten ones**, tying them into a bundle, and moving the bundle to where bundles of ten live. So 2,47,368 + 1,85,974 = **4,33,342** is exactly the sum of the raw place totals: 12 + 130 + 1,200 + 12,000 + 1,20,000 + 3,00,000 = 4,33,342.",{"id":108,"type":109,"prompt":110,"options":111,"explanation":124},"pred-left","prediction","Could you add 368 + 974 starting from the **left** (hundreds first) instead of the right?",[112,115,118,121],{"id":113,"label":114},"a","No, the answer would be wrong",{"id":116,"label":117},"b","Yes, if you are willing to go back and fix earlier digits when a carry appears",{"id":119,"label":120},"c","Only if there are no carries",{"id":122,"label":123},"d","Only for numbers with an even number of digits","**Yes.** From the left: hundreds 3 + 9 = 12 hundreds (1,200); tens 6 + 7 = 13 tens (130); ones 8 + 4 = 12. Then 1,200 + 130 + 12 = **1,342**. The only nuisance is that each new carry changes a digit you have already written, so you must go back and fix it. Starting from the right means each carry lands on a column you have **not yet** written, so you never need to go back. That is the whole reason we work right to left. Many mental calculators actually prefer left-to-right, because the big parts come first and give an estimate at once.",{"id":126,"type":127,"itemId":128,"prompt":129,"check":130,"hints":134,"feedback":136},"prac-carry-count","practice","four-operations.dp-carry-max","When you add two numbers column by column, the carry into any column is at most 1. What is the largest possible raw total in one column (two digits plus the carry)?",{"kind":131,"answer":132,"tolerance":133},"number",19,0,[135],"Use the biggest digits you can.",{"correct":137,"incorrect":138},"Right: 9 + 9 + 1 = 19, which is 1 ten and 9. So the carry out is again at most 1, and the argument keeps working column after column.","The biggest digits are 9 and 9, and the biggest carry in is 1, so the largest raw total is 19. Since 19 is less than 20, the next carry is never more than 1.",{"id":140,"type":47,"variant":141,"title":142,"markdown":143},"nuance-three","nuance","Adding three or more numbers","With two numbers each carry is 0 or 1. With many numbers the carry can be bigger: adding nine numbers that each end in 9 gives 81 ones, so you write 1 and carry **8**. Nothing about the method changes: 81 ones is 8 tens and 1 one. A kirana shopkeeper totalling a long bill is doing exactly this, often carrying 2, 3 or more.",{"id":145,"type":53,"title":146,"eyebrow":147,"navLabel":148},"ch2","Borrowing is regrouping too, and a second method that works","Chapter 02","2 Why borrowing works",{"id":150,"type":43,"markdown":151},"borrow-why","In 5,003 − 1,867, the ones column asks for 3 − 7, which is impossible with whole numbers. **Borrowing** (also called **decomposition** or **regrouping**) rewrites the top number without changing its value:\n\n5,003 = 5 thousands + 0 hundreds + 0 tens + 3 ones\n      = 4 thousands + 9 hundreds + 9 tens + 13 ones\n\nCheck: 4,000 + 900 + 90 + 13 = 5003. Same number, different packing. Now every column works: 13 − 7 = 6, 9 − 6 = 3, 9 − 8 = 1, 4 − 1 = 3, giving **3,136**.\n\nThe long chain of borrowing across zeros, which so many people find confusing, is just unpacking **one thousand** into 9 hundreds, 9 tens and 10 ones in one go.",{"id":153,"type":43,"markdown":154},"equal-add","There is a completely different way to subtract, called **equal additions**. Some European schools still teach it instead of borrowing (it is also called the *Austrian* or *additions* method), and older relatives may well have learnt to subtract this way. It uses the fact from the Investigate layer: **adding the same amount to both numbers does not change the difference.**\n\nFor 5,003 − 1,867:\n- Ones: 3 − 7 won't go. Add 10 ones to the top (13 − 7 = 6) **and** add 1 ten to the bottom (the 6 tens become 7 tens). Both numbers grew by 10.\n- Tens: 0 − 7 won't go. Add 10 tens to the top (10 − 7 = 3) and 1 hundred to the bottom (8 hundreds become 9).\n- Hundreds: 0 − 9 won't go. Add 10 hundreds to the top (10 − 9 = 1) and 1 thousand to the bottom (1 thousand becomes 2).\n- Thousands: 5 − 2 = 3.\n\nAnswer: **3,136**, with no crossing-out of zeros anywhere.",{"id":156,"type":62,"caption":157,"columns":158,"rows":162},"tab-two-methods","Two ways to subtract, both correct, for different reasons",[159,160,161],"Question","Decomposition (borrowing)","Equal additions",[163,167,171,175,179],[164,165,166],"What changes","Only the top number is repacked","Both numbers grow by the same amount",[168,169,170],"Why it's allowed","The top number keeps its value","The difference (gap) is unchanged",[172,173,174],"5,003 − 1,867","4 | 9 | 9 | 13 minus 1 | 8 | 6 | 7","5 | 10 | 10 | 13 minus 2 | 9 | 7 | 7",[176,177,178],"Zeros in the top number","Long chains of borrowing","No chains: each column is fixed on its own",[180,181,182],"Easy to model with bundles","Yes: unbundle a thousand","Harder to picture with objects",{"id":184,"type":127,"itemId":185,"prompt":186,"check":187,"hints":189,"feedback":192},"prac-equal-add","four-operations.dp-equal-additions","Use any method: 80,004 − 27,658 = ?",{"kind":131,"answer":188,"tolerance":133},52346,[190,191],"Borrowing: 80,004 = 7 ten-thousands + 9 thousands + 9 hundreds + 9 tens + 14 ones.","Or add 2 to both: 80,006 − 27,660, or add 342 to both: 80,346 − 28,000.",{"correct":193,"incorrect":194},"Correct: 52,346. Check: 52,346 + 27,658 = 80,004.","The answer is 52,346. A neat route: add 342 to both numbers to get 80,346 − 28,000 = 52,346.",{"id":196,"type":47,"variant":197,"title":198,"markdown":199},"mis-smaller-from-bigger","misconception","“Just take the smaller digit from the bigger one”","A very common error in 5,003 − 1,867 is to work each column as \"big minus small\": 7 − 3 = 4, 6 − 0 = 6, 8 − 0 = 8, 5 − 1 = 4, giving 4,864. That answer is not even close to 3,136. The mistake treats every column as if it were a separate subtraction, forgetting that the **top** number must stay on top. Subtraction is not commutative: 3 − 7 is not the same as 7 − 3. An estimate (5,000 − 2,000 = 3,000) catches it immediately.",{"id":201,"type":53,"title":202,"eyebrow":203,"navLabel":204},"ch3","Long multiplication is the distributive property in disguise","Chapter 03","3 Why rows shift",{"id":206,"type":43,"markdown":207},"dist-intro","The **distributive property** says that multiplying a sum is the same as multiplying each part and adding: a × (b + c) = a × b + a × c. A tiffin example: 6 children each get 3 idlis and 2 vadas. Total items = 6 × (3 + 2) = 6 × 3 + 6 × 2 = 18 + 12 = 30.\n\nLong multiplication is this property used again and again. Split each number into its places: 347 = 300 + 40 + 7 and 26 = 20 + 6. Then every place of one meets every place of the other, like the cells of a grid. The picture is a rectangle 347 long and 26 wide, cut into six smaller rectangles whose areas add up to the whole.",{"id":209,"type":62,"caption":210,"columns":211,"rows":217},"tab-grid","The grid (area) model for 347 × 26: six partial products",[212,213,214,215,216],"×","300","40","7","Row total",[218,224,230],[219,220,221,222,223],"20","6,000","800","140","6,940",[225,226,227,228,229],"6","1,800","240","42","2,082",[231,232,233,234,235],"Column total","7,800","1,040","182","**9,022**",{"id":237,"type":43,"markdown":238},"rows-shift","Now compare with the usual layout:\n\n```\n  347\n × 26\n-----\n2,082\n6,940\n-----\n9,022\n```\n\nThe first row, 2,082, is the bottom row of the grid: 347 × 6. The second row, 6,940, is the top row: 347 × **20**. Many people write it as 347 × 2 = 694 and \"shift one place left\" or \"put a zero first\". That shift is not a rule to memorise: it happens because the 2 in 26 is really **2 tens**, and multiplying by 20 is multiplying by 2 and then by 10. Multiplying by 10 moves every digit one place to the left, because each place is worth ten times the one to its right.",{"id":240,"type":47,"variant":104,"title":241,"markdown":242},"aha-shift","Why ×10 moves digits, not 'adds a zero'","\"Add a zero\" works for whole numbers but breaks for money: ₹12.50 × 10 is not ₹12.500. The real rule is that **every digit moves one place to the left**, so 1 ten becomes 1 hundred, 2 ones become 2 tens, 5 tenths become 5 ones: ₹125.00. In a whole number, the empty ones place then needs a 0 as a placeholder, which is why it *looks* like adding a zero. Multiplying by 10ⁿ (10, 100, 1,000…) moves every digit n places.",{"id":244,"type":245,"title":246,"problem":247,"steps":248},"we-long-mult","worked_example","A harder product with the reasons written in: 4,608 × 237","A seed company packs 4,608 seeds in every packet and ships 237 packets to a district. How many seeds is that?",[249,250,251,252,253,254,255,256],"**Estimate:** about 4,600 × 200 = 9,20,000, or with more care 4,600 × 240 ≈ 11,00,000. Expect about 11 lakh.","**Split the multiplier:** 237 = 200 + 30 + 7. By the distributive property, 4,608 × 237 = 4,608 × 7 + 4,608 × 30 + 4,608 × 200.","**× 7:** 4,608 × 7 = 32,256.","**× 30:** 4,608 × 3 = 13,824, then × 10 moves each digit one place left: 138,240.","**× 200:** 4,608 × 2 = 9,216, then × 100 moves each digit two places: 921,600.","**Add the partial products:** 32,256 + 138,240 + 921,600 = 1,092,096.","**Answer:** 10,92,096 seeds, about 10.9 lakh. That matches the estimate of about 11 lakh. ✓","**The zero in 4608:** 4608 has 0 tens, so every partial product has a 0 contribution from that place; it still has to be written to keep the places lined up.",{"id":258,"type":127,"itemId":259,"prompt":260,"check":261,"hints":263,"feedback":265},"prac-grid","four-operations.dp-grid-cell","In the grid model for 563 × 48, what is the partial product for the cell '500 × 40'?",{"kind":131,"answer":262,"tolerance":133},20000,[264],"5 × 4 = 20; then count the zeros.",{"correct":266,"incorrect":267},"Right: 500 × 40 = 20,000 (5 × 4 = 20, then 3 zeros from 500 and 40).","5 × 4 = 20. 500 has two zeros and 40 has one, so 500 × 40 = 20 followed by three zeros: 20,000.",{"id":269,"type":109,"prompt":270,"options":271,"explanation":280},"pred-digits","A 4-digit number is multiplied by a 3-digit number. How many digits can the product have?",[272,274,276,278],{"id":113,"label":273},"Exactly 7",{"id":116,"label":275},"6 or 7",{"id":119,"label":277},"7 or 8",{"id":122,"label":279},"Exactly 12","**6 or 7.** The smallest case is 1,000 × 100 = 100,000 (6 digits). The largest is 9,999 × 999 = 9,989,001 (7 digits), which is still less than 1,000 × 10,000 = 1,00,00,000 (8 digits). In general, an m-digit number times an n-digit number has m + n − 1 or m + n digits. This is a handy check on long multiplication: 4,608 × 237 has 7 digits. ✓",{"id":282,"type":53,"title":283,"eyebrow":284,"navLabel":285},"ch4","Long division is repeated subtraction in big chunks","Chapter 04","4 Why division works",{"id":287,"type":43,"markdown":288},"div-repeat","The most basic meaning of 8,736 ÷ 24 is: **how many times can you take 24 away from 8,736?** You could subtract 24 again and again, 364 times. That works, but it is slow. Long division is the same idea with **big chunks**: take away 24 hundreds at a time, then 24 tens at a time, then single 24s.",{"id":290,"type":62,"caption":291,"columns":292,"rows":296},"tab-chunk","Chunking 8,736 ÷ 24: subtract big multiples of 24",[293,294,295],"Subtraction","Chunk taken","Left over",[297,301,305,309],[298,299,300],"8,736 − 7,200","24 × 300","1,536",[302,303,304],"1,536 − 1,440","24 × 60","96",[306,307,308],"96 − 96","24 × 4","0",[310,311,312],"Total chunks","300 + 60 + 4","**364**",{"id":314,"type":43,"markdown":315},"bring-down","Now look at the usual long-division layout for 8,736 ÷ 24. It hides the zeros, but it is doing exactly the same chunks.\n\n- \"24 into 87\" really means **24 into 87 hundreds**: 3 hundreds, since 24 × 3 = 72. Take away 72 hundreds (7,200). 87 − 72 = 15 hundreds are left.\n- \"**Bring down** the 3\" means: 15 hundreds and 3 tens make 153 tens. 24 into 153 tens is 6 tens (24 × 6 = 144). 153 − 144 = 9 tens left.\n- Bring down the 6: 9 tens and 6 ones make 96 ones. 24 × 4 = 96, nothing left.\n\nSo \"bring down\" is not a mysterious move. It is **combining what is left over with the next place**, because leftover hundreds are worth ten times as many tens.",{"id":317,"type":47,"variant":104,"title":318,"markdown":319},"aha-chunk","Chunking lets you choose your own chunks","In chunking you are free to take away any multiple you like: 10 × 24 = 240 is easy, so take 240 over and over, then 24s at the end. It takes a few more lines, but no step can \"go wrong\" by picking too big a digit, and you can use only facts you are sure of (×1, ×2, ×5, ×10). Long division is simply chunking where you always take the **biggest possible** chunk for each place, which makes it the shortest route.",{"id":321,"type":245,"title":322,"problem":323,"steps":324},"we-short-div","Short division: 9,875 ÷ 7 with the reasoning written out","A temple kitchen has 9,875 laddoos to pack equally into 7 large baskets. How many in each basket, and how many are left?",[325,326,327,328,329,330],"**Thousands:** 9 thousands ÷ 7 = 1 thousand, with 2 thousands left over. Write 1.","**Hundreds:** the 2 leftover thousands are 20 hundreds, plus 8 hundreds = 28 hundreds. 28 ÷ 7 = 4 hundreds exactly. Write 4.","**Tens:** 7 tens ÷ 7 = 1 ten exactly. Write 1.","**Ones:** 5 ones ÷ 7 = 0 ones, with 5 left. **Write 0.** Forgetting this zero is the most common long-division mistake: it would make the answer 141 instead of 1,410.","**Answer:** 1,410 laddoos per basket, 5 left over.","**Check:** 7 × 1,410 + 5 = 9,870 + 5 = 9,875. ✓",{"id":332,"type":127,"itemId":333,"prompt":334,"check":335,"hints":337,"feedback":340},"prac-chunk","four-operations.dp-chunking","Use chunking: a school has ₹15,000 for library books costing ₹125 each. How many books can it buy?",{"kind":131,"answer":336,"tolerance":133},120,[338,339],"125 × 100 = 12,500. How much is left after that chunk?","What is 125 × 20?",{"correct":341,"incorrect":342},"Yes: 100 books (₹12,500) plus 20 books (₹2,500) = 120 books exactly.","Chunk 1: 125 × 100 = 12,500, leaving 2,500. Chunk 2: 125 × 20 = 2,500, leaving 0. So 100 + 20 = 120 books.",{"id":344,"type":345,"component":346,"componentVersion":5,"config":347,"objective":360,"textAlternative":361,"help":362},"lab-div2","interactive","arith-sprint",{"operations":348,"ranges":350,"rounds":357,"secondsTotal":133,"estimateFirst":358,"wordProblems":359},[349],"÷",{"a":351,"b":354},{"min":352,"max":353},1000,9999,{"min":355,"max":356},12,99,10,true,[],"Divide four-digit numbers by two-digit numbers, estimating the quotient first with rounded divisors.","This untimed sprint gives 10 divisions of a four-digit number by a two-digit divisor (12 to 99), each built so that it comes out exactly. Before the exact answer, you give an estimate.\n\nThe skill it trains is the hardest step in long division: **guessing each quotient digit**. Round the divisor to the nearest ten and use a times-table fact.\n\n- 1,547 ÷ 17: 17 is about 20, and 1,547 is about 1,600, so about 80. Try 17 × 90 = 1,530, which leaves 17: exactly **91**.\n- 2,695 ÷ 49: 49 is about 50, 2,695 ÷ 50 is about 54. Try 49 × 55 = 2,695: exactly **55**.\n- 5,293 ÷ 67: 67 is about 70, 5,293 ÷ 70 is about 75. 67 × 79 = 5,293: **79**.\n\nIf your first guess is too big, the product will be bigger than the number you are dividing: step down by one. If it is too small, the leftover will be at least the divisor: step up by one. Rounding the divisor **up** tends to make you guess too small; rounding it **down** makes you guess too big.",{"hints":363},[364,365],"Round the divisor to the nearest ten, then ask the times table.","Check each guess by multiplying back.",{"id":367,"type":53,"title":368,"eyebrow":369,"navLabel":370},"ch5","The division algorithm: one quotient, one remainder, every time","Chapter 05","5 Division algorithm",{"id":372,"type":43,"markdown":373},"da-state","Every division of whole numbers, whether it comes out exactly or not, fits one sentence. For any whole number **a** (the dividend) and any whole number **d** bigger than 0 (the divisor), there is **exactly one** pair of whole numbers **q** (the quotient) and **r** (the remainder) with\n\n**a = d × q + r, where r is 0 or more and less than d.