[{"data":1,"prerenderedAt":1267},["ShallowReactive",2],{"layer:order-of-operations:understand":3},{"layer":4,"contentHash":1248,"dependencyHashes":1249,"approval":1261,"releaseId":1266},{"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":1243,"reviewStatus":1244,"authoring":1245},1,"order-of-operations","en","understand","The rule, precisely","Terms, memory words, three kinds of brackets, “of”, word problems and error-spotting","Make the order of operations precise: split expressions into terms, see why DMAS, BODMAS and PEMDAS all mean one rule, handle nested brackets and \"of\", write expressions from word problems and find mistakes in working.",[13,14,15,16,17],"Split an expression into terms and use the term method to evaluate it.","Explain what DMAS, BODMAS, BIDMAS and PEMDAS stand for, and why D\u002FM and A\u002FS share a level.","Simplify expressions with nested ( ), { } and [ ] brackets and with the word \"of\".","Write a single expression for a multi-step word problem, using brackets where a total is shared or subtracted.","Identify and correct common order-of-operations mistakes, including on basic calculators.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Discover: brackets, × ÷ before + −",{"label":32,"value":33},"Chapters","13",{"label":35,"value":36},"Labs","2 order-ops games, sort game, match, sprint",{"label":38,"value":39},"Key idea","Three levels, partners left to right",[41,47,53,59,64,69,102,114,129,134,137,171,176,181,222,234,239,242,279,284,301,321,333,338,341,353,362,367,371,389,401,406,411,420,424,436,441,444,469,480,490,499,529,583,595,600,603,634,708,729,734,737,757,770,775,780,783,788,793,796,834,838,847,867,872,875,879,883,887,891,895,950,961,966,969,980,1006,1018,1023,1078,1203,1218,1224,1228,1233],{"id":42,"type":43,"markdown":44,"help":45},"intro","prose","In Discover you met the rule: **brackets first, then × and ÷, then + and −, with partners going left to right**. This layer makes that rule precise, shows you *why* each part of it is there, and gives you methods that work on long, messy expressions without mistakes.\n\nYou will learn to split an expression into **terms**, meet all the memory words (DMAS, BODMAS, BIDMAS, PEMDAS) and the trap hidden in each, use three kinds of brackets, handle the word **of**, turn word problems into expressions, and act as a checker who finds and fixes other people's mistakes. Along the way you will see why a cheap calculator and a scientific calculator can disagree.",{"simplerExplanation":46},"Same rule as before, but now with longer expressions, more kinds of brackets, and ways to check your work.",{"id":48,"type":49,"variant":50,"title":51,"markdown":52},"how-to-use","callout","observation","How to use this lesson","Keep a notebook open. Every worked example writes one step per line: copy that habit. Writing **one step per line** is the single best way to avoid order-of-operations mistakes, because you can see exactly what you did and where any slip happened.",{"id":54,"type":55,"title":56,"eyebrow":57,"navLabel":58},"ch1","chapter","Expressions, terms and values","Chapter 01","1 Terms",{"id":60,"type":43,"markdown":61,"help":62},"def-expression","An **arithmetic expression** is numbers joined by operation signs, possibly with brackets. It has a single **value**, the number you get when you work it out (we also say you **simplify** or **evaluate** it).\n\nThe + and − signs that are *not* inside brackets split an expression into pieces called **terms**. Each term is made only of numbers joined by × and ÷ (or a bracket).\n\nTake **12 + 18 ÷ 3 × 2 − 4 × 5**. Its terms are:\n\n- **12**\n- **18 ÷ 3 × 2**\n- **4 × 5** (with a − in front of it)\n\nThis gives a neat method: **work out each term on its own, then add and subtract the terms from left to right.**\n\n- Term 2: 18 ÷ 3 × 2 = 6 × 2 = 12\n- Term 3: 4 × 5 = 20\n- Put together: 12 + 12 − 20 = **4**.",{"simplerExplanation":63},"Terms are the chunks between the + and − signs. Work out each chunk, then add and take away the chunks.",{"id":65,"type":49,"variant":66,"title":67,"markdown":68},"def-term","definition","Term","A **term** of an expression is a part that is separated from the rest by a **+** or **−** sign that is not inside any brackets.\n\n- 5 × 6 − 3 has two terms: 5 × 6 and 3.\n- 7 + (2 + 8) × 4 has two terms: 7 and (2 + 8) × 4. The + inside the brackets does not split anything, because it is inside a bracket.\n- 100 ÷ 5 × 2 has just one term.