[{"data":1,"prerenderedAt":1171},["ShallowReactive",2],{"layer:order-of-operations:extend":3},{"layer":4,"contentHash":1150,"dependencyHashes":1151,"approval":1165,"releaseId":1170},{"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":1145,"reviewStatus":1146,"authoring":1147},1,"order-of-operations","en","extend","Puzzles, arguments and the wider world","Viral puzzles, four fours, the 24 game, olympiad problems, code and open questions","Take the order of operations further: why 8 ÷ 2(2 + 2) starts arguments, the four fours and 24 puzzles, olympiad problems, how code and spreadsheets differ, other notations, projects and open questions.",[13,14,15,16,17],"Explain why 8 ÷ 2(2 + 2) has two reasonable readings and rewrite it unambiguously.","Solve four fours and 24-game puzzles, and explain why some targets need extra tools or fractions.","Use structure (pairing, common factors, working backwards) to solve olympiad-style problems.","Compare how school maths, Python and spreadsheets handle precedence, including −3² and 2^3^2.","Plan and carry out a small project or survey about how people read expressions.",45,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Extend",{"label":26,"value":27},"Reading time","≈ 45 minutes",{"label":29,"value":30},"Prior knowledge","Deepen: reasons, powers, trees",{"label":32,"value":33},"Chapters","12",{"label":35,"value":36},"Labs","3 order-ops, sort, match, timed sprint",{"label":38,"value":39},"Big idea","Good notation leaves no doubt",[41,45,51,57,62,86,91,107,176,198,203,208,211,259,262,279,303,316,321,324,329,348,368,373,376,387,397,406,418,430,435,440,443,455,489,494,497,506,554,559,562,584,589,594,615,619,623,628,631,641,656,668,680,684,689,692,739,749,761,774,779,784,787,824,836,840,852,857,860,884,894,903,915,935,940,983,1101,1114,1120,1125,1129,1133],{"id":42,"type":43,"markdown":44},"intro","prose","You know the rule, why it works and where it came from. This layer takes it out into the wild: **viral puzzles** that start arguments online, **number games** that people have played for over a century, **olympiad problems**, how the rule lives inside **spreadsheets and programming languages**, projects you can run yourself, and questions that are still genuinely open.\n\nSome of these puzzles have no single right answer. That is the point: part of being good at maths is recognising when a question is badly posed, and saying so clearly.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"how-to","callout","observation","How to use this layer","Pick what interests you. The chapters are independent. Keep a puzzle notebook: write down every attempt, including failures. Many of the best puzzle solvers work backwards from the target, and many of the best puzzle *setters* start from an answer and hide it.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch1","chapter","The puzzle that broke the internet","Chapter 01","1 8 ÷ 2(2 + 2)",{"id":58,"type":43,"markdown":59,"help":60},"viral","Every few months a puzzle like this goes viral:\n\n**8 ÷ 2(2 + 2) = ?**\n\nHalf the comments say **16**. The other half say **1**. Both sides are sure, and both sides think the other side failed school maths. Who is right?\n\n**The \"16\" reading.** The bracket is 4. The 2(4) just means 2 × 4, an ordinary multiplication. So the expression is 8 ÷ 2 × 4, and ÷ and × go left to right: 8 ÷ 2 × (2 + 2) = 8 ÷ 2 × 4 = 4 × 4 = 16.\n\n**The \"1\" reading.** Writing 2(2 + 2) with no × sign, called **implied multiplication** or **multiplication by juxtaposition** (putting things side by side), makes 2(2 + 2) a single tight unit, like a term in algebra. So the expression is 8 ÷ [2 × (2 + 2)] = 8 ÷ 8 = **1**.\n\nThe honest answer: **the expression is badly written.** The basic school rule, applied strictly, gives 16. But many mathematicians, scientists and some calculators treat implied multiplication as binding more tightly, which gives 1. When a well-meaning expert could reasonably read it two ways, the fault is in the writing, not the reader.",{"simplerExplanation":61},"The rule you learned gives 16. But writing 2(2 + 2) without a × sign makes some people treat it as one unit, giving 1. The puzzle is badly written; good maths writing would add brackets to show which is meant.",{"id":63,"type":64,"caption":65,"columns":66,"rows":71},"tbl-viral","table","Two readings of the viral puzzles",[67,68,69,70],"Puzzle","Strict left to right","Implied × first","Clear way to write it",[72,77,81],[73,74,75,76],"8 ÷ 2(2 + 2)","16","1","(8 ÷ 2) × (2 + 2), or 8 ÷ [2 × (2 + 2)]",[78,79,75,80],"6 ÷ 2(1 + 2)","9","(6 ÷ 2) × (1 + 2), or 6 ÷ [2 × (1 + 2)]",[82,83,84,85],"48 ÷ 2(9 + 3)","288","2","use a fraction bar",{"id":87,"type":47,"variant":88,"title":89,"markdown":90},"nuance-implied","nuance","Implied multiplication is a real convention","In algebra, people really do treat side-by-side multiplication as tighter than ÷. A physicist who writes 1\u002F2x almost always means 1\u002F(2x), not (1\u002F2)x.\n\nSome calculators are built that way on purpose, and it is written down in their manuals. Casio's user's guide for the fx-82ES PLUS family lists \"multiplication where the multiplication sign is omitted\" **above** \"multiplication (×), division (÷)\" in its calculation-priority list — so those machines read 6 ÷ 2(1 + 2) as 6 ÷ 6 = 1. Even one maker can change its mind between models: a Texas Instruments TI-82 reads 1\u002F2x as 1\u002F(2x), while the TI-83 reads the same keystrokes as (1\u002F2)x.\n\nSo there are two reasonable conventions, and the viral puzzles sit exactly on the fault line between them. Professional mathematicians avoid the problem completely: they almost never use ÷ and write divisions as stacked fractions, where the bar shows exactly what is divided by what.",{"id":92,"type":93,"prompt":94,"options":95,"explanation":106},"predict-viral","prediction","A friend sends you **9 − 3 ÷ ⅓ + 1**, another viral puzzle. Using the standard rule, what is it?",[96,98,101,103],{"id":97,"label":75},"a",{"id":99,"label":100},"b","3",{"id":102,"label":79},"c",{"id":104,"label":105},"d","−2⅔","**1.** ÷ first: 3 ÷ ⅓ = 9 (how many thirds are in 3? Nine). Then left to right: 9 − 9 + 1 = 1. The common wrong answer is 3, from doing 9 − 3 = 6 first. This one is *not* ambiguous: it has no implied multiplication, just a trap.",{"id":108,"type":109,"component":110,"componentVersion":5,"config":111,"objective":171,"textAlternative":172,"help":173},"lab-sort-clear","interactive","sort-game",{"prompt":112,"bins":113,"items":123,"seconds":170},"How well written is each expression? Sort the cards.",[114,117,120],{"id":115,"label":116},"clear","Clear to everyone",{"id":118,"label":119},"conv","Clear by the rule; brackets would help",{"id":121,"label":122},"amb","Ambiguous: rewrite it",[124,128,132,135,139,143,147,151,155,158,162,166],{"id":125,"label":126,"bin":115,"why":127},"e1","(8 ÷ 2) × (2 + 2)","Every grouping is shown. Value 16.",{"id":129,"label":130,"bin":115,"why":131},"e2","8 ÷ [2 × (2 + 2)]","The divisor is fully bracketed. Value 1.",{"id":133,"label":73,"bin":121,"why":134},"e3","Implied multiplication: 16 by the strict rule, 1 if 2(2 + 2) is treated as a unit.",{"id":136,"label":137,"bin":121,"why":138},"e4","1\u002F2x","Could be (1\u002F2) × x or 1\u002F(2x). Scientists usually mean 1\u002F(2x).",{"id":140,"label":141,"bin":118,"why":142},"e5","24 ÷ 4 × 3","The rule gives 18 (left