**\n\nThis fact is called the **division algorithm** (although it is really a theorem, a proven statement, rather than a method). It says two things: such a q and r **exist**, and they are **unique**: there is only one correct pair.",{"id":375,"type":376,"items":377},"f-da","formulas",[378,381,384,387],{"expression":379,"caption":380},"a = d × q + r","dividend = divisor × quotient + remainder",{"expression":382,"caption":383},"0 ≤ r and r is less than d","the remainder is never negative and never as big as the divisor",{"expression":385,"caption":386},"r = 0 ⇔ d is a factor of a","an exact division means d divides a with no remainder",{"expression":388,"caption":389},"400 = 23 × 17 + 9","one example: 400 ÷ 23 = 17 remainder 9",{"id":391,"type":43,"markdown":392},"da-why","**Why a pair exists.** List the multiples of d: 0, d, 2d, 3d, … They go up in steps of d and eventually pass a. Let d × q be the **last multiple that is not bigger than a**. Then r = a − d × q is 0 or more. And r must be less than d: if r were d or more, then d × (q + 1) would still not be bigger than a, so d × q was not the last one after all.\n\n**Why the pair is unique.** Suppose two different pairs worked: a = d × q + r and a = d × Q + R, with both remainders between 0 and d − 1. Subtracting, d × (q − Q) = R − r. The right side is a gap between two remainders, so it is smaller than d (and bigger than −d). The left side is a multiple of d. The only multiple of d strictly between −d and d is **0**, so q = Q and r = R. There was only one pair all along.",{"id":394,"type":62,"caption":395,"columns":396,"rows":401},"tab-da","One dividend, several 'answers': only one satisfies the rule",[397,398,399,400],"Claimed answer for 100 ÷ 7","Check d × q + r","Is r less than 7?","Valid?",[402,407,412,417],[403,404,405,406],"14 remainder 2","7 × 14 + 2 = 100","yes","**Yes**",[408,409,410,411],"13 remainder 9","7 × 13 + 9 = 100","no (9 is not less than 7)","No: one more 7 fits",[413,414,415,416],"12 remainder 16","7 × 12 + 16 = 100","no","No",[418,419,420,421],"15 remainder −5","7 × 15 − 5 = 100","no (negative)","No: 7 × 15 is more than 100",{"id":423,"type":47,"variant":141,"title":424,"markdown":425},"nuance-da","Why the extra conditions matter","Without the condition \"r is less than d\", the equation a = d × q + r has many solutions: 100 = 7 × 13 + 9 = 7 × 12 + 16 = 7 × 0 + 100. The condition is what makes the quotient and remainder **well-defined**, so that everyone in the world gets the same answer to 100 ÷ 7. Calculators, computers and railway timetables all rely on this.",{"id":427,"type":43,"markdown":428},"div-zero","**Why you cannot divide by zero.** Dividing is the inverse of multiplying: 12 ÷ 3 = 4 because 3 × 4 = 12. So 12 ÷ 0 would have to be a number q with **0 × q = 12**. But 0 times anything is 0, never 12. **No** q works. The division has no answer.\n\nWhat about 0 ÷ 0? Now we need 0 × q = 0, and **every** number q works: 0 × 5 = 0, 0 × 1,000 = 0. An answer that could be anything is no answer at all. So mathematicians say division by zero is **undefined**, in both cases. Notice that the division algorithm already quietly excluded it by demanding that d is bigger than 0.",{"id":430,"type":47,"variant":197,"title":431,"markdown":432},"mis-zero","“Anything divided by zero is zero” (or “is infinity”)","Both are wrong for whole-number arithmetic. **Zero divided by a number** is fine: 0 ÷ 8 = 0, because 8 × 0 = 0 (8 people share no laddoos, each gets none). **A number divided by zero** has no answer, because no number times 0 gives 8. \"Infinity\" is not a whole number, and even in more advanced mathematics 8 ÷ 0 is left undefined. Your calculator will show an error, not 0.",{"id":434,"type":109,"prompt":435,"options":436,"explanation":444},"pred-rem-sum","When 53 is divided by 6 the remainder is 5, and when 38 is divided by 6 the remainder is 2. Without adding 53 and 38, predict the remainder when **53 + 38** is divided by 6.",[437,438,440,442],{"id":113,"label":215},{"id":116,"label":439},"1",{"id":119,"label":441},"5",{"id":122,"label":443},"You must calculate 91 ÷ 6 to know","**1.** Write 53 = 6 × 8 + 5 and 38 = 6 × 6 + 2. Adding: 91 = 6 × 14 + 7. But 7 is not less than 6, so one more 6 fits: 91 = 6 × 15 + **1**. Remainders add, then you 'tidy' them exactly like a carry. This idea, arithmetic of remainders, grows into modular arithmetic, which is used in calendars (what day of the week?) and in computer security.",{"id":446,"type":127,"itemId":447,"prompt":448,"check":449,"hints":451,"feedback":453},"prac-da","four-operations.dp-find-dividend","When a number is divided by 37, the quotient is 214 and the remainder is the largest possible. What is the number?",{"kind":131,"answer":450,"tolerance":133},7954,[452],"The largest possible remainder when dividing by 37 is 36.",{"correct":454,"incorrect":455},"Correct: 37 × 214 + 36 = 7,918 + 36 = 7,954.","The remainder must be less than 37, so the largest is 36. Then a = 37 × 214 + 36 = 7,918 + 36 = 7,954.",{"id":457,"type":345,"component":458,"componentVersion":5,"config":459,"objective":486,"textAlternative":487},"lab-match-da","match-pairs",{"prompt":460,"mode":461,"pairs":462},"Match each division-algorithm expression or division to its value.","connect",[463,466,469,472,475,478,481,484],{"a":464,"b":465},"7 × 45 + 3","318",{"a":467,"b":468},"12 × 25 + 11","311",{"a":470,"b":471},"9 × 111 + 8","1,007",{"a":473,"b":474},"15 × 60 + 14","914",{"a":476,"b":477},"318 ÷ 7","45 r 3",{"a":479,"b":480},"1,007 ÷ 9","111 r 8",{"a":482,"b":483},"99 ÷ 0","undefined",{"a":485,"b":308},"0 ÷ 99","Connect divisor × quotient + remainder expressions with dividends, and divisions with their quotient and remainder, including the zero cases.","This matching game has 8 pairs to connect.\n\n- 7 × 45 + 3 = **318**, and in reverse, 318 ÷ 7 = **45 r 3**.\n- 12 × 25 + 11 = **311** (so 311 ÷ 12 = 25 r 11; 11 is allowed because it is less than 12).\n- 9 × 111 + 8 = **1,007**, and 1,007 ÷ 9 = **111 r 8**.\n- 15 × 60 + 14 = **914**. Read backwards, 914 ÷ 60 = 15 r 14: 914 minutes is 15 hours 14 minutes.\n- 99 ÷ 0 is **undefined**: no number times 0 gives 99.\n- 0 ÷ 99 = **0**: 99 × 0 = 0.\n\nEach pair is the same fact seen from two sides: division and the multiplication-plus-remainder that undoes it.",{"id":489,"type":53,"title":490,"eyebrow":491,"navLabel":492},"ch6","Casting out nines: a 1,000-year-old check","Chapter 06","6 Casting out nines",{"id":494,"type":43,"markdown":495},"nines-intro","Long before calculators, accountants and astronomers in India, the Arab world and Europe checked their sums with a trick called **casting out nines**. Add the digits of a number, and keep adding the digits of the result until one digit is left (treat 9 as 0). Call this the **digit sum** (or digital root).\n\nFor 347 × 26 = 9,022:\n- digit sum of 347: 3 + 4 + 7 = 14, then 1 + 4 = **5**\n- digit sum of 26: 2 + 6 = **8**\n- multiply the digit sums: 5 × 8 = 40, digit sum **4**\n- digit sum of the answer 9,022: 9 + 0 + 2 + 2 = 13, then **4**\n\nThe two results match (4 = 4), so the multiplication passes the check. The same works for addition (add the digit sums) and subtraction (subtract them).",{"id":497,"type":43,"markdown":498},"nines-why","**Why does it work?** Look at what happens to powers of ten when you divide by 9:\n\n10 = 9 + 1, 100 = 99 + 1, 1,000 = 999 + 1, and so on. Every place value is **a multiple of 9, plus 1**.\n\nSo 347 = 3 × 100 + 4 × 10 + 7 = 3 × (99 + 1) + 4 × (9 + 1) + 7 = (a multiple of 9) + 3 + 4 + 7.