\n\nThe term method is the order of operations in disguise: working out terms first *is* doing × and ÷ before + and −.",{"id":70,"type":71,"caption":72,"columns":73,"rows":78},"tbl-terms","table","Splitting expressions into terms",[74,75,76,77],"Expression","Terms","Term values","Value",[79,84,89,92,97],[80,81,82,83],"5 × 6 − 3","5 × 6 ; 3","30 ; 3","27",[85,86,87,88],"7 + (2 + 8) × 4","7 ; (2 + 8) × 4","7 ; 40","47",[90,90,91,91],"100 ÷ 5 × 2","40",[93,94,95,96],"9 × 4 − 24 ÷ 6 + 3 × 7","9 × 4 ; 24 ÷ 6 ; 3 × 7","36 ; 4 ; 21","53",[98,99,100,101],"60 − 3 × 4 × 5 + 8","60 ; 3 × 4 × 5 ; 8","60 ; 60 ; 8","8",{"id":103,"type":104,"title":105,"problem":106,"steps":107},"we-terms","worked_example","The term method on a long expression","Evaluate **48 ÷ 6 + 7 × 3 − 45 ÷ 9 × 2 + 1**.",[108,109,110,111,112,113],"Mark the + and − signs: 48 ÷ 6 **+** 7 × 3 **−** 45 ÷ 9 × 2 **+** 1. There are four terms.","Term 1: 48 ÷ 6 = 8.","Term 2: 7 × 3 = 21.","Term 3: 45 ÷ 9 × 2 = 5 × 2 = 10 (left to right inside the term).","Term 4: 1.","Now left to right: 8 + 21 − 10 + 1 = 29 − 10 + 1 = 19 + 1 = **20**.",{"id":115,"type":116,"itemId":117,"prompt":118,"check":119,"hints":123,"feedback":126},"pr-terms","practice","order-of-operations.understand-terms","Evaluate **6 × 7 − 36 ÷ 4 + 2 × 5** using the term method.",{"kind":120,"answer":121,"tolerance":122},"number",43,0,[124,125],"Terms: 6 × 7, 36 ÷ 4, 2 × 5.","Term values: 42, 9, 10. Now 42 − 9 + 10.",{"correct":127,"incorrect":128},"Yes: 42 − 9 + 10 = 33 + 10 = 43.","Terms are 42, 9 and 10. Left to right: 42 − 9 = 33, then 33 + 10 = 43.",{"id":130,"type":55,"title":131,"eyebrow":132,"navLabel":133},"ch2","DMAS, BODMAS, BIDMAS, PEMDAS","Chapter 02","2 Memory words",{"id":135,"type":43,"markdown":136},"mnemonics","Different countries and textbooks remember the order of operations with different words. They all describe **the same rule**.\n\nThe important thing all of them hide: the letters are written in a line, but the rule has only **three levels** (four once powers appear in later classes):\n\n1. **Brackets** (and powers such as 3², which you will meet properly in Class 7 and 8).\n2. **× and ÷**, equal rank, left to right. (\"Of\" belongs here too, done just before the others.)\n3. **+ and −**, equal rank, left to right.\n\nSo DMAS does *not* mean \"all divisions, then all multiplications, then all additions, then all subtractions\". It means \"÷ and × together, then + and − together\".",{"id":138,"type":71,"caption":139,"columns":140,"rows":145},"tbl-mnemonics","The memory words and what their letters stand for",[141,142,143,144],"Word","Where it is common","Letters","Watch out",[146,151,156,161,166],[147,148,149,150],"DMAS","India, Pakistan (early school)","Division, Multiplication, Addition, Subtraction","D and M are one level; A and S are one level",[152,153,154,155],"BODMAS","India, UK, Australia","Brackets, Of, Division, Multiplication, Addition, Subtraction","Indian and UK books read O as \"of\" (½ of 10); others read it as \"Orders\", meaning powers",[157,158,159,160],"BIDMAS","UK","Brackets, Indices, Division, Multiplication, Addition, Subtraction","Indices are powers like 2³",[162,163,164,165],"PEMDAS","USA","Parentheses, Exponents, Multiplication, Division, Addition, Subtraction","Here M comes before D, yet the rule is the same",[167,168,169,170],"BEDMAS","Canada","Brackets, Exponents, Division, Multiplication, Addition, Subtraction","Same rule again",{"id":172,"type":49,"variant":173,"title":174,"markdown":175},"aha-pemdas","aha","PEMDAS says M before D, BODMAS says D before M","If the letters really gave the order, then PEMDAS (USA) and BODMAS (India) would disagree about 8 ÷ 4 × 2! PEMDAS-order would give 8 ÷ 8 = 1 and BODMAS-order would give 2 × 2 = 4.\n\nBut Indian and American mathematicians get **the same answer, 4**, because both words mean the same thing: × and ÷ share a level and go left to right. The fact that the two words put M and D in opposite orders is the best proof that the order of those letters does not matter.",{"id":177,"type":49,"variant":178,"title":179,"markdown":180},"misc-dmas-meaning","misconception","DMAS is not “Distributive, Multiplicative, Additive, Subtractive”","Some notes and syllabus lists expand DMAS as *Distributive, Multiplicative, Additive, Subtractive*. That is an easy mix-up to make, because those words are also real maths words, but it is not what DMAS means.\n\n**DMAS stands for Division, Multiplication, Addition, Subtraction**: the four operations in the order you meet them in the rule.