to right), but many people wrongly get 2. (24 ÷ 4) × 3 is kinder.",{"id":144,"label":145,"bin":118,"why":146},"e6","10 − 3 + 2","The rule gives 9, but the letter trap gives 5. It is correct as written; brackets are optional.",{"id":148,"label":149,"bin":118,"why":150},"e7","2 + 3 × 4","The rule gives 14. Basic calculators give 20, so 2 + (3 × 4) is safer in a shop.",{"id":152,"label":153,"bin":115,"why":154},"e8","A stacked fraction: 8 on top, 2 × (2 + 2) below","The fraction bar groups the whole top and the whole bottom. Value 1.",{"id":156,"label":78,"bin":121,"why":157},"e9","Same problem as the viral puzzle: 9 or 1.",{"id":159,"label":160,"bin":115,"why":161},"e10","(2 + 3) × 4","No rule needed beyond \"brackets first\". Value 20.",{"id":163,"label":164,"bin":121,"why":165},"e11","−3² (typed into a spreadsheet as =-3^2)","Maths convention gives −9; Excel gives 9, because its precedence table applies the minus sign before the power. Write −(3²) or (−3)².",{"id":167,"label":168,"bin":121,"why":169},"e12","2^3^2 (typed into a program)","Maths convention stacks powers from the top: 2^(3^2) = 512. Excel treats ^ like any equal-rank operator and works left to right: (2^3)^2 = 64.",0,"Classify expressions as clear, clear only by convention, or genuinely ambiguous, and learn how to rewrite the ambiguous ones.","A sorting game with 12 cards and 3 bins.\n\nClear to everyone: (8 ÷ 2) × (2 + 2) = 16; 8 ÷ [2 × (2 + 2)] = 1; a stacked fraction with 8 over 2 × (2 + 2) = 1; (2 + 3) × 4 = 20.\n\nClear by the rule; brackets would help: 24 ÷ 4 × 3 = 18; 10 − 3 + 2 = 9; 2 + 3 × 4 = 14.\n\nAmbiguous: 8 ÷ 2(2 + 2) (16 or 1); 1\u002F2x ((1\u002F2)x or 1\u002F(2x)); 6 ÷ 2(1 + 2) (9 or 1); −3² in a spreadsheet (−9 by maths convention, 9 in Excel); 2^3^2 (512 by maths convention, 64 in Excel).",{"hints":174},[175],"Ask: could two careful, well-trained people read it differently?",{"id":177,"type":178,"itemId":179,"prompt":180,"check":181,"hints":193,"feedback":195},"pr-implied","practice","order-of-operations.extend-implied","A science worksheet writes **1\u002F2x** and means a single fraction. Which reading do scientists usually intend?",{"kind":182,"options":183,"correct":192},"choice",[184,186,188,190],{"id":97,"label":185},"(1\u002F2) × x",{"id":99,"label":187},"1 ÷ (2 × x)",{"id":102,"label":189},"1 ÷ 2 + x",{"id":104,"label":191},"12 × x",[99],[194],"Implied multiplication (2x) is usually treated as one unit.",{"correct":196,"incorrect":197},"Right: 2x is treated as a unit, so 1\u002F2x usually means 1\u002F(2x). Writing it as a stacked fraction removes all doubt.","Most scientists read 2x as a single unit, so 1\u002F2x means 1 ÷ (2x). The strict school rule would give (1\u002F2) × x, which is why the notation is risky.",{"id":199,"type":47,"variant":200,"title":201,"markdown":202},"try-survey","try_it","Mini project: survey your family","Write **8 ÷ 2(2 + 2)** on a card. Ask 10 people (family, neighbours, teachers) for the answer, and note their age group and whether they used a calculator.\n\n1. Make a tally and a bar chart of the answers (16, 1, other).\n2. Ask each person to explain their method in one sentence.\n3. Then show them the table above. Does anyone change their mind?\n\nThis is a real data-handling project: you are collecting, organising and interpreting data about how people read maths.",{"id":204,"type":53,"title":205,"eyebrow":206,"navLabel":207},"ch2","The four fours puzzle","Chapter 02","2 Four fours",{"id":209,"type":43,"markdown":210},"fourfours","A puzzle that has been played for well over a hundred years: **make every whole number you can using exactly four 4s**, with + − × ÷ and brackets. You may also join 4s into **44** or **444**.\n\nHere are 0 to 10:",{"id":212,"type":64,"caption":213,"columns":214,"rows":218},"tbl-ff","Four fours: 0 to 10 (one solution each; there are many more)",[215,216,217],"Number","Four fours","Check",[219,223,226,229,232,236,240,244,248,252,255],[220,221,222],"0","44 − 44","44 − 44 = 0",[75,224,225],"44 ÷ 44","44 ÷ 44 = 1",[84,227,228],"4 ÷ 4 + 4 ÷ 4","4 ÷ 4 + 4 ÷ 4 = 1 + 4 ÷ 4 = 1 + 1 = 2",[100,230,231],"(4 + 4 + 4) ÷ 4","(4 + 4 + 4) ÷ 4 = (8 + 4) ÷ 4 = 12 ÷ 4 = 3",[233,234,235],"4","4 × (4 − 4) + 4","4 × (4 − 4) + 4 = 4 × 0 + 4 = 0 + 4 = 4",[237,238,239],"5","(4 × 4 + 4) ÷ 4","(4 × 4 + 4) ÷ 4 = (16 + 4) ÷ 4 = 20 ÷ 4 = 5",[241,242,243],"6","(4 + 4) ÷ 4 + 4","(4 + 4) ÷ 4 + 4 = 8 ÷ 4 + 4 = 2 + 4 = 6",[245,246,247],"7","44 ÷ 4 − 4","44 ÷ 4 − 4 = 11 − 4 = 7",[249,250,251],"8","4 + 4 + 4 − 4","4 + 4 + 4 − 4 = 8 + 4 − 4 = 12 − 4 = 8",[79,253,254],"4 + 4 + 4 ÷ 4","4 + 4 + 4 ÷ 4 = 4 + 4 + 1 = 8 + 1 = 9",[256,257,258],"10","(44 − 4) ÷ 4","(44 − 4) ÷ 4 = 40 ÷ 4 = 10",{"id":260,"type":43,"markdown":261},"ff-limits","Now try 11. And 13. You will struggle, and not because you are missing a trick.\n\nFix the rules exactly: exactly four 4s, joined into 4, 44, 444 or 4444 if you wish; only + − × ÷ and brackets; no decimal point, no √, no factorial, and no minus sign in front of a number on its own. Under **those** rules a computer can check every possible arrangement — every way of splitting the 4s, every order, every bracketing — and the numbers from 0 to 20 that **cannot** be made are exactly **11, 13, 14, 18, 19**.\n\nChange the rules and the list changes: if you are not allowed to join 4s at all, 10 becomes impossible too. That is the real lesson — an \"impossible\" claim in a puzzle only means anything once the allowed moves are written down.\n\nTo reach the missing numbers, puzzle fans allow extra tools:\n\n- **√4 = 2** (square root): 13 = 44 ÷ 4 + √4 = 11 + 2.\n- **.4 = 0.4** (a decimal): 11 = 4 ÷ .4 + 4 ÷ 4 = 10 + 1; 18 = 44 × .4 + .4 = 17.6 + 0.4.\n- **4! = 4 × 3 × 2 × 1 = 24** (factorial): 19 = 4! − 4 − 4 ÷ 4 = 24 − 4 − 1.\n- 14 = 4 × (√4 + √4) − √4 = 16 − 2.\n\nEvery one of these still relies on the order of operations to mean exactly one thing.",{"id":263,"type":109,"component":264,"componentVersion":5,"config":265,"objective":273,"textAlternative":274,"help":275},"lab-orderops-ff","order-ops",{"expressions":266,"showRuleCard":272},[267,224,227,230,268,238,242,269,270,253,271],"44 - 44","4 × (4 - 4) + 4","44 ÷ 4 - 4","4 + 4 + 4 - 4","(44 - 4) ÷ 4",true,"Check the four fours solutions for 0 to 10 step by step, and see how brackets and the order of operations shape each one.","Eleven four-fours expressions, one for each number from 0 to 10 (44 counts as two 4s joined):\n\n- 44 − 44 = 0\n- 44 ÷ 44 = 1\n- 4 ÷ 4 + 4 ÷ 4 = 2\n- (4 + 4 + 4) ÷ 4 = 3\n- 4 × (4 − 4) + 4 = 4\n- (4 × 4 + 4) ÷ 4 = 5\n- (4 + 4) ÷ 4 + 4 = 6\n- 44 ÷ 4 − 4 = 7\n- 4 + 4 + 4 − 4 = 8\n- 4 + 4 + 4 ÷ 4 = 9\n- (44 − 4) ÷ 4 = 10\n\nFor example (4 × 4 + 4) ÷ 4 = (16 + 4) ÷ 4 = 20 ÷ 4 = 5 and 44 ÷ 4 − 4 = 11 − 4 = 7. Try to find a second solution for each number.",{"hints":276},[277,278],"4 ÷ 4 = 1 is a useful building block.","44 ÷ 4 = 11 is another.",{"id":280,"type":109,"component":281,"componentVersion":5,"config":282,"objective":301,"textAlternative":302},"lab-match-ff","match-pairs",{"prompt":283,"mode":284,"pairs":285},"Four fours memory game: match each expression to the number it makes.","memory",[286,287,288,289,290,291,292,293,295,298],{"a":224,"b":75},{"a":227,"b":84},{"a":230,"b":100},{"a":238,"b":237},{"a":246,"b":245},{"a":253,"b":79},{"a":257,"b":256},{"a":294,"b":33},"(44 + 4) ÷ 4",{"a":296,"b":297},"4 × 4 + 4 ÷ 4","17",{"a":299,"b":300},"(4 + 4 ÷ 4) × 4","20","Match four-fours expressions to their values, working each out mentally with the order of operations.","A memory game with 20 cards: ten four-fours expressions and their values.