\n\nThat means **a number and its digit sum leave the same remainder when divided by 9**. The digit sum is really \"the remainder on division by 9\". And remainders behave well: the remainder of a product is the remainder of the product of the remainders (the same tidying you saw with 53 + 38 and 6 in the last chapter). So if the digit sums disagree, the calculation is **definitely** wrong.",{"id":500,"type":47,"variant":501,"title":502,"markdown":503},"model-nines","model_limit","What casting out nines cannot catch","If the digit sums **disagree**, there is certainly a mistake. If they **agree**, the answer is only *probably* right. The check is blind to any error that changes the answer by a multiple of 9, including:\n\n- **Swapped digits:** 9,202 instead of 9,022 has the same digit sum (4), so it passes.\n- **A missing or extra zero:** 922 or 90,220 also have digit sum 4.\n- **A partial product in the wrong row:** shifting a row changes the answer by a multiple of 9 × (that row), which also slips through.\n\nSo casting out nines catches most careless slips, but it is a filter, not a proof. Pair it with an estimate, which catches exactly the place-value errors that nines miss.",{"id":505,"type":245,"title":506,"problem":507,"steps":508},"we-nines","Checking a big bill with nines","A shopkeeper adds three bills: ₹48,375 + ₹29,618 + ₹7,904 and writes ₹85,997. Check with casting out nines, then find the truth.",[509,510,511,512,513],"Digit sums: 48,375 → 4+8+3+7+5 = 27 → **9** (a 9, which counts as 0); 29,618 → 26 → **8**; 7,904 → 20 → **2**.","Add them: 9 + 8 + 2 = 19 → digit sum **1**.","The shopkeeper's answer: 85,997 → 8+5+9+9+7 = 38 → **2**.","1 ≠ 2, so the total is **wrong**.","Correct sum: 48,375 + 29,618 + 7,904 = **₹85,897**, digit sum 1. ✓ The shopkeeper was off by -100: one carry was dropped in the tens column.",{"id":515,"type":127,"itemId":516,"prompt":517,"check":518,"hints":530,"feedback":532},"prac-nines","four-operations.dp-nines-check","Which of these answers to 736 × 58 is **certainly wrong**, according to casting out nines? (Digit sums: 736 → 7, 58 → 4.)",{"kind":519,"options":520,"correct":529},"choice",[521,523,525,527],{"id":113,"label":522},"42,688",{"id":116,"label":524},"42,868",{"id":119,"label":526},"42,698",{"id":122,"label":528},"4,268",[119],[531],"7 × 4 = 28, so the answer's digit sum must be 2 + 8 = 10 → 1.",{"correct":533,"incorrect":534},"Right: 42,698 has digit sum 29 → 11 → 2, not 1, so it must be wrong. The true answer is 42,688. Notice that 42,868 (swapped digits) and 4,268 (missing digit) both pass the check even though they are wrong!","The digit sums multiply to 7 × 4 = 28 → 1. Only 42,698 has a different digit sum (2), so only it is certainly wrong. The true product is 42,688; 42,868 and 4,268 pass the nines check but fail an estimate.",{"id":536,"type":537,"conceptId":538,"relation":539,"explanation":540},"conn-prime","connection","prime-and-composite","related_to","Digit sums also give the divisibility tests for 3 and 9, and the division algorithm is the tool for testing whether any number is a factor.",{"id":542,"type":53,"title":543,"eyebrow":544,"navLabel":545},"ch7","The unitary method and the rule of three","Chapter 07","7 Unitary method",{"id":547,"type":43,"markdown":548},"unitary","*\"12 notebooks cost ₹540. What do 7 notebooks cost?\"* The **unitary method** goes through one: 1 notebook costs ₹540 ÷ 12 = ₹45, so 7 notebooks cost 7 × ₹45 = **₹315**. It is called *unitary* because it finds the value of one **unit** first.\n\nIndian mathematicians had a name for this kind of problem more than 1,400 years ago: **trairāśika**, the \"rule of three\", because three quantities are known (12 notebooks, ₹540, 7 notebooks) and the fourth is found. Aryabhata described it around 499 CE, and Brahmagupta and Bhāskara II both used it. Traders carried the rule to the Arab world and then to Europe, where for centuries it was taught as the most important rule in commercial arithmetic.",{"id":550,"type":245,"title":551,"problem":552,"steps":553},"we-unitary-order","Multiply first or divide first?","A tractor ploughs 45 acres using 36 litres of diesel. How much diesel for 70 acres?",[554,555,556,557,558],"**Unitary route:** 1 acre uses 36 ÷ 45 litres. That is not a whole number (0.8 L), which is awkward.","**Rule-of-three route:** multiply first, divide last: 36 × 70 ÷ 45 = 2520 ÷ 45 = **56 litres**.","**Why is the order allowed?** 36 × 70 ÷ 45 means (36 ÷ 45) × 70, and multiplying and dividing can be done in either order when the numbers are exact: 2520 ÷ 45 = 56 and 0.8 × 70 = 56.","**Smart simplification:** 70 and 45 share a factor 5, and 36 and 45 share 9: 36 × 70 ÷ 45 = (36 ÷ 9) × (70 ÷ 5) = 4 × 14 = 56.","**Sense check:** 70 acres is a bit more than 1½ × 45, so the diesel should be a bit more than 1½ × 36 = 54 L. 56 L fits. ✓",{"id":560,"type":47,"variant":501,"title":561,"markdown":562},"model-unitary","When the unitary method gives nonsense","The unitary method only works when the two quantities are **proportional**: double one and the other doubles. Many real situations are not like that.\n\n- One singer sings a song in 3 minutes. Three singers together still take **3 minutes**, not 1.\n- A cricketer scores 50 in one match. That does not mean 500 in ten matches.\n- One kettle boils water in 4 minutes. Four kettles boiling four lots of water take 4 minutes, not 16.\n- Some prices are not proportional: a 1 kg packet often costs less than twice a 500 g packet.\n\nBefore using it, ask: **if I had twice as much of the first thing, would I really have exactly twice as much of the second?**",{"id":564,"type":43,"markdown":565},"inverse-prop","Some problems are proportional in the **opposite** direction. *\"6 workers build a wall in 10 days. How long would 15 workers take?\"* More workers means **fewer** days. Count the total work in **worker-days**: 6 × 10 = 60 worker-days. Shared among 15 workers: 60 ÷ 15 = **4 days**. This is called **inverse proportion**. (It still assumes every worker is equally fast and nobody gets in anyone's way, which real builders might dispute.)",{"id":567,"type":127,"itemId":568,"prompt":569,"check":570,"hints":573,"feedback":575},"prac-unitary","four-operations.dp-unitary","A train covers 318 km in 6 hours at a steady speed. How far does it go in 11 hours at the same speed?",{"kind":131,"answer":571,"tolerance":133,"unit":572},583,"km",[574],"Find the distance in 1 hour first.",{"correct":576,"incorrect":577},"Correct: 318 ÷ 6 = 53 km per hour, and 53 × 11 = 583 km.","1 hour: 318 ÷ 6 = 53 km. 11 hours: 53 × 11 = 583 km.",{"id":579,"type":127,"itemId":580,"prompt":581,"check":582,"hints":593,"feedback":595},"prac-proportional","four-operations.dp-proportional-or-not","Which situation can **not** be solved with the unitary method?",{"kind":519,"options":583,"correct":592},[584,586,588,590],{"id":113,"label":585},"5 kg of rice costs ₹310; find the cost of 8 kg",{"id":116,"label":587},"A car uses 4 L of petrol for 60 km; find petrol for 150 km",{"id":119,"label":589},"A photo takes 2 seconds to print; how long for 2 people to look at it?",{"id":122,"label":591},"8 pens cost ₹96; find the cost of 3 pens",[119],[594],"Ask: does doubling one quantity double the other?",{"correct":596,"incorrect":597},"Right: the time to look at a photo does not depend on how many people look. The others are all proportional.","Rice cost, petrol use and pen cost all double when the amount doubles. Looking at a photo does not take twice as long for two people, so (c) is not proportional.",{"id":599,"type":53,"title":600,"eyebrow":601,"navLabel":602},"ch8","Multi-step problems: bar models and working backwards","Chapter 08","8 Multi-step problems",{"id":604,"type":43,"markdown":605},"bar-intro","Hard word problems are rarely hard because of the arithmetic. They are hard because there are **several steps** and it is not obvious which comes first. Two strategies from strong problem-solvers help.