\n\nThe **distributive property** is a different (and very useful) idea: 3 × (4 + 5) = 3 × 4 + 3 × 5. It belongs to the topic [Properties of numbers](\u002Ftopics\u002Fproperties-of-numbers). It does help explain *why* the order of operations is the way it is (you will see that in the Deepen layer), but it is not one of the letters.",{"id":182,"type":183,"component":184,"componentVersion":5,"config":185,"objective":216,"textAlternative":217,"help":218},"lab-match-mnemonic","interactive","match-pairs",{"prompt":186,"mode":187,"pairs":188},"Connect each letter or word to what it stands for.","connect",[189,192,195,198,201,204,207,210,213],{"a":190,"b":191},"D in DMAS","Division",{"a":193,"b":194},"M in DMAS","Multiplication",{"a":196,"b":197},"A in DMAS","Addition",{"a":199,"b":200},"S in DMAS","Subtraction",{"a":202,"b":203},"B in BODMAS","Brackets",{"a":205,"b":206},"O in BODMAS","Of (as in ½ of 10)",{"a":208,"b":209},"I in BIDMAS","Indices (powers like 2³)",{"a":211,"b":212},"P in PEMDAS","Parentheses (round brackets)",{"a":214,"b":215},"E in PEMDAS","Exponents (powers)","Match each letter of the order-of-operations memory words to the word it stands for.","A connect-the-pairs game with 9 pairs.\n\n- DMAS: D = Division, M = Multiplication, A = Addition, S = Subtraction.\n- BODMAS: B = Brackets, O = Of (as in ½ of 10).\n- BIDMAS: I = Indices, meaning powers such as 2³.\n- PEMDAS: P = Parentheses (round brackets), E = Exponents, another name for powers.\n\nRemember: in every one of these words, D and M share one level (left to right) and A and S share one level (left to right).",{"hints":219},[220,221],"Parentheses is the American word for round brackets.","Indices and exponents both mean powers, like the 3 in 2³.",{"id":223,"type":116,"itemId":224,"prompt":225,"check":226,"hints":228,"feedback":231},"pr-pemdas","order-of-operations.understand-md-equal","Evaluate **60 ÷ 5 × 3** the way both PEMDAS and BODMAS users would.",{"kind":120,"answer":227,"tolerance":122},36,[229,230],"÷ and × share a level: go left to right.","60 ÷ 5 = 12 first.",{"correct":232,"incorrect":233},"Yes: 60 ÷ 5 = 12, then 12 × 3 = 36.","Left to right: 60 ÷ 5 = 12, then 12 × 3 = 36. Doing 5 × 3 first (to get 4) is the letter trap.",{"id":235,"type":55,"title":236,"eyebrow":237,"navLabel":238},"ch3","Left to right, and why","Chapter 03","3 Left to right",{"id":240,"type":43,"markdown":241},"why-lr","Why must partners go left to right? Because **− and ÷ belong to the number right after them**.\n\nIn 10 − 3 + 2, the − sign says \"take away **3**\". It does not say \"take away 3 + 2\". The 2 is being *added*. So the story is: start with 10, take away 3, add 2. That gives **9**.\n\nIn 8 ÷ 4 × 2, the ÷ sign says \"divide by **4**\". It does not say \"divide by 4 × 2\". The story is: start with 8, share into 4 groups (2 each), then double: **4**.\n\nFor + and × alone, order never matters: 2 + 3 + 4 and 4 + 3 + 2 are both 9. The trouble only comes with − and ÷, because 10 − 3 is not 3 − 10 and 8 ÷ 4 is not 4 ÷ 8. Going left to right keeps every − and ÷ attached to the correct number.",{"id":243,"type":71,"caption":244,"columns":245,"rows":249},"tbl-lr","Left to right versus the trap",[74,246,247,248],"Correct: left to right","Trap","Trap gives",[250,255,260,265,270,275],[251,252,253,254],"10 − 3 + 2","10 − 3 + 2 = 7 + 2 = 9","10 − (3 + 2)","5",[256,257,258,259],"8 ÷ 4 × 2","8 ÷ 4 × 2 = 2 × 2 = 4","8 ÷ (4 × 2)","1",[261,262,263,264],"36 ÷ 6 ÷ 2","36 ÷ 6 ÷ 2 = 6 ÷ 2 = 3","36 ÷ (6 ÷ 2)","12",[266,267,268,269],"20 − 5 − 3","20 − 5 − 3 = 15 − 3 = 12","20 − (5 − 3)","18",[271,272,273,274],"50 − 20 + 10 − 5","50 − 20 + 10 − 5 = 30 + 10 − 5 = 40 − 5 = 35","50 − (20 + 10) − 5","15",[276,277,278,259],"48 ÷ 8 × 2 ÷ 3","48 ÷ 8 × 2 ÷ 3 = 6 × 2 ÷ 3 = 12 ÷ 3 = 4","48 ÷ (8 × 2) ÷ 3",{"id":280,"type":49,"variant":281,"title":282,"markdown":283},"nuance-reorder","nuance","You may move terms, if you carry their signs","It is fine to rearrange a line of + and − if **each number keeps the sign in front of it**. 10 − 3 + 2 can be rewritten as 10 + 2 − 3 = 9, because the 3 is still being taken away and the 2 is still being added.\n\nThe same is true for × and ÷: 8 ÷ 4 × 2 = 8 × 2 ÷ 4 = 16 ÷ 4 = 4.\n\nWhat you may **not** do is let a − or ÷ \"grab\" more than the one number after it. That is exactly what the trap does.",{"id":285,"type":286,"prompt":287,"options":288,"explanation":300},"predict-lr","prediction","What is **100 ÷ 10 × 5 − 20 + 5**?",[289,291,294,297],{"id":290,"label":88},"a",{"id":292,"label":293},"b","35",{"id":295,"label":296},"c","25",{"id":298,"label":299},"d","−13","**35.