\n\n- 44 ÷ 44 = 1\n- 4 ÷ 4 + 4 ÷ 4 = 2\n- (4 + 4 + 4) ÷ 4 = 3\n- (4 × 4 + 4) ÷ 4 = 5\n- 44 ÷ 4 − 4 = 7\n- 4 + 4 + 4 ÷ 4 = 9\n- (44 − 4) ÷ 4 = 10\n- (44 + 4) ÷ 4 = 12\n- 4 × 4 + 4 ÷ 4 = 17\n- (4 + 4 ÷ 4) × 4 = 20\n\nWatch 4 × 4 + 4 ÷ 4: both the × and the ÷ happen before the +, giving 16 + 1 = 17.",{"id":304,"type":178,"itemId":305,"prompt":306,"check":307,"hints":310,"feedback":313},"pr-ff","order-of-operations.extend-four-fours","Using exactly four 4s, which number does **4 × 4 + 4 + 4** make? (Then try to make the same number with a different four-fours expression.)",{"kind":308,"answer":309,"tolerance":170},"number",24,[311,312],"× first: 4 × 4 = 16.","16 + 4 + 4.",{"correct":314,"incorrect":315},"Yes: 16 + 4 + 4 = 24. If you allow a square root, (4 + 4 + 4) × √4 = 12 × 2 = 24 is another way.","4 × 4 = 16 first, then 16 + 4 + 4 = 24.",{"id":317,"type":53,"title":318,"eyebrow":319,"navLabel":320},"ch3","Make 24","Chapter 03","3 Make 24",{"id":322,"type":43,"markdown":323},"m24","The **24 game** is played with four numbers (often from playing cards, 1 to 13). Use each number exactly once, with + − × ÷ and brackets, to make exactly **24**. Why 24? Because it has so many factors (1, 2, 3, 4, 6, 8, 12, 24) that most sets of four numbers can reach it.\n\nThe key strategy is **working backwards**: 24 = 3 × 8 = 4 × 6 = 2 × 12 = 1 × 24, or 20 + 4, or 30 − 6. Look at your numbers and ask which of these you can build.\n\n- **4, 7, 7, 7**: 7 ÷ 7 = 1, 7 − 1 = 6, 4 × 6 = 24. So 4 × (7 − 7 ÷ 7) = 4 × (7 − 1) = 4 × 6 = 24.\n- **1, 2, 3, 4**: 1 × 2 × 3 × 4 = 2 × 3 × 4 = 6 × 4 = 24.\n- **2, 3, 4, 9**: a tempting try is (9 − 3 − 2) × 4 = (6 − 2) × 4 = 4 × 4 = 16, which misses. Work backwards instead: 24 = 48 ÷ 2, and 48 = 12 × 4 = (9 + 3) × 4. So (9 + 3) × 4 ÷ 2 = 12 × 4 ÷ 2 = 48 ÷ 2 = 24.",{"id":325,"type":47,"variant":326,"title":327,"markdown":328},"aha-fractions","aha","Some sets need fractions on the way","A few sets are famous for being hard because the only solutions go through a fraction:\n\n- **3, 3, 8, 8**: 8 ÷ (3 − 8 ÷ 3) = 8 ÷ (3 − 2⅔) = 8 ÷ ⅓ = **24**.\n- **1, 5, 5, 5**: 5 × (5 − 1 ÷ 5) = 5 × 4⅘ = **24**.\n- **1, 3, 4, 6**: 6 ÷ (1 − 3 ÷ 4) = 6 ÷ ¼ = **24**.\n\nAnd some sets are impossible: **1, 1, 1, 1** can make at most (1 + 1) × (1 + 1) = 4. A computer can check every possibility to prove a set is impossible.",{"id":330,"type":109,"component":264,"componentVersion":5,"config":331,"objective":343,"textAlternative":344,"help":345},"lab-orderops-24",{"expressions":332,"showRuleCard":272},[333,334,335,336,337,338,339,340,341,342],"4 × (7 - 7 ÷ 7)","(10 - 4) × (3 + 1)","(12 - 8) × (9 - 3)","(9 - 5 + 2) × 4","8 × (1 + 1 + 1)","(7 - 3) × (2 + 4)","(13 - 1) × (5 - 3)","[11 - (6 - 1)] × 4","(9 + 3) × 4 ÷ 2","6 × 5 - 12 ÷ 2","Step through ten solutions of the 24 game and confirm that each really makes 24.","Ten 24-game solutions (only whole numbers along the way):\n\n- 4 × (7 − 7 ÷ 7) = 24\n- (10 − 4) × (3 + 1) = 24\n- (12 − 8) × (9 − 3) = 24\n- (9 − 5 + 2) × 4 = 24\n- 8 × (1 + 1 + 1) = 24\n- (7 − 3) × (2 + 4) = 24\n- (13 − 1) × (5 − 3) = 24\n- [11 − (6 − 1)] × 4 = 24\n- (9 + 3) × 4 ÷ 2 = 24\n- 6 × 5 − 12 ÷ 2 = 24\n\nFor example 6 × 5 − 12 ÷ 2 = 30 − 12 ÷ 2 = 30 − 6 = 24. Notice how often the last step is 4 × 6, 3 × 8, 2 × 12 or 30 − 6.",{"hints":346},[347],"Look at the last operation in each: it is usually one of the factor pairs of 24.",{"id":349,"type":178,"itemId":350,"prompt":351,"check":352,"hints":363,"feedback":365},"pr-24","order-of-operations.extend-make-24","Which expression makes **24** from **2, 2, 6, 8**, using each number once?",{"kind":182,"options":353,"correct":362},[354,356,358,360],{"id":97,"label":355},"(8 − 6) × 2 × 2",{"id":99,"label":357},"8 × 6 ÷ (2 × 2)",{"id":102,"label":359},"(8 + 6 − 2) × 2",{"id":104,"label":361},"8 + 6 + 2 × 2",[102],[364],"Work out each one using the rule.",{"correct":366,"incorrect":367},"Yes: (8 + 6 − 2) × 2 = 12 × 2 = 24. The others: (a) = 8, (b) = 12, (d) = 18.","Check each: (8 − 6) × 2 × 2 = 8, 8 × 6 ÷ (2 × 2) = 12, (8 + 6 − 2) × 2 = 24, 8 + 6 + 2 × 2 = 18.",{"id":369,"type":53,"title":370,"eyebrow":371,"navLabel":372},"ch4","Olympiad-style problems","Chapter 04","4 Olympiad problems",{"id":374,"type":43,"markdown":375},"olymp","Olympiad problems on this topic rarely ask you to grind through a long expression. They ask you to **see structure**: pair terms, spot a zero, count cleverly, or search systematically.",{"id":377,"type":378,"title":379,"problem":380,"steps":381},"we-olymp1","worked_example","A long alternating sum","Evaluate **2026 − 2025 + 2024 − 2023 + … + 4 − 3 + 2 − 1**.",[382,383,384,385,386],"Going left to right term by term would take 2,025 steps. Look for structure instead.","Group in pairs from the left: (2026 − 2025) + (2024 − 2023) + … + (2 − 1). Grouping this way is allowed because each − stays attached to its own number.","Every pair equals 1.","How many pairs? The numbers 1 to 2026 make 2026 ÷ 2 = 1,013 pairs.","So the sum is 1 × 1,013 = **1,013**.",{"id":388,"type":378,"title":389,"problem":390,"steps":391},"we-olymp2","Best place for brackets","Insert brackets into **1 + 2 × 3 + 4 × 5 + 6** (keeping the order) to make the value as **large** as possible.",[392,393,394,395,396],"Without brackets: 1 + 2 × 3 + 4 × 5 + 6 = 1 + 6 + 4 × 5 + 6 = 1 + 6 + 20 + 6 = 7 + 20 + 6 = 27 + 6 = 33.","Idea: + makes small numbers bigger before they are multiplied. So bracket each addition that sits next to a ×: (1 + 2), (3 + 4), (5 + 6).","Then multiply: (1 + 2) × (3 + 4) × (5 + 6) = 3 × (3 + 4) × (5 + 6) = 3 × 7 × (5 + 6) = 3 × 7 × 11 = 21 × 11 = 231.","Is that the best? There are 42 ways to bracket six numbers. A computer check finds 22 different values, the biggest being **231** and the smallest 33 (which is the no-bracket value!).","So the greedy idea, \"add small numbers before multiplying\", wins here.",{"id":398,"type":378,"title":399,"problem":400,"steps":401},"we-olymp3","The digits to 100","Put + or − signs (and join digits if you like) between **1 2 3 4 5 6 7 8 9**, in order, to make **100**.",[402,403,404,405],"This is a classic from old recreational puzzle books. Joining digits lets you make big numbers like 78 or 123.","One solution: 1 + 2 + 3 − 4 + 5 + 6 + 78 + 9 = 100.","A solution with only three signs: 123 − 45 − 67 + 89 = 100.","With only + and − there is no order-of-operations question at all (left to right always works). Now try allowing × too: the order of operations suddenly matters, and there are many more solutions, such as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 × 9 = 100.",{"id":407,"type":178,"itemId":408,"prompt":409,"check":410,"hints":412,"feedback":415},"pr-olymp","order-of-operations.extend-olympiad","Using **1, 2, 3, 4, 5 in order**, put + or × between them (no brackets). Which filling gives **25**? Enter the value of **1 × 2 + 3 + 4 × 5** to check it.",{"kind":308,"answer":411,"tolerance":170},25,[413,414],"× first: 1 × 2 = 2 and 4 × 5 = 20.","2 + 3 + 20.",{"correct":416,"incorrect":417},"Yes: 2 + 3 + 20 = 25. It is the only filling of the 16 that gives 25.","1 × 2 = 2 and 4 × 5 = 20 first; then 2 + 3 + 20 = 25.",{"id":419,"type":178,"itemId":420,"prompt":421,"check":422,"hints":424,"feedback":427},"pr-missing","order-of-operations.extend-missing-number","Find the missing number: **60 − ▢ × 7 = 11**.",{"kind":308,"answer":423,"tolerance":170},7,[425,426],"The − is done last, so undo it first: ▢ × 7 = 60 − 11.","▢ × 7 = 49.",{"correct":428,"incorrect":429},"Yes: ▢ × 7 = 60 − 11 = 49, so ▢ = 7. Check: 60 − 49 = 11.","Undo in reverse order: ▢ × 7 = 60 − 11 = 49, so ▢ = 49 ÷ 7 = 7.",{"id":431,"type":47,"variant":432,"title":433,"markdown":434},"example-missing-number","example","Working backwards through the order","**Find the missing number:** ▢ × 4 + 3 × 5 = 47.