\n\nA **bar model** draws each quantity as a strip. Equal parts get equal strips. The picture shows you which operation to use without any keyword hunting.\n\n**Working backwards** starts from the end of the story and undoes each step with its inverse operation: undo \"spent ₹40\" by adding ₹40, undo \"halved\" by doubling.",{"id":607,"type":245,"title":608,"problem":609,"steps":610},"we-backwards","Working backwards: Ravi's pocket money","Ravi spent half his money on a book, then ₹40 on a pen. He had ₹60 left. How much did he start with?",[611,612,613,614],"**Last step first.** After buying the pen he had ₹60. Undo 'spent ₹40' by adding: 60 + 40 = ₹100. That is what he had after the book.","**Undo the halving.** He had spent half and kept half, so ₹100 was half his money. Double it: 100 × 2 = **₹200**.","**Check forwards:** ₹200 → half on the book leaves ₹100 → minus ₹40 leaves ₹60. ✓","**Bar model view:** draw a bar for his money, cut into two equal halves. One half is the book. The other half is split into ₹40 (pen) and ₹60 (left), so each half is ₹100.",{"id":616,"type":245,"title":617,"problem":618,"steps":619},"we-trip","A school-trip budget in four steps","132 students and 8 teachers go to Mysuru Palace. Each bus seats 50 and costs ₹6,500 for the day. Entry is ₹40 per person and lunch is ₹75 per person. The teachers' share is paid by the school, but the whole cost is to be recovered from the students equally. How much should each student pay?",[620,621,622,623,624,625],"**People:** 132 + 8 = 140.","**Buses:** 140 ÷ 50 = 2 r 40. The 40 extra people still need a seat: round **up** to 3 buses. Bus cost: 3 × ₹6,500 = ₹19,500.","**Per-person costs:** entry + lunch = ₹40 + ₹75 = ₹115; for 140 people: 140 × 115 = ₹16,100.","**Total:** ₹19,500 + ₹16,100 = ₹35,600.","**Per student:** 35,600 ÷ 132 = 269 r 92. Charging ₹269 would leave ₹92 unpaid, so round **up** to ₹270. Collected: 132 × 270 = ₹35,640, which covers the cost with ₹40 to spare.","**Estimate check:** 3 buses ≈ ₹20,000, and 140 × ₹115 ≈ ₹16,000, total ≈ ₹36,000. Divide by about 130: roughly ₹280. ✓",{"id":627,"type":62,"caption":628,"columns":629,"rows":634},"tab-kirana","A kirana bill worked out in full (the multi-step idea in table form; the rates are made-up round figures, not real shop prices)",[630,631,632,633],"Item","Quantity","Rate","Amount",[635,640,645,650,654,659,664],[636,637,638,639],"Sona masoori rice","5 kg","₹62 per kg","₹310",[641,642,643,644],"Groundnut oil","2 L","₹145 per L","₹290",[646,647,648,649],"Toor dal","3 kg","₹128 per kg","₹384",[651,225,652,653],"Biscuit packets","₹30 each","₹180",[655,656,657,658],"Total","4 kinds","—","**₹1,164**",[660,661,662,663],"Paid with","3 notes","two ₹500 + one ₹200","₹1,200",[665,657,666,667],"Change","₹1,200 − total","**₹36**",{"id":669,"type":127,"itemId":670,"prompt":671,"check":672,"hints":675,"feedback":678},"prac-yield","four-operations.dp-crop-yield","Two farmers compare harvests. Lakshmi's 3-hectare field gave 13,500 kg of paddy. Suresh's 5-hectare field gave 21,000 kg. How many **more kg per hectare** did Lakshmi's field give?",{"kind":131,"answer":673,"tolerance":133,"unit":674},300,"kg",[676,677],"Find each farm's yield per hectare first.","Then subtract.",{"correct":679,"incorrect":680},"Right: 13,500 ÷ 3 = 4,500 kg per hectare, 21,000 ÷ 5 = 4,200 kg per hectare, difference 300 kg.","Per hectare: Lakshmi 13,500 ÷ 3 = 4,500 kg and Suresh 21,000 ÷ 5 = 4,200 kg. Lakshmi's field gave 4,500 − 4,200 = 300 kg more per hectare, even though Suresh's total was bigger.",{"id":682,"type":47,"variant":104,"title":683,"markdown":684},"aha-rate","Totals versus rates","Suresh harvested more rice in total, yet Lakshmi's field was more **productive**. Comparing totals answers \"who has more?\"; dividing to get a **rate** (per hectare, per person, per hour, per over) answers \"who did better for their size?\" Run rates in cricket, marks per subject, rainfall per day and population per square kilometre are all division turning totals into fair comparisons.",{"id":686,"type":53,"title":687,"eyebrow":688,"navLabel":689},"ch9","Arithmetic with India-sized numbers","Chapter 09","9 India-sized numbers",{"id":691,"type":43,"markdown":692},"census","India's 2011 Census counted **1,21,08,54,977** people (about 121 crore) in its published final totals. Of these, 62,32,70,258 were male and 58,75,84,719 female. Check the addition yourself: 62,32,70,258 + 58,75,84,719 = 1,21,08,54,977. ✓\n\nThe difference, 62,32,70,258 − 58,75,84,719 = **3,56,85,539**, is about 3.6 crore, more than the whole population of many countries. Numbers this large are where careful column work and good estimation both matter: a single slipped digit in the crores place is an error of crores of people.",{"id":694,"type":245,"title":695,"problem":696,"steps":697},"we-sex-ratio","Females per 1,000 males","India reports its sex ratio as the number of females per 1,000 males. Using the 2011 figures, estimate it and then find it to the nearest whole number.",[698,699,700,701,702],"**The idea:** divide females by males to get females per male, then multiply by 1,000.","**Estimate:** about 58.8 crore ÷ 62.3 crore. That is a bit less than 1, roughly 0.94, so about 940 per 1,000.","**More exactly:** 58,75,84,719 × 1,000 ÷ 62,32,70,258 = 942.7.","**Rounded:** **943 females per 1,000 males**, which is the figure reported for the 2011 Census.","**Why multiply before dividing?** Multiplying by 1,000 first keeps the numbers whole for longer; the order does not change the result.",{"id":704,"type":43,"markdown":705},"elections","In the 2024 Lok Sabha election, India had **about 96.8 crore** registered voters (electors), the largest electorate in the world, spread across **543 constituencies**. On average that is about 96,80,00,000 ÷ 543 ≈ **17,82,689** electors per constituency (about 17.8 lakh).\n\nThat average hides huge differences: some Himalayan and island constituencies have well under 2 lakh electors, while some city constituencies have well over 30 lakh. An average is just the total divided equally, which never happens in real life. Still, it is the right number for a first estimate: for example, if each polling station serves roughly 1,000 to 1,500 electors, you would need something like 96.8 crore ÷ 1,000 ≈ 9.7 lakh polling stations at most, and indeed the Election Commission set up **about 10 lakh**.",{"id":707,"type":47,"variant":141,"title":708,"markdown":709},"nuance-precision","How many digits should you give?","If the electorate is only known as \"about 96.8 crore\", it makes no sense to say each constituency has exactly 1,782,688 electors. The input had three meaningful digits, so the answer deserves about three: **17.8 lakh**. A good rule: **your answer can't be more precise than the numbers you started with.** Census counts like 1,21,08,54,977 are exact on the day they describe, but even those were already out of date the next morning, as babies were born.",{"id":711,"type":345,"component":712,"componentVersion":5,"config":713,"objective":719,"textAlternative":720},"lab-round-big","rounding-race",{"roundTo":714,"range":717,"rounds":357,"secondsPerRound":133},[715,716],10000,100000,{"min":716,"max":718},9999999,"Round populations and vote counts in the lakhs to the nearest ten thousand or lakh, the precision used in news reports.","This game shows a number between 1,00,000 (one lakh) and 99,99,999 (just under one crore) on a number line and asks you to round it to the nearest ten thousand or the nearest lakh. 10 untimed rounds.\n\nThe rule is the same as for small numbers, just further left. To the nearest **lakh**, look at the ten-thousands digit: 5 or more rounds up. To the nearest **ten thousand**, look at the thousands digit.\n\n- 17,82,688 to the nearest lakh: ten-thousands digit 8, so **18,00,000**. To the nearest ten thousand: thousands digit 2, so **17,80,000**.\n- 43,49,999 to the nearest lakh: ten-thousands digit 4, so **43,00,000**, even though it is very close to 43.5 lakh.\n- 99,60,000 to the nearest lakh: round up to **1,00,00,000**, one crore.