** × and ÷ first, left to right: 100 ÷ 10 = 10, then 10 × 5 = 50. Now + and −, left to right: 50 − 20 = 30, then 30 + 5 = 35. Full chain: 100 ÷ 10 × 5 − 20 + 5 = 10 × 5 − 20 + 5 = 50 − 20 + 5 = 30 + 5 = 35.",{"id":302,"type":183,"component":303,"componentVersion":5,"config":304,"objective":314,"textAlternative":315,"help":316},"lab-orderops-lr","order-ops",{"expressions":305,"showRuleCard":313},[306,256,261,307,308,309,276,310,311,312],"10 - 3 + 2","20 - 5 - 3","24 ÷ 4 × 3 - 2","50 - 20 + 10 - 5","100 ÷ 10 × 5 - 20 + 5","7 + 18 ÷ 3 × 2 - 6","90 - 45 ÷ 5 × 3 + 12",true,"Practise the left-to-right rule on expressions where the letter trap would give a wrong answer.","Ten expressions where doing + before −, or × before ÷, would go wrong. Tap the operation to do next; the expression reduces one step at a time.\n\n- 10 − 3 + 2 = 7 + 2 = 9\n- 8 ÷ 4 × 2 = 2 × 2 = 4\n- 36 ÷ 6 ÷ 2 = 6 ÷ 2 = 3\n- 20 − 5 − 3 = 15 − 3 = 12\n- 24 ÷ 4 × 3 − 2 = 6 × 3 − 2 = 18 − 2 = 16\n- 50 − 20 + 10 − 5 = 30 + 10 − 5 = 40 − 5 = 35\n- 48 ÷ 8 × 2 ÷ 3 = 6 × 2 ÷ 3 = 12 ÷ 3 = 4\n- 100 ÷ 10 × 5 − 20 + 5 = 10 × 5 − 20 + 5 = 50 − 20 + 5 = 30 + 5 = 35\n- 7 + 18 ÷ 3 × 2 − 6 = 7 + 6 × 2 − 6 = 7 + 12 − 6 = 19 − 6 = 13\n- 90 − 45 ÷ 5 × 3 + 12 = 90 − 9 × 3 + 12 = 90 − 27 + 12 = 63 + 12 = 75\n\nIn each one, the leftmost of two partner operations is done first.",{"hints":317},[318,319,320],"Two partners side by side? Tap the one on the left.","Finish every × and ÷ before touching + or −.","If there are several × or ÷ signs, the game expects the leftmost one first.",{"id":322,"type":116,"itemId":323,"prompt":324,"check":325,"hints":327,"feedback":330},"pr-lr2","order-of-operations.understand-left-right","Evaluate **72 ÷ 9 ÷ 2 × 5**.",{"kind":120,"answer":326,"tolerance":122},20,[328,329],"Only ÷ and ×: left to right.","72 ÷ 9 = 8, then 8 ÷ 2 = 4.",{"correct":331,"incorrect":332},"Yes: 72 ÷ 9 = 8, 8 ÷ 2 = 4, 4 × 5 = 20.","Left to right: 72 ÷ 9 = 8, 8 ÷ 2 = 4, 4 × 5 = 20.",{"id":334,"type":55,"title":335,"eyebrow":336,"navLabel":337},"ch4","Three kinds of brackets","Chapter 04","4 Brackets",{"id":339,"type":43,"markdown":340},"bracket-kinds","When one bracket sits inside another, it gets hard to see which closing bracket belongs to which opening one. So people use **different shapes**:\n\n- ( ) **round brackets** (also called parentheses or small brackets)\n- { } **curly brackets** (also called braces)\n- [ ] **square brackets** (also called box brackets)\n\nIn many Indian textbooks the usual nesting is round inside curly inside square: **[ { ( ) } ]**. But the shapes do not have a ranking of their own. The rule is simply:\n\n> **Work out the innermost brackets first, then move outwards.**\n\nWhatever shape the innermost bracket has, it goes first.",{"id":342,"type":104,"title":343,"problem":344,"steps":345,"help":351},"we-nested","Three layers of brackets","Simplify **[50 − {12 + (18 − 6) ÷ 4}] × 2**.",[346,347,348,349,350],"Innermost: the round brackets. 18 − 6 = 12. The expression is [50 − {12 + 12 ÷ 4}] × 2.","Next: the curly brackets hold 12 + 12 ÷ 4. Inside them, ÷ first: 12 ÷ 4 = 3, so they hold 12 + 3 = 15. The expression is [50 − 15] × 2.","Next: the square brackets. 50 − 15 = 35. The expression is 35 × 2.","Finally: 35 × 2 = **70**.","Notice that inside each bracket, the usual order still applies: in {12 + 12 ÷ 4} the ÷ was done before the +.",{"simplerExplanation":352},"Peel the brackets like an onion: from the inside out. Inside each layer, use × ÷ before + −.",{"id":354,"type":104,"title":355,"problem":356,"steps":357},"we-nested2","When brackets sit side by side","Simplify **(7 + 5) × (9 − 4) − [36 ÷ (2 + 4)]**.",[358,359,360,361],"There are three innermost brackets: (7 + 5), (9 − 4) and (2 + 4). They do not overlap, so they can be done in any order.","7 + 5 = 12, 9 − 4 = 5, 2 + 4 = 6. The expression is 12 × 5 − [36 ÷ 6].","Square bracket: 36 ÷ 6 = 6. The expression is 12 × 5 − 6.","× first: 12 × 5 = 60. Then 60 − 6 = **54**.",{"id":363,"type":49,"variant":364,"title":365,"markdown":366},"careful-pairs","careful","Count your brackets","Every opening bracket needs exactly one matching closing bracket of the same shape. Before you start, check:\n\n- Number of ( equals number of ).\n- The shapes close in the reverse order they opened: [ { ( ) } ] is fine; [ { ( ] } ) is broken.