\n\nUndo the operations in the *reverse* of the order they are done. The + is done last, so undo it first: ▢ × 4 = 47 − 3 × 5 = 47 − 15 = 32. Then undo the ×: ▢ = 32 ÷ 4 = **8**. Check: 8 × 4 + 3 × 5 = 32 + 3 × 5 = 32 + 15 = 47.\n\nThis \"undo in reverse order\" idea is exactly how you will solve equations in algebra.",{"id":436,"type":53,"title":437,"eyebrow":438,"navLabel":439},"ch5","Inside spreadsheets and code","Chapter 05","5 Code and sheets",{"id":441,"type":43,"markdown":442},"code","Every programming language has an **operator precedence table**, its own written-down order of operations. For ordinary arithmetic they almost all agree with school maths: in Python, Java, C, JavaScript, Scratch and spreadsheets, `2 + 3 * 4` is 14.\n\nBut at the edges they can differ, and professional programmers learn to add brackets whenever there is the slightest doubt.\n\n- **Symbols.** Computers use * for × and \u002F for ÷. Powers are ** in Python and ^ in spreadsheets.\n- **Whole-number division.** In Python, 7 \u002F 2 is 3.5, but 7 \u002F\u002F 2 is 3 (the quotient) and 7 % 2 is 1 (the remainder). \u002F\u002F and % share a level with * and \u002F.\n- **Negative squares.** In maths, −3² means −(3²) = −9. Python agrees: -3**2 is −9. But Microsoft Excel gives **9** for =-3^2: its published precedence table ranks negation *above* exponentiation, so the minus is applied first. Spreadsheets that follow Excel's rules behave the same way.\n- **Stacked powers.** In maths, 2^3^2 means 2^(3^2) = 2⁹ = 512, and Python agrees. Excel treats ^ like any other equal-rank operator and works left to right, so =2^3^2 is (2³)² = **64**.",{"id":444,"type":93,"prompt":445,"options":446,"explanation":454},"predict-python","In Python, what does **-3**2** give? (In Python, ** means 'to the power of'.)",[447,448,450,452],{"id":97,"label":79},{"id":99,"label":449},"−9",{"id":102,"label":451},"−6",{"id":104,"label":453},"an error","**−9.** Python follows the maths convention: the power is done before the minus sign, so −3² = −(3 × 3) = −9. Excel gives 9 for =-3^2, because its precedence table applies the minus sign first. Same symbols, different rules: add brackets and write (-3)**2 if you mean 9.",{"id":456,"type":64,"caption":457,"columns":458,"rows":463},"tbl-code","The same idea in different systems",[459,460,461,462],"You mean","School maths","Python","Spreadsheet (Excel)",[464,468,471,475,479,484],[149,465,466,467],"14","2 + 3 * 4 → 14","=2+3*4 → 14",[160,300,469,470],"(2 + 3) * 4 → 20","=(2+3)*4 → 20",[472,233,473,474],"8 ÷ 4 × 2","8 \u002F 4 * 2 → 4.0","=8\u002F4*2 → 4",[476,449,477,478],"−3²","-3**2 → -9","=-3^2 → 9 (!)",[480,481,482,483],"2 to the power 3², i.e. 2⁹","512","2**3**2 → 512","=2^3^2 → 64 (!)",[485,486,487,488],"quotient of 7 ÷ 2","3 (remainder 1)","7 \u002F\u002F 2 → 3","=QUOTIENT(7,2) → 3",{"id":490,"type":47,"variant":491,"title":492,"markdown":493},"careful-code","careful","The golden rule of professional coders","When in doubt, **add brackets**. They cost nothing, they make the meaning obvious to the next reader, and they protect you when code is moved between languages or spreadsheets with slightly different rules. Many software teams require brackets around mixed operations even when the precedence rules would give the right answer anyway.",{"id":495,"type":43,"markdown":496},"careers","Who uses this every day?\n\n- **Accountants and shopkeepers** build spreadsheets of bills, taxes and discounts. A missing bracket in a GST formula such as price × (1 + rate ÷ 100) can mis-price every item in a shop.\n- **Software engineers** write expressions in code millions of times; the precedence table of their language is second nature.\n- **Engineers and scientists** plug numbers into formulas like distance = speed × time + ½ × a × t². The order of operations decides what the formula means.\n- **Data analysts** compute averages: (sum of values) ÷ (count). Forgetting the brackets and dividing only the last value is one of the most common spreadsheet errors.\n- **Teachers and textbook writers** have to write expressions that cannot be misread, which is harder than it looks!",{"id":498,"type":378,"title":499,"problem":500,"steps":501},"we-gst","A GST bill in one formula","An item costs ₹500 before tax. GST is 18%. Write one formula for the price with tax and evaluate it. What goes wrong without the brackets?",[502,503,504,505],"Price with tax = price × (100 + rate) ÷ 100.","= 500 × (100 + 18) ÷ 100 = 500 × 118 ÷ 100 = 59,000 ÷ 100 = **₹590**.","Without brackets: 500 × 100 + 18 ÷ 100 = 50,000 + 0.18 = 50,000.18. Nobody would pay ₹50,000 for a ₹500 item!","Estimating (\"about ₹600\") catches the error immediately.",{"id":507,"type":109,"component":508,"componentVersion":5,"config":509,"objective":549,"textAlternative":550,"help":551},"lab-sprint-challenge","arith-sprint",{"operations":510,"ranges":515,"rounds":522,"secondsTotal":523,"estimateFirst":524,"wordProblems":525},[511,512,513,514],"+","-","×","÷",{"a":516,"b":519},{"min":517,"max":518},10,99,{"min":520,"max":521},2,12,16,180,false,[526,530,533,536,539,542,544,546],{"prompt":527,"answer":528,"unit":529,"operation":514},"An item costs ₹500; add 18% GST: 500 × (100 + 18) ÷ 100. Price with tax?",590,"₹",{"prompt":531,"answer":532,"operation":512},"2026 − 2025 + 2024 − 2023 + … + 2 − 1 = ?",1013,{"prompt":534,"answer":535,"operation":513},"(1 + 2) × (3 + 4) × (5 + 6) = ?",231,{"prompt":537,"answer":538,"operation":514},"Find ▢ if ▢ × 4 + 3 × 5 = 47.",8,{"prompt":540,"answer":541,"operation":513},"37 × 64 + 37 × 36 = ? (Use the distributive property.)",3700,{"prompt":543,"answer":517,"operation":514},"Four fours: (44 − 4) ÷ 4 = ?",{"prompt":545,"answer":522,"operation":513},"8 ÷ 2 × (2 + 2) by the strict left-to-right rule = ?",{"prompt":547,"answer":548,"operation":514},"Mean of 12, 15 and 18: (12 + 15 + 18) ÷ 3 = ?",15,"A timed challenge mixing fast two-digit arithmetic with olympiad-style expression problems from this layer.","A 16-round challenge with a 3-minute timer (switch the timer off in settings if you prefer). Generated questions use a two-digit number and a number from 2 to 12; every division is exact. Mixed in at random are problems from this layer:\n\n- 500 × (100 + 18) ÷ 100 = ₹590 (GST)\n- 2026 − 2025 + … + 2 − 1 = 1,013\n- (1 + 2) × (3 + 4) × (5 + 6) = 231\n- ▢ × 4 + 3 × 5 = 47 gives ▢ = 8\n- 37 × 64 + 37 × 36 = 3,700\n- (44 − 4) ÷ 4 = 10\n- 8 ÷ 2 × (2 + 2) = 16\n- (12 + 15 + 18) ÷ 3 = 15",{"hints":552},[553],"Look for structure before calculating: pairs, common factors, zeros.",{"id":555,"type":53,"title":556,"eyebrow":557,"navLabel":558},"ch6","Other ways to write the same thing","Chapter 06","6 Other notations",{"id":560,"type":43,"markdown":561},"notations","Our way of writing, with the operation *between* its numbers (**infix**), is the reason we need an order of operations at all. Other notations avoid the problem:\n\n- **Prefix (Polish) notation**: the operation comes first. 