\n\nNews reports say \"about 17.8 lakh voters\" or \"18 lakh voters\": these are rounded values like the ones in this game.",{"id":722,"type":345,"component":346,"componentVersion":5,"config":723,"objective":777,"textAlternative":778},"lab-sprint-big",{"operations":724,"ranges":727,"rounds":732,"secondsTotal":133,"estimateFirst":358,"wordProblems":733},[725,726,212,349],"+","-",{"a":728,"b":730},{"min":352,"max":729},99999,{"min":731,"max":356},11,24,[734,738,742,746,749,753,756,759,763,767,770,773],{"prompt":735,"answer":736,"unit":737,"operation":725},"A district has 18,45,630 people in towns and 32,78,445 in villages. What is the total population?",5124075,"people",{"prompt":739,"answer":740,"unit":741,"operation":726},"A state produced 1,24,50,000 tonnes of rice this year and 98,76,500 tonnes last year. How many more tonnes this year?",2573500,"tonnes",{"prompt":743,"answer":744,"unit":745,"operation":726},"A constituency has 17,82,688 electors and 11,64,250 voted. How many did not vote?",618438,"electors",{"prompt":747,"answer":748,"unit":572,"operation":212},"A train runs 1,384 km each way, 312 trips a year. How many km does it cover in a year?",431808,{"prompt":750,"answer":751,"unit":752,"operation":212},"A cricket stadium sold 42,500 tickets at ₹650 each. What was the total ticket income in rupees?",27625000,"₹",{"prompt":754,"answer":755,"unit":752,"operation":349},"₹85,05,000 of relief money is shared equally among 3,150 families. How much does each family get?",2700,{"prompt":757,"answer":758,"unit":752,"operation":726},"A school collects ₹270 from each of 132 students and pays ₹35,600 for a trip. How much is left over?",40,{"prompt":760,"answer":761,"unit":762,"operation":349},"A factory makes 2,48,640 bulbs and packs them 48 to a carton. How many cartons?",5180,"cartons",{"prompt":764,"answer":765,"unit":766,"operation":349},"6 workers build a wall in 10 days. How many days would 15 workers take, at the same rate?",4,"days",{"prompt":768,"answer":769,"unit":752,"operation":726},"A shop buys 45 kg of cashews for ₹31,500 and sells them at ₹820 per kg. What is the profit?",5400,{"prompt":771,"answer":772,"unit":674,"operation":212},"A farmer's 3 hectares gave 13,500 kg. At that rate, what would 8 hectares give?",36000,{"prompt":774,"answer":775,"unit":776,"operation":212},"A water board supplies 135 L per person per day to a town of 2,40,000. How many litres a day?",32400000,"L","Solve large-number and multi-step problems from census, elections, farming and railways, estimating before calculating.","This untimed sprint has 24 rounds: 12 large-number word problems mixed at random with generated sums on numbers up to 99,999 (estimate first on the generated ones). The word problems and their answers:\n\n1. Town + village: 18,45,630 + 32,78,445 = **51,24,075** people.\n2. Rice: 1,24,50,000 − 98,76,500 = **25,73,500** tonnes more.\n3. Did not vote: 17,82,688 − 11,64,250 = **6,18,438**.\n4. Train: 1,384 × 312 = **4,31,808** km.\n5. Tickets: 42,500 × ₹650 = **₹2,76,25,000**.\n6. Relief: ₹85,05,000 ÷ 3,150 = **₹2,700** per family.\n7. Trip: 132 × ₹270 − ₹35,600 = **₹40** left.\n8. Bulbs: 2,48,640 ÷ 48 = **5,180** cartons.\n9. Wall: 6 × 10 ÷ 15 = **4** days (inverse proportion).\n10. Cashews: 45 × ₹820 − ₹31,500 = **₹5,400** profit.\n11. Yield: 13,500 ÷ 3 × 8 = **36,000** kg.\n12. Water: 135 × 2,40,000 = **3,24,00,000** litres a day.\n\nSeveral are two-step problems: decide the order of steps first, as in the bar-model chapter.",{"id":780,"type":537,"conceptId":781,"relation":782,"explanation":783},"conn-number","number-system","helps_understand","Reading, writing and rounding numbers in lakhs and crores in both systems is what makes India-sized arithmetic manageable.",{"id":785,"type":53,"title":786,"eyebrow":787,"navLabel":788},"ch10","Who worked this out? A short history of calculating","Chapter 10","10 History",{"id":790,"type":43,"markdown":791},"hist-intro","Every method in this topic depends on **place value with a zero**: the idea that the same digit is worth different amounts in different places, and that an empty place needs a symbol. That idea took shape in India, probably by the 5th or 6th century CE, and it changed calculation forever. With Roman numerals (MMCDLXXV × CCCVI) column arithmetic is almost impossible; with Indian numerals it is something a ten-year-old can learn.",{"id":793,"type":794,"title":795,"items":796},"tl-history","timeline","From counting boards to column methods",[797,801,805,809,813,817,821,825],{"time":798,"title":799,"text":800},"c. 2000 BCE","Babylon and Egypt","Babylonian scribes use a base-60 place system (without a true zero at first) and tables for multiplying; Egyptian scribes multiply by repeated doubling.",{"time":802,"title":803,"text":804},"3rd-7th c.","Bakhshali manuscript","An Indian birch-bark manuscript of arithmetic problems that uses a dot as a place-holder for zero. Its date is genuinely unsettled: scholars have argued for 200-400 CE, and also for a 7th-century original surviving only in a later copy.",{"time":806,"title":807,"text":808},"499 CE","Aryabhata","Aryabhata finishes the Aryabhatiya at the age of 23. It sets out arithmetic including methods for square and cube roots, and historians read it as working with a decimal place-value system and a zero.",{"time":810,"title":811,"text":812},"628 CE","Brahmagupta","The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, with rules such as 'a number minus itself is zero'. On division he said zero divided by zero is zero, and did not commit himself on a number divided by zero — neither is the modern rule.",{"time":814,"title":815,"text":816},"c. 820s CE","Al-Khwarizmi","In Baghdad, under the caliph al-Ma'mun, al-Khwarizmi writes the first systematic account of calculating with Indian numerals. The Arabic original is lost; Europe knew it as *Algoritmi de numero Indorum*, and his name gives us the word 'algorithm'.",{"time":818,"title":819,"text":820},"c. 1150 CE","Bhāskara II, Līlāvatī","A celebrated Sanskrit arithmetic in thirteen chapters, written in verse, covering the four operations, the rule of three and compound proportions. Its problems are everyday stories about kings, elephants and a broken pearl necklace.",{"time":822,"title":823,"text":824},"1202","Fibonacci, Liber Abaci","Leonardo of Pisa, who learnt Hindu–Arabic numerals in North Africa, teaches them to European merchants.",{"time":826,"title":827,"text":828},"1400s-1800s","Galley to long division","The 'galley' (scratch) method, with digits crossed out as the work goes, is the usual way to divide in Europe before 1600. Calandri prints a long-division example in 1491 and Henry Briggs introduces the modern layout around 1600, but the galley method stays popular into the 1700s.",{"id":830,"type":47,"variant":141,"title":831,"markdown":832},"nuance-brahmagupta","Even great mathematicians can be partly wrong","Brahmagupta was the first known writer to treat zero as a number and to state rules such as \"zero added to a number leaves it unchanged\" and \"a number multiplied by zero is zero\". He also tried to say what a number divided by zero is, and what 0 ÷ 0 is (he said 0). Today we say both are **undefined**, for the reasons in Chapter 5. Later Indian mathematicians, including Bhāskara II, discussed division by zero again. Mathematics grows by people building on and correcting each other, across centuries and countries.",{"id":834,"type":43,"markdown":835},"lilavati","Here is a problem written in the spirit of the Līlāvatī — not one taken from the book itself: *\"A merchant buys 3 pearls for 2 coins and sells 5 pearls for 4 coins. How many pearls must he trade to make a profit of 14 coins?