\n\nIf a question has a broken bracket pattern, it has no single meaning. Point it out instead of guessing.",{"id":368,"type":49,"variant":281,"title":369,"markdown":370},"nuance-widget","About the games on this site","The step-by-step order game on this site uses round ( ) and square [ ] brackets only, and only whole numbers. Curly brackets { } and the word \"of\" appear in the worked examples instead. The rule is the same for every shape: innermost first.",{"id":372,"type":183,"component":303,"componentVersion":5,"config":373,"objective":383,"textAlternative":384,"help":385},"lab-orderops-brackets",{"expressions":374,"showRuleCard":313},[375,376,377,378,379,380,381,382],"[20 - (3 + 5)] ÷ 4","3 × [12 - (2 + 4)]","[(7 + 5) × 2 - 4] ÷ 5","100 - [6 × (8 - 3) + 10]","(9 - 4) × (6 + 2) - 12","[36 ÷ (2 + 4)] × [15 - 3 × 4]","2 × [5 + 3 × (8 - 6)]","[(40 - 4) ÷ 6 + 4] × 3","Work out expressions with round brackets inside square brackets, always starting from the innermost.","Eight expressions with ( ) and [ ] brackets. The innermost bracket is always worked out first; inside any bracket, × and ÷ still come before + and −.\n\n- [20 − (3 + 5)] ÷ 4 = [20 − 8] ÷ 4 = 12 ÷ 4 = 3\n- 3 × [12 − (2 + 4)] = 3 × [12 − 6] = 3 × 6 = 18\n- [(7 + 5) × 2 − 4] ÷ 5 = [12 × 2 − 4] ÷ 5 = [24 − 4] ÷ 5 = 20 ÷ 5 = 4\n- 100 − [6 × (8 − 3) + 10] = 100 − [6 × 5 + 10] = 100 − [30 + 10] = 100 − 40 = 60\n- (9 − 4) × (6 + 2) − 12 = 5 × (6 + 2) − 12 = 5 × 8 − 12 = 40 − 12 = 28\n- [36 ÷ (2 + 4)] × [15 − 3 × 4] = [36 ÷ 6] × [15 − 3 × 4] = 6 × [15 − 3 × 4] = 6 × [15 − 12] = 6 × 3 = 18\n- 2 × [5 + 3 × (8 − 6)] = 2 × [5 + 3 × 2] = 2 × [5 + 6] = 2 × 11 = 22\n- [(40 − 4) ÷ 6 + 4] × 3 = [36 ÷ 6 + 4] × 3 = [6 + 4] × 3 = 10 × 3 = 30",{"hints":386},[387,388],"Find a bracket with no other bracket inside it.","Inside a bracket, the normal order still applies.",{"id":390,"type":116,"itemId":391,"prompt":392,"check":393,"hints":395,"feedback":398},"pr-brackets","order-of-operations.understand-nested","Simplify **5 × [30 − (4 + 2) × 4]**.",{"kind":120,"answer":394,"tolerance":122},30,[396,397],"Round brackets first: 4 + 2 = 6.","Inside the square bracket: 30 − 6 × 4 = 30 − 24.",{"correct":399,"incorrect":400},"Yes: (4 + 2) = 6; 6 × 4 = 24; 30 − 24 = 6; 5 × 6 = 30.","Innermost first: 4 + 2 = 6. Square bracket: 30 − 6 × 4 = 30 − 24 = 6. Finally 5 × 6 = 30.",{"id":402,"type":55,"title":403,"eyebrow":404,"navLabel":405},"ch5","The little word “of”","Chapter 05","5 The word “of”",{"id":407,"type":43,"markdown":408,"help":409},"of","The **O** in BODMAS is read two ways. Indian and British school books usually take it to mean **of**, as in \"half **of** 10\" or \"one-third **of** 27\"; other books (and sites such as Math is Fun) read it as **Orders**, meaning powers and roots. \"Of\" means multiply: ½ of 10 = ½ × 10 = 5.\n\nSo why does \"of\" get its own letter? Because BODMAS-style textbooks follow a **convention**: \"of\" is done **before** ordinary × and ÷. The idea is that \"½ of 16\" is one tightly-bound quantity (8), not two separate pieces. This is a textbook agreement, not a law of mathematics — plenty of books and every programming language simply treat \"of\" as an ordinary ×.\n\n- 40 ÷ ½ of 16 (BODMAS convention): first ½ of 16 = 8, then 40 ÷ 8 = **5**.\n- 40 ÷ ½ × 16 (with × instead of \"of\"): left to right, 40 ÷ ½ = 80, then 80 × 16 = **1,280**.\n\nThat is a huge difference from one small word — which is exactly why most mathematicians avoid writing \"÷ ½ of\" at all and use brackets instead: 40 ÷ (½ × 16) says the same as the first line with no room for doubt.\n\nThis site follows the BODMAS convention whenever the word \"of\" appears, and says so in the question.",{"simplerExplanation":410},"'Of' means times, but it sticks tightly to its numbers: work out '½ of 16' first, as if it were in brackets.",{"id":412,"type":104,"title":413,"problem":414,"steps":415},"we-of","Using “of” in a word problem","A class of 36 students goes on a trip. One-third of the class travels by bus; the rest go by train, 4 to a compartment. How many compartments are needed?",[416,417,418,419],"Students by bus: ⅓ of 36 = 12.","Students by train: 36 − ⅓ of 36 = 36 − 12 = 24. (\"Of\" first, then subtract.)","Compartments: (36 − ⅓ of 36) ÷ 4. The brackets are needed because the whole remainder is shared.","24 ÷ 4 = **6 compartments**.",{"id":421,"type":49,"variant":281,"title":422,"markdown":423},"nuance-of-percent","“Of” in percentages and fractions","You will meet \"of\" again with