2 + 3 × 4 is + 2 × 3 4. The programming language Lisp writes it as (+ 2 (* 3 4)).\n- **Postfix (Reverse Polish)**: the operation comes last. 2 3 4 × +. No brackets are ever needed.\n- **Fraction bars and stacked layouts**: the position on the page shows the grouping.\n- **Expression trees**: a picture of the structure, with the last operation at the top.\n\nWhy do we still use infix? Because it is compact and close to how we speak (\"two plus three times four\"), and once everyone knows the convention it is fast to read. It is a trade-off: a convenient notation that needs a shared rule.",{"id":563,"type":64,"caption":564,"columns":565,"rows":570},"tbl-notations","One expression, four notations",[566,567,568,569],"Infix (school)","Prefix (Polish)","Postfix (RPN)","Value",[571,574,577,580],[149,572,573,465],"+ 2 × 3 4","2 3 4 × +",[160,575,576,300],"× + 2 3 4","2 3 + 4 ×",[145,578,579,79],"+ − 10 3 2","10 3 − 2 +",[581,582,583,300],"(7 − 2) × (1 + 3)","× − 7 2 + 1 3","7 2 − 1 3 + ×",{"id":585,"type":47,"variant":586,"title":587,"markdown":588},"question-spoken","question","How do you say it out loud?","Spoken maths is ambiguous too. \"Two plus three times four\" could be heard either way. Radio quiz hosts and teachers often add pauses: \"two… plus three-times-four\". In some languages, the natural word order puts the operation last, like postfix. How would you say (2 + 3) × 4 in English, Hindi or your home language so that nobody could misunderstand?",{"id":590,"type":53,"title":591,"eyebrow":592,"navLabel":593},"ch7","Projects to try","Chapter 07","7 Projects",{"id":595,"type":596,"title":597,"items":598},"steps-projects","steps","Four projects, from a weekend to a month",[599,603,607,611],{"title":600,"tag":601,"text":602},"Puzzle book","weekend","Write 10 \"insert the brackets\" puzzles with answers. Start from an answer and hide it. Swap books with a friend.",{"title":604,"tag":605,"text":606},"Four fours chart","1–2 weeks","Find four-fours expressions for 0 to 50. Record which need √, decimals or factorials. Which is the first number you cannot make?",{"title":608,"tag":609,"text":610},"Calculator audit","a week","Test every calculator, app and spreadsheet you can find with 2 + 3 × 4, 8 ÷ 2(2 + 2), −3² and 2^3^2. Publish a table for your class.",{"title":612,"tag":613,"text":614},"Kirana bill checker","a month","Design a spreadsheet for a small shop: quantity × price per line, a subtotal, GST and change from a note. Test it with a shopkeeper.",{"id":616,"type":617,"prompt":618},"reflect-rule","reflection","Imagine you could redesign mathematical notation from scratch for children who have never seen any maths. Would you keep the order of operations as it is, change it, or replace infix notation completely? Give one advantage and one disadvantage of your choice.",{"id":620,"type":47,"variant":586,"title":621,"markdown":622},"open-questions","Open questions for curious learners","Some of these have partial answers; none has a single agreed one.\n\n- Which is the smallest whole number that cannot be made with four 4s if √, decimals and factorials are all allowed? (It depends on exactly which tools you allow, and people still argue about the rules.)\n- Should school textbooks stop using the ÷ sign altogether, as some mathematicians suggest?\n- Should implied multiplication, as in 2(2 + 2), officially bind more tightly than ÷? Who would decide?\n- Is there a notation that is as short as infix but never needs a precedence rule?\n- What fraction of four-number sets from 1 to 13 can make 24? (A computer can answer this; can you design the search?)",{"id":624,"type":53,"title":625,"eyebrow":626,"navLabel":627},"ch-countdown","The numbers game: big targets","Chapter 08","8 Numbers game",{"id":629,"type":43,"markdown":630},"countdown","A famous television numbers game (called *Countdown* in the UK, and copied in many countries) gives players **six numbers** and a **three-digit target**. Using + − × ÷ and brackets, each number at most once, and every step a whole number, they have 30 seconds to hit the target, or get as close as possible.\n\nSmall numbers are chosen from 1 to 10 and big numbers from 25, 50, 75 and 100. A typical round: **75, 4, 7, 2, 9, 1**, target **307**.\n\nWork backwards: 307 is close to 300 = 4 × 75. The difference is 7, and we have a 7! So 4 × 75 + 7 = 300 + 7 = 307. Three numbers are left over, which is allowed.\n\nThe order of operations is what lets you write the answer as one short line: 4 × 75 + 7 needs no brackets, because the × is done first anyway.",{"id":632,"type":378,"title":633,"problem":634,"steps":635},"we-952","A legendary solution: 952","In a famous 1997 episode, a contestant was given **25, 50, 75, 100, 3, 6** and the target **952**. Show that ((100 + 6) × 3 × 75 − 50) ÷ 25 = 952.",[636,637,638,639,640],"Innermost bracket: 100 + 6 = 106.","Then × and ÷ inside the big bracket, left to right: 106 × 3 = 318, and 318 × 75 = 23,850.","Still inside the big bracket: 23,850 − 50 = 23,800.","Finally: 23,800 ÷ 25 = **952**.","The idea behind it: 952 × 25 = 23,800, which is 50 more than 23,850 = 106 × 225, and 225 = 3 × 75. Working backwards from the target with multiplication and division is what made it possible.",{"id":642,"type":109,"component":264,"componentVersion":5,"config":643,"objective":651,"textAlternative":652,"help":653},"lab-orderops-countdown",{"expressions":644,"showRuleCard":272},[645,646,647,648,649,650],"4 × 75 + 7","3 × 6 × (50 - 25)","[(100 + 6) × 3 × 75 - 50] ÷ 25","(75 - 7) × 4 + 2","100 × 9 - 75 ÷ 25","(50 + 25) × (6 + 3) - 100","Step through big-target numbers-game solutions, including the famous 952, and see how brackets carry each plan.","Six numbers-game solutions:\n\n- 4 × 75 + 7 = 307\n- 3 × 6 × (50 − 25) = 450\n- [(100 + 6) × 3 × 75 − 50] ÷ 25 = 952\n- (75 − 7) × 4 + 2 = 274\n- 100 × 9 − 75 ÷ 25 = 897\n- (50 + 25) × (6 + 3) − 100 = 575\n\nIn full: 3 × 6 × (50 − 25) = 3 × 6 × 25 = 18 × 25 = 450. The 952 solution passes through 23,850 and 23,800 before the final division.",{"hints":654},[655],"Start from the innermost bracket; inside the big bracket, the leftmost × comes first.",{"id":657,"type":178,"itemId":658,"prompt":659,"check":660,"hints":662,"feedback":665},"pr-countdown","order-of-operations.extend-numbers-game","Numbers **50, 25, 6, 3**. A player writes **3 × 6 × (50 − 25)**. What target did she reach?",{"kind":308,"answer":661,"tolerance":170},450,[663,664],"Bracket first: 50 − 25 = 25.","3 × 6 × 25.",{"correct":666,"incorrect":667},"Yes: 3 × 6 = 18 and 18 × 25 = 450.","Bracket: 25. Then left to right: 3 × 6 = 18, 18 × 25 = 450.",{"id":669,"type":178,"itemId":670,"prompt":671,"check":672,"hints":674,"feedback":677},"pr-countdown2","order-of-operations.extend-numbers-game-2","Numbers **100, 9, 75, 25**. Evaluate **100 × 9 − 75 ÷ 25** to see which target it reaches.",{"kind":308,"answer":673,"tolerance":170},897,[675,676],"× and ÷ first: 100 × 9 and 75 ÷ 25.","900 − 3.",{"correct":678,"incorrect":679},"Yes: 900 − 3 = 897.","× and ÷ first: 100 × 9 = 900 and 75 ÷ 25 = 3. Then 900 − 3 = 897. (Left to right would give (900 − 75) ÷ 25 = 33, a very different number.)",{"id":681,"type":47,"variant":200,"title":682,"markdown":683},"tryit-countdown","Play it at home","Write 1–10 (twice) and 25, 50, 75, 100 on slips of paper. One player picks six slips; another rolls three dice to make a target (for example 4, 2, 6 → 426). Everyone has two minutes.