\"*\n\nCost of one pearl: 2 ÷ 3 coin. Selling price of one pearl: 4 ÷ 5 coin. To keep everything whole, think in groups of **15 pearls** (15 is a common multiple of 3 and 5): 15 pearls cost 10 coins and sell for 12 coins, a profit of 2 coins. For 14 coins of profit he needs 7 such groups: **7 × 15 = 105 pearls**. The trick of choosing a common multiple is exactly what the LCM topic studies.",{"id":837,"type":537,"conceptId":838,"relation":839,"explanation":840},"conn-hcf","hcf-and-lcm","applied_in","Choosing a common multiple of 3 and 5 (here 15) to keep a rule-of-three problem in whole numbers is an everyday use of the LCM.",{"id":842,"type":53,"title":843,"eyebrow":844,"navLabel":845},"ch11","Words, checks and a quiz","Chapter 11","11 Wrap-up",{"id":847,"type":848,"title":849,"terms":850},"glossary-deepen","glossary","Words for why methods work",[851,855,859,863,867,871,875,879,883,887,890,893,897,901,905,909,913,917,921,925],{"term":852,"meaning":853,"example":854},"expanded form","A number written as the sum of its place values.","2,47,368 = 2,00,000 + 40,000 + 7,000 + 300 + 60 + 8",{"term":856,"meaning":857,"example":858},"regrouping","Exchanging 10 of one place for 1 of the next place up (carrying), or 1 of a place for 10 of the next place down (borrowing). The number's value does not change.","13 tens = 1 hundred + 3 tens",{"term":860,"meaning":861,"example":862},"decomposition","The borrowing method of subtraction, in which the top number is repacked into different places.","5,003 = 4,000 + 900 + 90 + 13",{"term":864,"meaning":865,"example":866},"equal additions","A subtraction method that adds the same amount to both numbers, column by column, so the difference is unchanged.","13 − 7 in the ones, then add 1 ten to the bottom",{"term":868,"meaning":869,"example":870},"distributive property","Multiplying a sum equals multiplying each part and adding: a × (b + c) = a × b + a × c.","347 × 26 = 347 × 20 + 347 × 6",{"term":872,"meaning":873,"example":874},"partial product","One of the pieces multiplied separately in long multiplication or the grid model, before adding.","In 347 × 26, the partial products are 6,940 and 2,082.",{"term":876,"meaning":877,"example":878},"grid (area) model","A table or rectangle that splits both factors by place value; each cell is a partial product.","300 × 20 = 6,000 is one cell of 347 × 26",{"term":880,"meaning":881,"example":882},"chunking","Dividing by subtracting convenient multiples of the divisor (chunks) and adding up how many were taken.","8,736 ÷ 24: take 7,200, then 1,440, then 96",{"term":884,"meaning":885,"example":886},"short division","Long division written compactly, carrying each remainder to the next digit mentally.","9,875 ÷ 7 = 1,410 r 5",{"term":888,"meaning":889,"example":388},"division algorithm","For whole a and d bigger than 0, there is exactly one q and r with a = d × q + r and r less than d.",{"term":483,"meaning":891,"example":892},"Having no meaningful answer. Division by zero is undefined.","12 ÷ 0 is undefined",{"term":894,"meaning":895,"example":896},"digit sum (digital root)","Add a number's digits repeatedly until one digit is left (9 counts as 0); it equals the remainder on division by 9.","9,022 → 13 → 4",{"term":898,"meaning":899,"example":900},"casting out nines","Checking a calculation by doing the same operation on the digit sums and comparing with the answer's digit sum.","5 × 8 = 40 → 4 matches 9,022 → 4",{"term":902,"meaning":903,"example":904},"unitary method","Solving a proportional problem by first finding the value for one unit.","12 cost ₹540 → 1 costs ₹45",{"term":906,"meaning":907,"example":908},"rule of three (trairāśika)","The ancient Indian method: from three known quantities in proportion, find the fourth.","36 × 70 ÷ 45 = 56",{"term":910,"meaning":911,"example":912},"proportional","Two quantities are proportional if doubling (tripling…) one doubles (triples…) the other.","Cost of rice and its weight",{"term":914,"meaning":915,"example":916},"inverse proportion","When one quantity doubles and the other halves, so their product stays the same.","6 workers × 10 days = 15 workers × 4 days",{"term":918,"meaning":919,"example":920},"rate","A quantity per one unit of something else, found by dividing.","4,500 kg per hectare",{"term":922,"meaning":923,"example":924},"bar model","A drawing of quantities as strips, used to see the structure of a word problem.","Half the bar is the book; the other half is ₹40 + ₹60",{"term":926,"meaning":927,"example":928},"counter-example","One example that shows a general claim is false.","Three singers don't sing a song in 1 minute.",{"id":930,"type":930,"title":931,"questions":932},"quiz","Check your reasoning",[933,946,959,972,982,995,1008,1021,1034,1047],{"itemId":934,"prompt":935,"options":936,"correct":116,"why":945},"four-operations.dp-quiz-carry","In column addition, what does 'carry 1' into the hundreds column actually move?",[937,939,941,943],{"id":113,"label":938},"One extra one",{"id":116,"label":940},"Ten tens, bundled as one hundred",{"id":119,"label":942},"A spare digit from the answer",{"id":122,"label":944},"Nothing; it is just a reminder","A carry is 10 of one place bundled into 1 of the next place. Carrying into the hundreds moves ten tens, which is one hundred.",{"itemId":947,"prompt":948,"options":949,"correct":116,"why":958},"four-operations.dp-quiz-equal","The equal-additions method of subtraction works because…",[950,952,954,956],{"id":113,"label":951},"the top number keeps its value",{"id":116,"label":953},"adding the same amount to both numbers leaves the difference unchanged",{"id":119,"label":955},"subtraction is commutative",{"id":122,"label":957},"zeros can be ignored","Equal additions add 10 of a place to the top and 1 of the next place (also 10 of the first) to the bottom, so the gap stays the same.",{"itemId":960,"prompt":961,"options":962,"correct":116,"why":971},"four-operations.dp-quiz-shift","Why is the second row of 347 × 26 written one place to the left?",[963,965,967,969],{"id":113,"label":964},"It is a tradition",{"id":116,"label":966},"Because the 2 in 26 means 2 tens, so that row is 347 × 20",{"id":119,"label":968},"To leave room for the carry",{"id":122,"label":970},"Because 2 is even","The distributive property splits 26 into 20 + 6. The second row is 347 × 20 = 6,940, and multiplying by 10 moves every digit one place left.",{"itemId":973,"prompt":974,"options":975,"correct":113,"why":981},"four-operations.dp-quiz-digits","How many digits can a 5-digit number times a 2-digit number have?",[976,977,978,979],{"id":113,"label":275},{"id":116,"label":273},{"id":119,"label":277},{"id":122,"label":980},"10","Smallest: 10,000 × 10 = 100,000 (6 digits). Largest: 99,999 × 99 = 9,899,901 (7 digits).",{"itemId":983,"prompt":984,"options":985,"correct":116,"why":994},"four-operations.dp-quiz-da","Which statement of 250 ÷ 7 follows the division algorithm?",[986,988,990,992],{"id":113,"label":987},"250 = 7 × 34 + 12",{"id":116,"label":989},"250 = 7 × 35 + 5",{"id":119,"label":991},"250 = 7 × 36 − 2",{"id":122,"label":993},"250 = 7 × 30 + 40","All four equations are true, but only 7 × 35 + 5 has a remainder that is at least 0 and less than 7.",{"itemId":996,"prompt":997,"options":998,"correct":116,"why":1007},"four-operations.dp-quiz-zero","Why is 15 ÷ 0 undefined?",[999,1001,1003,1005],{"id":113,"label":1000},"Because the answer is 0",{"id":116,"label":1002},"Because no number multiplied by 0 gives 15",{"id":119,"label":1004},"Because 15 is odd",{"id":122,"label":1006},"Because the answer is too big to write","Division undoes multiplication: 15 ÷ 0 = q would need 0 × q = 15, and 0 times any number is 0.",{"itemId":1009,"prompt":1010,"options":1011,"correct":116,"why":1020},"four-operations.dp-quiz-nines","Casting out nines cannot detect which error?",[1012,1014,1016,1018],{"id":113,"label":1013},"A wrong digit in the tens place",{"id":116,"label":1015},"Writing 6,390 instead of 6,930",{"id":119,"label":1017},"Forgetting a carry of 1",{"id":122,"label":1019},"Adding 1 to the answer","Swapping digits keeps the digit sum the same (9 and 9), so the check passes even though the answer is