percentages: 10% of ₹250 = 10\u002F100 × 250 = ₹25. Every time, \"of\" means multiply, and it is safest to work out the \"of\" part first, as a chunk: ₹250 − 10% of ₹250 = 250 − 25 = ₹225 (a 10% discount).",{"id":425,"type":116,"itemId":426,"prompt":427,"check":428,"hints":430,"feedback":433},"pr-of","order-of-operations.understand-of","Simplify **60 − ½ of 40 + 5**.",{"kind":120,"answer":429,"tolerance":122},45,[431,432],"Do the \"of\" first: ½ of 40 = 20.","Now 60 − 20 + 5, left to right.",{"correct":434,"incorrect":435},"Yes: ½ of 40 = 20; 60 − 20 = 40; 40 + 5 = 45.","\"Of\" first: ½ of 40 = 20. Then left to right: 60 − 20 = 40, and 40 + 5 = 45.",{"id":437,"type":55,"title":438,"eyebrow":439,"navLabel":440},"ch6","From word problems to expressions","Chapter 06","6 Word problems",{"id":442,"type":43,"markdown":443},"wp-method","Turning a story into a single expression is a skill in its own right. It forces you to decide *what is multiplied by what* and *what must be worked out first*, which is exactly what the order of operations encodes.\n\nA reliable method:\n\n1. **Find the chunks.** Each \"number of things × price (or size) of one\" is a chunk. Write each chunk as a product.\n2. **Join the chunks** with + or −.\n3. **Ask: is a whole total being taken away, shared or multiplied?** If yes, wrap that total in brackets.\n4. **Evaluate** using the order of operations.\n5. **Check** that the answer makes sense (size, units).",{"id":445,"type":446,"title":447,"items":448},"steps-wp","steps","Writing an expression from a story",[449,453,457,461,465],{"title":450,"tag":451,"text":452},"Find the chunks","how many × each","3 pens at ₹12 → 3 × 12. 2 notebooks at ₹40 → 2 × 40.",{"title":454,"tag":455,"text":456},"Join them","+ or −","Total cost: 3 × 12 + 2 × 40.",{"title":458,"tag":459,"text":460},"Bracket whole totals","shared \u002F taken away","Change from ₹500: 500 − (3 × 12 + 2 × 40).",{"title":462,"tag":463,"text":464},"Evaluate","the rule","500 − (36 + 80) = 500 − 116 = 384.",{"title":466,"tag":467,"text":468},"Check","does it make sense?","Bill ₹116 is under ₹500, so change must be under ₹500 and over ₹300. ₹384 ✓",{"id":470,"type":104,"title":471,"problem":472,"steps":473},"we-wp-shop","The school supplies bill","A shop sells pens at ₹12 and notebooks at ₹40. Asha buys **3 pens and 2 notebooks** for herself, and the **same again** for her brother. She pays with ₹500. Write one expression for her change and evaluate it.",[474,475,476,477,478,479],"One set: 3 × 12 + 2 × 40.","Two sets: 2 × (3 × 12 + 2 × 40). The brackets are needed: the whole set is doubled, not just the pens.","Change: 500 − 2 × (3 × 12 + 2 × 40).","Brackets: 3 × 12 = 36, 2 × 40 = 80, 36 + 80 = 116.","Then × before −: 2 × 116 = 232.","500 − 232 = **₹268**.",{"id":481,"type":104,"title":482,"problem":483,"steps":484},"we-wp-train","Railway tickets","A family of 2 adults and 3 children travel by train. An adult ticket costs ₹245 and a child ticket is ₹125. They also pay a ₹30 booking fee once. What is the total?",[485,486,487,488,489],"Chunks: adults 2 × 245, children 3 × 125, fee 30.","Expression: 2 × 245 + 3 × 125 + 30.","× first: 2 × 245 = 490, 3 × 125 = 375.","Add: 490 + 375 + 30 = **₹895**.","Check: roughly 2 × 250 + 3 × 125 ≈ 500 + 375 = 875, plus the fee, so ₹895 is sensible. ✓",{"id":491,"type":104,"title":492,"problem":493,"steps":494},"we-wp-share","Sharing the prize money","A school team wins ₹5,000. They spend ₹1,400 on a party and share the rest equally among 12 players. How much does each player get?",[495,496,497,498],"The *rest* is shared, so the subtraction must happen first: (5,000 − 1,400) ÷ 12.","Brackets: 5,000 − 1,400 = 3,600.","3,600 ÷ 12 = **₹300** each.","Without brackets, 5,000 − 1,400 ÷ 12 would divide only the party money, and 1,400 ÷ 12 is not even a whole number of rupees. That is a hint that the expression is wrong.",{"id":500,"type":71,"caption":501,"columns":502,"rows":504},"tbl-wp","More stories and their expressions",[503,74,77],"Story",[505,509,513,517,521,525],[506,507,508],"Cricket: 6 fours, 2 sixes and 9 singles","6 × 4 + 2 × 6 + 9","45 runs",[510,511,512],"5 rows of chairs with 8 in each, then 6 more chairs","5 × 8 + 6","46 chairs",[514,515,516],"4 friends split a ₹360 bill and a ₹40 tip","(360 + 40) ÷ 4","₹100 each",[518,519,520],"Rangoli: 7 rows of 9 dots, minus the 5 corner and centre dots","7 × 9 − 5","58 dots",[522,523,524],"3 packs of 10 balloons; 4 burst; the rest tied in bunches of 2","(3 × 10 − 