\n\nScore 10 points for hitting the target exactly, 7 for being within 5, and 5 for being within 10, but **only if your written line of working is correct under the order of operations**. Swap papers and check each other's lines.",{"id":685,"type":53,"title":686,"eyebrow":687,"navLabel":688},"ch-discount","Discounts, GST and the order of a bill","Chapter 09","9 Discounts and GST",{"id":690,"type":43,"markdown":691},"discount","A shop has a **₹1,000** shirt on sale. There are two things to apply: a **₹100 discount** and **18% GST**. Does the order matter?\n\n- **Discount first, then GST:** (1,000 − 100) × 118 ÷ 100 = 900 × 1.18 = **₹1,062**.\n- **GST first, then discount:** 1,000 × 118 ÷ 100 − 100 = 1,180 − 100 = **₹1,080**.\n\nThey differ by ₹18, which is exactly the 18% GST on the ₹100 discount. The brackets decide whether the tax is charged on the discounted price or on the full price. In India the first way is the usual one: the GST Council’s own guidance says that \"pre-supply discounts i.e. discounts recorded in the invoice\" are left out when working out the taxable value, so a discount printed on the bill is taken off **before** GST is added.\n\nNow try a **10% discount** instead of ₹100. Discount first: 1,000 × 90 ÷ 100 × 118 ÷ 100 = ₹1,062. GST first: 1,000 × 118 ÷ 100 × 90 ÷ 100 = ₹1,062. **The same!** Percentage changes are multiplications, and multiplications can be done in any order. A flat discount is a subtraction, and mixing it with a multiplication makes the order matter.",{"id":693,"type":694,"title":695,"prompt":696,"options":697},"explorer-discount","explorer","Discount and tax: which order?","Choose a pair of price changes to see if the order matters.",[698,709,720,730],{"id":699,"label":700,"chain":701,"badge":705,"note":708},"flat-then-gst","₹100 off, then 18% GST",[702,703,704],"1,000 − 100 = 900","900 × 118 ÷ 100","₹1,062",{"text":706,"tone":707},"Usual invoice order","yes","Expression: (1,000 − 100) × 118 ÷ 100. The bracket is essential: without it, 1,000 − 100 × 118 ÷ 100 = 1,000 − 118 = 882, which takes the tax off the discount instead.",{"id":710,"label":711,"chain":712,"badge":716,"note":719},"gst-then-flat","18% GST, then ₹100 off",[713,714,715],"1,000 × 118 ÷ 100 = 1,180","1,180 − 100","₹1,080",{"text":717,"tone":718},"₹18 more","no","Expression: 1,000 × 118 ÷ 100 − 100. No brackets needed, because × and ÷ come first anyway. The customer pays tax on the ₹100 that was discounted.",{"id":721,"label":722,"chain":723,"badge":727,"note":729},"pct-pct","10% off and 18% GST",[724,725,726],"× 90 ÷ 100","× 118 ÷ 100","₹1,062 either way",{"text":728,"tone":707},"Order does not matter","1,000 × 90 ÷ 100 × 118 ÷ 100 = 1,062, and swapping the two changes gives the same. Two percentage changes are both multiplications, so they commute.",{"id":731,"label":732,"chain":733,"badge":737,"note":738},"flat-flat","₹100 off, then ₹50 coupon",[734,735,736],"1,000 − 100","900 − 50","₹850 either way",{"text":728,"tone":707},"1,000 − 100 − 50 = 850 = 1,000 − 50 − 100. Two subtractions can swap, as long as each keeps its − sign.",{"id":740,"type":378,"title":741,"problem":742,"steps":743},"we-invoice","A kirana invoice with GST","An invoice lists **3 packets of biscuits at ₹40**, **2 bottles of juice at ₹90**, and a **₹20 discount** on the whole bill. GST of **5%** is charged on the discounted total. Write one expression and evaluate it.",[744,745,746,747,748],"Items: 3 × 40 + 2 × 90 = 120 + 180 = 300.","Discounted total: (3 × 40 + 2 × 90 − 20) = 280. The whole thing must be taxed, so it stays in brackets.","Add 5% GST: multiply by 105 and divide by 100.","Expression: **(3 × 40 + 2 × 90 − 20) × 105 ÷ 100**.","= 280 × 105 ÷ 100 = 29,400 ÷ 100 = **₹294**.",{"id":750,"type":178,"itemId":751,"prompt":752,"check":753,"hints":755,"feedback":758},"pr-discount","order-of-operations.extend-discount-gst","A ₹2,000 phone cover has a **₹200 discount**, then **18% GST** is added to the discounted price. Using **(2000 − 200) × 118 ÷ 100**, what is the final price?",{"kind":308,"answer":754,"tolerance":170,"unit":529},2124,[756,757],"Bracket first: 1,800.","1,800 × 118 = 2,12,400; then ÷ 100.",{"correct":759,"incorrect":760},"Yes: 1,800 × 118 ÷ 100 = ₹2,124.","Bracket: 2,000 − 200 = 1,800. Then 1,800 × 118 = 2,12,400 and ÷ 100 gives ₹2,124.",{"id":762,"type":93,"prompt":763,"options":764,"explanation":773},"predict-coupon","A ₹600 order gets **20% off**, then a **₹50 coupon**. Which expression gives the price, and is it different if the coupon is used first?",[765,767,769,771],{"id":97,"label":766},"600 × 80 ÷ 100 − 50 = ₹430; coupon first gives (600 − 50) × 80 ÷ 100 = ₹440",{"id":99,"label":768},"Both orders give ₹430",{"id":102,"label":770},"600 − 20 − 50 = ₹530",{"id":104,"label":772},"(600 − 50) × 20 ÷ 100 = ₹110","**(a).** Percentage first: 600 × 80 ÷ 100 = 480, then 480 − 50 = ₹430. Coupon first: (600 − 50) × 80 ÷ 100 = 550 × 0.8 = ₹440. Mixing a flat amount with a percentage makes the order matter, and the shop's rules decide which is used. (c) treats 20% as ₹20; (d) works out the discount, not the price.",{"id":775,"type":47,"variant":776,"title":777,"markdown":778},"misc-percent-subtract","misconception","Real mistake: “10% off then 10% on gets you back where you started”","Many adults believe this. Try it on ₹1,000: 10% off gives 1,000 × 90 ÷ 100 = 900; then 10% on gives 900 × 110 ÷ 100 = **990**, not 1,000. The second 10% is taken of a smaller number.\n\nAs one expression: 1,000 × 90 ÷ 100 × 110 ÷ 100 = 1,000 × 99 ÷ 100 = 990. Writing it as one line makes the ₹10 loss obvious.",{"id":780,"type":53,"title":781,"eyebrow":782,"navLabel":783},"ch-catering","Cooking for a crowd","Chapter 10","10 Cooking for a crowd",{"id":785,"type":43,"markdown":786},"catering","A gurdwara **langar** or a wedding caterer may cook for thousands of people. Big numbers make the order of operations even more important, because a slip is multiplied too.\n\nA caterer plans for **1,200 guests**. Each plate gets **2 rotis, 150 g of rice and 120 g of dal**. The kitchen also cooks **10% extra** for safety.\n\n- Rotis: 1,200 × 2 × 110 ÷ 100 = 2,640.\n- Rice in kg: 1,200 × 150 × 110 ÷ 100 ÷ 1,000 = 198 kg.\n- Dal in kg: 1,200 × 120 × 110 ÷ 100 ÷ 1,000 = 792\u002F5 kg.