wrong.",{"itemId":1022,"prompt":1023,"options":1024,"correct":116,"why":1033},"four-operations.dp-quiz-nines-why","Casting out nines works because every place value (10, 100, 1,000…) is…",[1025,1027,1029,1031],{"id":113,"label":1026},"a multiple of 9",{"id":116,"label":1028},"one more than a multiple of 9",{"id":119,"label":1030},"one less than a multiple of 9",{"id":122,"label":1032},"a power of 9","10 = 9 + 1, 100 = 99 + 1, 1,000 = 999 + 1, so a number and its digit sum leave the same remainder when divided by 9.",{"itemId":1035,"prompt":1036,"options":1037,"correct":116,"why":1046},"four-operations.dp-quiz-unitary","Which problem is proportional, so the unitary method applies?",[1038,1040,1042,1044],{"id":113,"label":1039},"A boy is 4 feet tall at age 8; how tall at 16?",{"id":116,"label":1041},"3 kg of onions cost ₹105; cost of 7 kg?",{"id":119,"label":1043},"One clock strikes 6 in 5 seconds; how long to strike 12?",{"id":122,"label":1045},"One cook makes tea in 5 min; how long for 5 cooks?","Onion cost doubles when the weight doubles: ₹105 ÷ 3 = ₹35 per kg, so 7 kg cost ₹245. Height is not proportional to age; a clock striking 6 has 5 gaps, not 6; and extra cooks don't make tea faster.",{"itemId":1048,"prompt":1049,"options":1050,"correct":113,"why":1059},"four-operations.dp-quiz-backwards","After spending ₹150 and then half of what was left, Meera has ₹200. What did she start with?",[1051,1053,1055,1057],{"id":113,"label":1052},"₹550",{"id":116,"label":1054},"₹500",{"id":119,"label":1056},"₹350",{"id":122,"label":1058},"₹700","Work backwards: ₹200 was half of what was left, so ₹400 before the half. Add back the ₹150: ₹550. Check: 550 − 150 = 400, half is 200. ✓",{"id":1061,"type":1062,"prompt":1063},"reflect","reflection","Pick one method from this layer (carrying, borrowing, long multiplication, long division or casting out nines). Explain to a younger friend **why** it works, using only place value and one real example. Which part of your explanation was hardest to put into words?",{"id":1065,"type":1066,"title":1067,"points":1068},"cheat-sheet","summary","Cheat sheet",[1069,1070,1071,1072,1073,1074,1075,1076,1077,1078,1079],"**Carrying = regrouping:** add place by place (expanded form), then trade 10 of a place for 1 of the next. We go right to left so carries never change digits already written.","**Borrowing = regrouping:** repack the top number (5,003 = 4,000 + 900 + 90 + 13). **Equal additions** instead adds the same to both numbers, keeping the difference.","**Long multiplication = distributive property:** 347 × 26 = 347 × 20 + 347 × 6. Rows shift because the multiplier's digits are tens, hundreds… An m-digit × n-digit product has m + n − 1 or m + n digits.","**Long division = chunked repeated subtraction.** 'Bring down' combines leftover hundreds (as tens) with the next digit. Never skip a 0 in the quotient.","**Division algorithm:** a = d × q + r with 0 ≤ r and r less than d; q and r exist and are unique for every d bigger than 0.","**Dividing by zero is undefined:** 8 ÷ 0 has no answer (0 × q is never 8); 0 ÷ 0 has too many. But 0 ÷ 8 = 0.","**Casting out nines:** digit sum = remainder on division by 9, because every place value is a multiple of 9 plus 1. Disagreement proves an error; agreement doesn't prove correctness (swapped digits slip through).","**Unitary method \u002F rule of three:** find one unit, or multiply first then divide. Only for proportional quantities; for inverse proportion keep the product fixed (worker-days).","**Multi-step problems:** bar models show structure; working backwards undoes each step with its inverse.","**Big numbers:** estimate first, give answers no more precise than the data, and use rates (per hectare, per 1,000, per constituency) to compare fairly.","**History:** place value with zero developed in India; the Aryabhatiya (499 CE), Brahmagupta's rules for zero (628 CE), Bhāskara II's Līlāvatī (1150); Fibonacci taught the methods to European merchants in 1202.",{"id":1081,"type":537,"conceptId":1082,"relation":782,"explanation":1083},"conn-properties","properties-of-numbers","The commutative, associative and distributive properties are the reasons every column method in this layer is allowed.",{"id":1085,"type":537,"conceptId":1086,"relation":539,"explanation":1087},"conn-order","order-of-operations","Rule-of-three calculations like 36 × 70 ÷ 45 rely on knowing when multiplication and division can be done in either order.",{"id":1089,"type":1089,"sourceIds":1090},"sources",[1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108],"four-operations-wiki-long-division","four-operations-wiki-brahmagupta","four-operations-britannica-arithmetic","four-operations-khan-arithmetic","four-operations-mathsisfun-long-multiplication","four-operations-mathsisfun-long-division","four-operations-ncert-math-6","four-operations-wiki-census-2011","four-operations-wiki-election-2024","four-operations-mactutor-indian-numerals","four-operations-mactutor-bakhshali","four-operations-mactutor-aryabhata","four-operations-mactutor-al-khwarizmi","four-operations-wiki-lilavati","four-operations-wiki-galley-division","four-operations-wiki-arithmetic","four-operations-censusindia-2011-final","four-operations-wiki-subtraction",[1091,1092,1093,1094,1095,1096,1097,1098,1099,1100,1101,1102,1103,1104,1105,1106,1107,1108],"needs_review",{"generatedBy":1112,"notes":1113},"claude-code","Draft generated locally with Python-checked numbers; pending owner review.","0ae83bdc06f688c680520940e07470bafd0e3a7da7d897d1ee03f92e46ad6f01",{"logic:practice":1116,"component:arith-sprint@1":1117,"component:match-pairs@1":1118,"component:rounding-race@1":1119,"source:four-operations-britannica-arithmetic":1120,"source:four-operations-censusindia-2011-final":1121,"source:four-operations-khan-arithmetic":1122,"source:four-operations-mactutor-al-khwarizmi":1123,"source:four-operations-mactutor-aryabhata":1124,"source:four-operations-mactutor-bakhshali":1125,"source:four-operations-mactutor-indian-numerals":1126,"source:four-operations-mathsisfun-long-division":1127,"source:four-operations-mathsisfun-long-multiplication":1128,"source:four-operations-ncert-math-6":1129,"source:four-operations-wiki-arithmetic":1130,"source:four-operations-wiki-brahmagupta":1131,"source:four-operations-wiki-census-2011":1132,"source:four-operations-wiki-election-2024":1133,"source:four-operations-wiki-galley-division":1134,"source:four-operations-wiki-lilavati":1135,"source:four-operations-wiki-long-division":1136,"source:four-operations-wiki-subtraction":1137},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","6c2f2d540b01b2d7cd170d87a127ebd380947b02b030b91412433582d09e54f1","66e407fe40575b09f96c71af650fbd183b777a5b7047ed8fcc6b693b637804f6","fb5ffe392392d45ac0baca9fda577da046d3f0005f9ad55558553ff955f8217c","ef6b590392c4299a9ce1be700f25412c2a52dea7c431f6dd28b6d7c1ce15374d","358218dd269e4c1c427294ad29796c47982d5ce4c84d55c0e5afe038afb32f58","ffe9fff3f3462e46fd683d5db3901365eedac1a7f748dfed02e109ec54397cae","10368cd0767ab2cd0105ea28d43f843cf2d19a131d9955a358cea2fd9f3e7462","e5cf10b627ae5e1c1d0cb78cbdabf4a54d8dba727ef143b6e7088d5ca2bb4555","794b6ef61d62320200593d8724a8525d6a81168da99708651826563b823043a5","f673458eca6d084a854c7e5fd91ec78c1ed561c54faf5e27538711d81efda7a5","c66e4c408e97e6c5758c1b1160148cfabaf1f5e9ca4caba87383f8ca4e389c95","3fd5b7fcbfec31c612f8356e67126230dae21df1a69f1c1a06216d510015a231","1db8b2f0f76ee0a79d803c4141d87b8198945baeebe2130a015dd376b05aa9b3","8ba18aecc6b48bd05854e49bc36f8655f5b8327b0952c4bfdae46051fb03ed50","a1a93797032706e1d199249d49fe94af4635bca81a62c0a0e03d69ccafa9062d","72e6ef79e0dc90768278cc22338cc136fdc6d8ba028bf31b4966d3910e29ca80","648ccbfcd50467926989bfe91c0cf2c67d7862a2120b585065123a76914a1087","8dc9aada436d8e4ebc8c48665c6a2ac0a3b19a65a83e09f99e95fe119815f054","40c70d4a1c69f3c608afe351817f3c7b192cc10915db61b683c16eb3cc3affb0",{"state":1139,"reviewer":1140,"selfReview":358,"reviewedAt":1141,"method":1142},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899598109]