4) ÷ 2","13 bunches",[526,527,528],"Monthly milk: 2 litres a day at ₹56 a litre for 30 days","2 × 56 × 30","₹3,360",{"id":530,"type":183,"component":531,"componentVersion":5,"config":532,"objective":577,"textAlternative":578,"help":579},"lab-sprint-wp","arith-sprint",{"operations":533,"ranges":538,"rounds":326,"secondsTotal":122,"estimateFirst":313,"wordProblems":544},[534,535,536,537],"+","-","×","÷",{"a":539,"b":542},{"min":540,"max":541},2,50,{"min":540,"max":543},12,[545,549,552,555,558,562,565,567,571,574],{"prompt":546,"answer":547,"unit":548,"operation":536},"Asha buys 3 pens at ₹12 and 2 notebooks at ₹40, and the same again for her brother. What is the total bill?",232,"₹",{"prompt":550,"answer":551,"unit":548,"operation":534},"2 adult train tickets at ₹245, 3 child tickets at ₹125 and a ₹30 fee. What is the total?",895,{"prompt":553,"answer":554,"unit":548,"operation":537},"A team shares ₹5,000 minus ₹1,400 party costs equally among 12 players. How much each?",300,{"prompt":556,"answer":557,"unit":548,"operation":537},"4 friends split a ₹360 bill plus a ₹40 tip equally. How much does each pay?",100,{"prompt":559,"answer":560,"unit":561,"operation":537},"3 packs of 10 balloons, 4 burst, the rest tied in bunches of 2. How many bunches?",13,"bunches",{"prompt":563,"answer":429,"unit":564,"operation":534},"A batter hits 6 fours, 2 sixes and 9 singles. How many runs?","runs",{"prompt":566,"answer":326,"unit":548,"operation":535},"Change from ₹500 after buying 4 kg rice at ₹65 a kg and 2 kg dal at ₹110 a kg?",{"prompt":568,"answer":569,"unit":570,"operation":535},"A hall has 12 rows of 15 seats. 25 seats are broken. How many seats can be used?",155,"seats",{"prompt":572,"answer":573,"unit":548,"operation":537},"A ₹1,200 gift is bought by 6 friends; 2 more friends join and they re-share it equally. How much does each pay now?",150,{"prompt":575,"answer":576,"unit":548,"operation":536},"Milk: 2 litres a day at ₹56 a litre. What is the bill for 30 days?",3360,"Estimate first, then solve multi-step shopping, travel and cricket problems by writing one expression for each.","A 20-round practice game. Generated questions with numbers up to 50 ask for an estimate first, then the exact answer. Mixed in at random are word problems from this list (these skip the estimate):\n\n- Two sets of 3 pens (₹12) and 2 notebooks (₹40): 2 × (3 × 12 + 2 × 40) = ₹232.\n- 2 × 245 + 3 × 125 + 30 = ₹895 for train tickets.\n- (5,000 − 1,400) ÷ 12 = ₹300 each.\n- (360 + 40) ÷ 4 = ₹100 each.\n- (3 × 10 − 4) ÷ 2 = 13 bunches.\n- 6 × 4 + 2 × 6 + 9 = 45 runs.\n- 500 − (4 × 65 + 2 × 110) = ₹20 change.\n- 12 × 15 − 25 = 155 seats.\n- 1,200 ÷ (6 + 2) = ₹150 each.\n- 2 × 56 × 30 = ₹3,360 for milk.\n\nWrite the expression before you calculate, and ask whether any total needs brackets.",{"hints":580},[581,582],"Estimate by rounding each price to a friendly number.","If a whole total is shared or taken away, bracket it.",{"id":584,"type":116,"itemId":585,"prompt":586,"check":587,"hints":589,"feedback":592},"pr-wp","order-of-operations.understand-word-problem","A kirana shop sells sugar at ₹45 a kg and tea at ₹120 a packet. Ravi buys **5 kg of sugar and 2 packets of tea** and pays with a ₹500 note. How much change, in rupees?",{"kind":120,"answer":588,"tolerance":122,"unit":548},35,[590,591],"Bill: 5 × 45 + 2 × 120.","Change = 500 − (bill). Bill = 225 + 240 = 465.",{"correct":593,"incorrect":594},"Yes: 500 − (225 + 240) = 500 − 465 = ₹35.","Bill = 5 × 45 + 2 × 120 = 225 + 240 = 465. Change = 500 − 465 = ₹35. Did you forget the brackets around the bill?",{"id":596,"type":55,"title":597,"eyebrow":598,"navLabel":599},"ch7","Spot the mistake, fix the working","Chapter 07","7 Fix the working",{"id":601,"type":43,"markdown":602},"mistakes","Most wrong answers in this topic come from a small number of mistakes. If you know them by name, you can check your own work quickly.\n\n1. **Reading like a sentence.** Doing operations strictly left to right, ignoring × ÷ first. 5 + 5 × 5 becomes 50 instead of 30.\n2. **The letter trap.** Doing + before − or ÷ before × because of the order of letters. 10 − 3 + 2 becomes 5 instead of 9.\n3. **Dropping brackets.** Losing a bracket halfway through, especially after a − sign. 20 − (5 + 3) becomes 20 − 5 + 3 = 18 instead of 12.\n4. **Doing two steps at once in your head** and mixing them up. Write one step per line.\n5. **Wrong brackets in a word problem.