\n\nEach is a long chain of × and ÷ done left to right. Multiplying first and dividing last keeps every step a whole number, which is useful when working by hand.",{"id":788,"type":64,"caption":789,"columns":790,"rows":795},"tbl-catering","Planning for 1,200 guests (with 10% extra)",[791,792,793,794],"Item","Per plate or basis","Expression","Total",[796,800,805,810,814,819],[797,84,798,799],"Rotis","1,200 × 2 × 110 ÷ 100","2,640",[801,802,803,804],"Rice","150 g","1,200 × 150 × 110 ÷ 100 ÷ 1,000","198 kg",[806,807,808,809],"Dal","120 g","1,200 × 120 × 110 ÷ 100 ÷ 1,000","158.4 kg",[811,812,798,813],"Sweets","2 pieces","2,640 pieces",[815,816,817,818],"Cost of rice at ₹60\u002Fkg","198 kg of rice","198 × 60","₹11,880",[820,821,822,823],"Cost per guest if total is ₹1,80,000","whole menu","1,80,000 ÷ 1,200","₹150",{"id":825,"type":178,"itemId":826,"prompt":827,"check":828,"hints":830,"feedback":833},"pr-cost-guest","order-of-operations.extend-cost-per-guest","A langar kitchen spends **₹48,000 on grain**, **₹22,000 on vegetables** and **₹10,000 on gas** to feed **1,600 people**. What is the cost per person? Use **(48000 + 22000 + 10000) ÷ 1600**.",{"kind":308,"answer":829,"tolerance":170,"unit":529},50,[831,832],"Total first, in the bracket: 80,000.","80,000 ÷ 1,600.",{"correct":834,"incorrect":835},"Yes: 80,000 ÷ 1,600 = ₹50 per person.","Bracket: 48,000 + 22,000 + 10,000 = 80,000. Then 80,000 ÷ 1,600 = ₹50. Without the brackets, 48,000 + 22,000 + 10,000 ÷ 1,600 would be over ₹70,000 per person.",{"id":837,"type":47,"variant":491,"title":838,"markdown":839},"careful-units","Units hide inside the order","In 1,200 × 150 ÷ 1,000, the ÷ 1,000 turns grams into kilograms. If someone writes 1,200 × (150 ÷ 1,000) they get the same 180 kg, but 1,200 ÷ (1,000 × 150), a misreading of 1,200 ÷ 1,000 × 150, gives 0.008, off by a factor of more than 20,000. Always write the unit change as its own clear step, and check the size: rice for 1,200 people should be a few hundred kilograms, not a few grams.",{"id":841,"type":178,"itemId":842,"prompt":843,"check":844,"hints":846,"feedback":849},"pr-catering","order-of-operations.extend-catering","A school feeds **850 students**. Each gets **3 idlis**, and the kitchen makes **20% extra**. How many idlis? Use **850 × 3 × 120 ÷ 100**.",{"kind":308,"answer":845,"tolerance":170},3060,[847,848],"Left to right: 850 × 3 = 2,550.","2,550 × 120 = 3,06,000; ÷ 100.",{"correct":850,"incorrect":851},"Yes: 850 × 3 × 120 ÷ 100 = 3,060 idlis.","Left to right: 850 × 3 = 2,550; 2,550 × 120 = 3,06,000; 3,06,000 ÷ 100 = 3,060.",{"id":853,"type":53,"title":854,"eyebrow":855,"navLabel":856},"ch-olymp2","More olympiad-style challenges","Chapter 11","11 Olympiad set 2",{"id":858,"type":43,"markdown":859},"olymp2","Here are four more problems of the kind that appear in maths olympiads and puzzle contests. Try each before reading the solution. The skills they reward are the ones from this whole topic: knowing exactly what an expression means, working systematically, and spotting structure.",{"id":861,"type":64,"caption":862,"columns":863,"rows":865},"tbl-one-pair","One pair of brackets in 3 + 4 × 5 − 2 × 6 (value 11 without brackets)",[864,569],"Brackets placed",[866,869,872,875,878,881],[867,868],"(3 + 4) × 5 − 2 × 6","23",[870,871],"(3 + 4 × 5) − 2 × 6","11",[873,874],"(3 + 4 × 5 − 2) × 6","126",[876,877],"3 + (4 × 5 − 2) × 6","111",[879,880],"3 + 4 × (5 − 2) × 6","75",[882,883],"3 + 4 × (5 − 2 × 6)","−25",{"id":885,"type":378,"title":886,"problem":887,"steps":888},"we-one-pair","One pair of brackets, many values","Place **exactly one pair of brackets** in **3 + 4 × 5 − 2 × 6** to make **23**. Then find the largest value one pair can make.",[889,890,891,892,893],"Without brackets: 3 + 20 − 12 = 11.","Be systematic: a pair of brackets starts before one number and ends after a later one. With 5 numbers there are 4 + 3 + 2 + 1 = 10 placements.","Several placements change nothing (they surround something done first anyway), leaving six different results: 23, 11, 126, 111, 75 and −25 (see the table).","For 23: (3 + 4) × 5 − 2 × 6 = 35 − 12 = **23**.","Largest: (3 + 4 × 5 − 2) × 6 = (3 + 20 − 2) × 6 = 21 × 6 = **126**. Making the whole of the start one big number, then multiplying it by 6, wins.",{"id":895,"type":378,"title":896,"problem":897,"steps":898},"we-sum-products","A sum of products","Evaluate **1 × 2 + 2 × 3 + 3 × 4 + … + 9 × 10**.",[899,900,901,902],"Each term is a product, done first: 2, 6, 12, 20, 30, 42, 56, 72, 90.","Add them: 2 + 6 = 8, + 12 = 20, + 20 = 40, + 30 = 70, + 42 = 112, + 56 = 168, + 72 = 240, + 90 = **330**.","A cleverer route: the running totals 2, 8, 20, 40, 70, 112, 168, 240, 330 are n × (n + 1) × (n + 2) ÷ 3 for n = 1, 2, 3, … Check n = 9: 9 × 10 × 11 ÷ 3 = 990 ÷ 3 = 330 ✓.","Without the order of operations, 1 × 2 + 2 × 3 + … could not be written so compactly; it would need nine pairs of brackets.",{"id":904,"type":178,"itemId":905,"prompt":906,"check":907,"hints":909,"feedback":912},"pr-one-pair","order-of-operations.extend-one-pair","Place one pair of brackets in **3 + 4 × 5 − 2 × 6** to make the value **75**. Enter the value of **3 + 4 × (5 − 2) × 6** to confirm.",{"kind":308,"answer":908,"tolerance":170},75,[910,911],"Bracket first: 5 − 2 = 3.","4 × 3 × 6 = 72.",{"correct":913,"incorrect":914},"Yes: 3 + 4 × 3 × 6 = 3 + 72 = 75.","Bracket: 5 − 2 = 3. Then 4 × 3 × 6 = 72, and 3 + 72 = 75.",{"id":916,"type":178,"itemId":917,"prompt":918,"check":919,"hints":930,"feedback":932},"pr-olymp-parity","order-of-operations.extend-parity","Signs + or − are placed between **1 2 3 4 5 6 7 8 9 10** (one sign in each gap, before 2 up to before 10). Can the result ever be **0**?",{"kind":182,"options":920,"correct":929},[921,923,925,927],{"id":97,"label":922},"Yes, with the right choice of signs",{"id":99,"label":924},"No: the result is always odd",{"id":102,"label":926},"No: the result is always a multiple of 10",{"id":104,"label":928},"Yes, in exactly one way",[99],[931],"1 + 2 + … + 10 = 55. What happens to the total when a + becomes a −?",{"correct":933,"incorrect":934},"Right: 1 + 2 + … + 10 = 55 is odd. Changing +n to −n subtracts 2n, an even number, so the result stays odd and can never be 0.","Start from 1 + 2 + … + 10 = 55. Changing any +n into −n lowers the total by 2n, which is even. So every result is odd, and 0 is impossible.",{"id":936,"type":53,"title":937,"eyebrow":938,"navLabel":939},"ch8","Words, check-up and summary","Chapter 12","12 Wrap-up",{"id":941,"type":942,"title":943,"terms":944},"glossary-extend","glossary","Vocabulary for the wider world",[945,949,952,955,958,961,964,968,971,975,979],{"term":946,"meaning":947,"example":948},"Implied multiplication","Multiplication shown by writing things side by side, with no × sign.","2(2 + 2), 3x",{"term":950,"meaning":951},"Juxtaposition","Placing symbols next to each other. In maths it usually means multiplication.",{"term":953,"meaning":954,"example":73},"Ambiguous","Able to be read in more than one reasonable way.",{"term":956,"meaning":957},"Operator precedence","The order-of-operations table built into a programming language.",{"term":959,"meaning":960},"Infix notation","Writing the operation between its two numbers, as in 2 + 3.",{"term":962,"meaning":963},"Prefix (Polish) notation","Writing the operation before its numbers, as in + 2 3.",{"term":965,"meaning":966,"example":967},"Factorial","n! is the product of the whole numbers from 1 to n.","4! = 4 × 3 × 2 × 1 = 24",{"term":216,"meaning":969,"example":970},"A puzzle: make numbers using exactly four 4s and chosen operations.","(44 − 4) ÷ 4 = 10",{"term":972,"meaning":973,"example":974},"24 game","A puzzle: combine four given numbers with + − × ÷ to make 24.","8 ÷ (3 − 8 ÷ 3) = 24",{"term":976,"meaning":977,"example":978},"Quotient","The whole-number result of a division, ignoring the remainder.","7 \u002F\u002F 2 = 3 in Python",{"term":980,"meaning":981,"example":982},"GST","Goods and Services Tax, added to prices in India as a percentage.","₹500 at 18% → ₹590",{"id":984,"type":985,"title":986,"questions":987},"quiz-extend","quiz","Extend check-up",[988,997,1010,1020,1031,1044,1055,1064,1075,1088],{"itemId":989,"prompt":990,"options":991,"correct":99,"why":996},"order-of-operations.eq-1","By the strict school rule (implied × treated as ordinary ×), what is 8 ÷ 2(2 + 2)?",[992,993,994,995],{"id":97,"label":75},{"id":99,"label":74},{"id":102,"label":233},{"id":104,"label":249},"(2 + 2) = 4; then 8 ÷ 2 × 4 left to right: 4 × 4 = 16.",{"itemId":998,"prompt":999,"options":1000,"correct":99,"why":1009},"order-of-operations.eq-2","Why is 8 ÷ 2(2 + 2) called badly written?",[1001,1003,1005,1007],{"id":97,"label":1002},"It has no answer",{"id":99,"label":1004},"Careful readers can reasonably get 16 or 1",{"id":102,"label":1006},"It divides by zero",{"id":104,"label":1008},"It uses too many 2s","Implied multiplication is treated as tighter than ÷ by some conventions and calculators, so the expression has two reasonable readings.",{"itemId":1011,"prompt":1012,"options":1013,"correct":97,"why":1019},"order-of-operations.eq-3","What is 9 − 3 ÷ ⅓ + 1?",[1014,1015,1016,1017],{"id":97,"label":75},{"id":99,"label":100},{"id":102,"label":79},{"id":104,"label":1018},"−1","3 ÷ ⅓ = 9; then 9 − 9 + 1 = 1.",{"itemId":1021,"prompt":1022,"options":1023,"correct":97,"why":1030},"order-of-operations.eq-4","Which is a four-fours expression for 5?",[1024,1025,1027,1028],{"id":97,"label":238},{"id":99,"label":1026},"4 + 4 ÷ 4",{"id":102,"label":246},{"id":104,"label":1029},"4 × 4 ÷ 4 + 4","(16 + 4) ÷ 4 = 5. (b) uses three 4s; (c) = 7; (d) = 8.",{"itemId":1032,"prompt":1033,"options":1034,"correct":102,"why":1043},"order-of-operations.eq-5","Which set can NOT make 24?",[1035,1037,1039,1041],{"id":97,"label":1036},"1, 2, 3, 4",{"id":99,"label":1038},"3, 3, 8, 8",{"id":102,"label":1040},"1, 1, 1, 1",{"id":104,"label":1042},"4, 7, 7, 7","The biggest value from four 1s is (1 + 1) × (1 + 1) = 4.",{"itemId":1045,"prompt":1046,"options":1047,"correct":102,"why":1054},"order-of-operations.eq-6","What is 2026 − 2025 + 2024 − 2023 + … + 2 − 1?",[1048,1049,1050,1052],{"id":97,"label":220},{"id":99,"label":75},{"id":102,"label":1051},"1,013",{"id":104,"label":1053},"2,026","Pair from the left: each pair is 1, and there are 1,013 pairs.",{"itemId":1056,"prompt":1057,"options":1058,"correct":99,"why":1063},"order-of-operations.eq-7","In Microsoft Excel, =-3^2 gives…",[1059,1060,1061,1062],{"id":97,"label":449},{"id":99,"label":79},{"id":102,"label":451},{"id":104,"label":453},"Excel’s precedence table applies the minus sign before the power: (−3)² = 9. School maths reads −3² as −(3²) = −9.",{"itemId":1065,"prompt":1066,"options":1067,"correct":99,"why":1074},"order-of-operations.eq-8","In postfix (RPN), 2 3 + 4 × means…",[1068,1069,1070,1072],{"id":97,"label":149},{"id":99,"label":160},{"id":102,"label":1071},"2 × 3 + 4",{"id":104,"label":1073},"2 + 3 + 4","2 and 3 are added first, then the result is multiplied by 4: 20.",{"itemId":1076,"prompt":1077,"options":1078,"correct":99,"why":1087},"order-of-operations.eq-9","A ₹500 item with 18% GST: which formula is right?",[1079,1081,1083,1085],{"id":97,"label":1080},"500 × 100 + 18 ÷ 100",{"id":99,"label":1082},"500 × (100 + 18) ÷ 100",{"id":102,"label":1084},"500 + 18 ÷ 100",{"id":104,"label":1086},"(500 × 100 + 18) ÷ 100","500 × 118 ÷ 100 = ₹590.",{"itemId":1089,"prompt":1090,"options":1091,"correct":102,"why":1100},"order-of-operations.eq-10","Brackets in 1 + 2 × 3 + 4 × 5 + 6 (order kept) can make at most…",[1092,1094,1096,1098],{"id":97,"label":1093},"33",{"id":99,"label":1095},"99",{"id":102,"label":1097},"231",{"id":104,"label":1099},"720","(1 + 2) × (3 + 4) × (5 + 6) = 3 × 7 × 11 = 231.",{"id":1102,"type":1103,"title":1104,"points":1105},"cheat-extend","summary","Cheat sheet",[1106,1107,1108,1109,1110,1111,1112,1113],"**8 ÷ 2(2 + 2):** 16 by the strict rule, 1 if implied multiplication binds tighter. It is badly written: add brackets or use a fraction bar.","**Implied multiplication** (2x, 2(…)) is often treated as tighter than ÷ in algebra and science; calculators differ, and some (Casio fx-82ES PLUS, TI-82) document it.","**Four fours:** with + − × ÷, brackets and joined 4s, 0–10 are all possible; from 0 to 20 only 11, 13, 14, 18 and 19 are impossible, and they need extra tools (√, decimals, !).","**24 game:** work backwards from 3 × 8, 4 × 6, 2 × 12. Hard sets like 3, 3, 8, 8 need fractions; 1, 1, 1, 1 is impossible.","**Olympiad habits:** pair terms, spot zeros and common factors, and undo operations in reverse order to find missing numbers.","**Code and spreadsheets** mostly follow school rules, but Excel differs on −3² (9) and 2^3^2 (64). Professionals add brackets whenever in doubt.","**Notations:** infix needs precedence rules; prefix, postfix and fraction bars do not.","**Real work:** GST, bills, averages and science formulas all depend on brackets in the right places.",{"id":1115,"type":1116,"conceptId":1117,"relation":1118,"explanation":1119},"conn-data","connection","data-handling","applied_in","Surveying how people read 8 ÷ 2(2 + 2), and computing a mean as (sum) ÷ (count), both use this topic.",{"id":1121,"type":1116,"conceptId":1122,"relation":1123,"explanation":1124},"conn-patterns","patterns","related_to","Alternating sums like 2026 − 2025 + … − 1 and bracket counts (Catalan numbers) are patterns revealed by grouping.",{"id":1126,"type":1116,"conceptId":1127,"relation":1118,"explanation":1128},"conn-four-ops","four-operations","Four fours and the 24 game are practice in all four operations, chained by the order of operations.",{"id":1130,"type":1116,"conceptId":1131,"relation":1123,"explanation":1132},"conn-number-system","number-system","Reading large results such as 3,700 or 59,000 and writing them in Indian and international forms connects to the Number system topic.",{"id":1134,"type":1135,"sourceIds":1136},"sources-extend","sources",[1137,1138,1139,1140,1141,1142,1143,1144],"order-of-operations-wikipedia","order-of-operations-wikipedia-four-fours","order-of-operations-wikipedia-24-game","order-of-operations-mathsisfun-pemdas","order-of-operations-miller-grouping","order-of-operations-casio-priority","order-of-operations-excel-precedence","order-of-operations-gst-valuation",[1137,1138,1139,1140,1141,1142,1143,1144],"needs_review",{"generatedBy":1148,"notes":1149},"claude-code","Draft generated with Python-checked arithmetic (× ÷ left to right, then + − left to right); pending owner review.","829250834682990125cbc87c97deb1ddc1244ef7cc121bb770749b3a15109755",{"component:sort-game@1":1152,"logic:practice":1153,"component:order-ops@1":1154,"component:match-pairs@1":1155,"component:arith-sprint@1":1156,"source:order-of-operations-casio-priority":1157,"source:order-of-operations-excel-precedence":1158,"source:order-of-operations-gst-valuation":1159,"source:order-of-operations-mathsisfun-pemdas":1160,"source:order-of-operations-miller-grouping":1161,"source:order-of-operations-wikipedia":1162,"source:order-of-operations-wikipedia-24-game":1163,"source:order-of-operations-wikipedia-four-fours":1164},"b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","2df5e6142624f3ba1931c70a8d8ac63b09cd1eb649fb1ae8757bb36f96452720","7c20b35268a810137167a94207d30541be7d945d944d4fd6ac2383362ba2f153","12aa32ec7e994e396bce93f4ca65ff67dde9e6236169cbdfaa8a3693729155c2","7d20cb0b6dc5cf2de499e3df9053481fefcc10be6aebbdbd74e72664e0b2a45a","b8e2b4b4403541ce11f0ad7556454fb58e597e57642a953ab207231ad44c83c1","88ccf3653f7d9723bbb33a67a6548ccff64980f752cba0cd94a32bc60f72bd44","1022e875bb64d7d92d0b0eb62e2dfa81ff1c82aef5e67c620cafa77e2d0e8e1e","3f1d6a760aa170524009986575f77b1a9af08bf736fab494703b115246a5fded",{"state":1166,"reviewer":1167,"selfReview":272,"reviewedAt":1168,"method":1169},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597468]