** Taking away only part of a bill, or sharing only part of an amount.",{"id":604,"type":71,"caption":605,"columns":606,"rows":610},"tbl-mistakes","Errors in working, and the fix",[607,608,609],"Working shown","Which mistake","Correct working",[611,615,618,622,626,630],[612,613,614],"5 + 5 × 5 = 10 × 5 = 50","Reading like a sentence","5 + 5 × 5 = 5 + 25 = 30",[616,617,252],"10 − 3 + 2 = 10 − 5 = 5","Letter trap (+ before −)",[619,620,621],"20 − (5 + 3) = 20 − 5 + 3 = 18","Dropped the bracket","20 − (5 + 3) = 20 − 8 = 12",[623,624,625],"48 ÷ 4 × 2 = 48 ÷ 8 = 6","Letter trap (× before ÷)","48 ÷ 4 × 2 = 12 × 2 = 24",[627,628,629],"3 × (4 + 2) = 12 + 2 = 14","Only multiplied the first number in the bracket","3 × (4 + 2) = 3 × 6 = 18",[631,632,633],"2 × [10 − (3 + 1)] = 2 × [7 + 1] = 16","Dropped the bracket inside","2 × [10 − (3 + 1)] = 2 × [10 − 4] = 2 × 6 = 12",{"id":635,"type":183,"component":636,"componentVersion":5,"config":637,"objective":702,"textAlternative":703,"help":704},"lab-sort-working","sort-game",{"prompt":638,"bins":639,"items":645,"seconds":122},"Is each line of working correct or incorrect? Sort the cards.",[640,642],{"id":641,"label":609},"ok",{"id":643,"label":644},"bad","Has a mistake",[646,650,654,658,662,666,670,674,678,682,686,690,694,698],{"id":647,"label":648,"bin":641,"why":649},"w1","6 + 4 × 3 = 6 + 12 = 18","× first: 4 × 3 = 12, then 6 + 12 = 18.",{"id":651,"label":652,"bin":643,"why":653},"w2","6 + 4 × 3 = 10 × 3 = 30","The + was done first. × must come first: 6 + 12 = 18.",{"id":655,"label":656,"bin":643,"why":657},"w3","15 − 5 + 2 = 15 − 7 = 8","+ was done before −. Left to right: 15 − 5 = 10, then 10 + 2 = 12.",{"id":659,"label":660,"bin":641,"why":661},"w4","15 − 5 + 2 = 10 + 2 = 12","Left to right: 15 − 5 = 10, then + 2 = 12.",{"id":663,"label":664,"bin":643,"why":665},"w5","24 ÷ 6 × 2 = 24 ÷ 12 = 2","× was done before ÷. Left to right: 24 ÷ 6 = 4, then 4 × 2 = 8.",{"id":667,"label":668,"bin":641,"why":669},"w6","24 ÷ 6 × 2 = 4 × 2 = 8","÷ and × share a level; 24 ÷ 6 is on the left, so it goes first.",{"id":671,"label":672,"bin":641,"why":673},"w7","30 − (8 + 2) = 30 − 10 = 20","Brackets first: 8 + 2 = 10. Then 30 − 10 = 20.",{"id":675,"label":676,"bin":643,"why":677},"w8","30 − (8 + 2) = 22 + 2 = 24","The bracket was dropped. The whole (8 + 2) must be taken away: 30 − 10 = 20.",{"id":679,"label":680,"bin":643,"why":681},"w9","4 × (5 + 1) = 20 + 1 = 21","Only the 5 was multiplied. Brackets first: 5 + 1 = 6, then 4 × 6 = 24.",{"id":683,"label":684,"bin":641,"why":685},"w10","2 + 18 ÷ 3 × 2 = 2 + 6 × 2 = 2 + 12 = 14","÷ and × left to right (18 ÷ 3 = 6, 6 × 2 = 12), then 2 + 12 = 14.",{"id":687,"label":688,"bin":643,"why":689},"w11","2 + 18 ÷ 3 × 2 = 2 + 18 ÷ 6 = 2 + 3 = 5","× was done before ÷. Left to right gives 18 ÷ 3 = 6, 6 × 2 = 12, so the answer is 14.",{"id":691,"label":692,"bin":641,"why":693},"w12","[12 − (2 + 4)] × 3 = [12 − 6] × 3 = 18","Innermost first: 2 + 4 = 6; 12 − 6 = 6; 6 × 3 = 18.",{"id":695,"label":696,"bin":641,"why":697},"w13","100 − 4 × 5 × 2 = 100 − 40 = 60","× first: 4 × 5 × 2 = 40. Then 100 − 40 = 60.",{"id":699,"label":700,"bin":643,"why":701},"w14","100 − 4 × 5 × 2 = 96 × 10 = 960","The − was done first. Multiplications first: 4 × 5 × 2 = 40, then 100 − 40 = 60.","Judge each piece of working: is it correct, or does it contain one of the common order-of-operations mistakes?","A sorting game with 14 cards and 2 bins: correct working and working with a mistake.\n\nCorrect: 6 + 4 × 3 = 18; 15 − 5 + 2 = 12; 24 ÷ 6 × 2 = 8; 30 − (8 + 2) = 20; 2 + 18 ÷ 3 × 2 = 14; [12 − (2 + 4)] × 3 = 18; 100 − 4 × 5 × 2 = 60.\n\nMistakes: 6 + 4 × 3 = 30 (added first); 15 − 5 + 2 = 8 (+ before −); 24 ÷ 6 × 2 = 2 (× before ÷); 30 − (8 + 2) = 24 (dropped bracket); 4 × (5 + 1) = 21 (multiplied only the 5); 2 + 18 ÷ 3 × 2 = 5 (× before ÷); 100 − 4 × 5 × 2 = 960 (subtracted first).",{"hints":705},[706,707],"Redo each line yourself, one step at a time, then compare.","Look closely at the first step: that is where most mistakes start.",{"id":709,"type":116,"itemId":710,"prompt":711,"check":712,"hints":724,"feedback":726},"pr-mistake","order-of-operations.understand-spot-mistake","Neha's working: **45 − 3 × (2 + 8) = 45 − 3 × 2 + 8 = 45 − 6 + 8 = 47**. Where did she go wrong?",{"kind":713,"options":714,"correct":723},"choice",[715,717,719,721],{"id":290,"label":716},"She should have done 45 − 3 first",{"id":292,"label":718},"She dropped the brackets: (2 + 8) should become 10",{"id":295,"label":720},"She should have done + before −",{"id":298,"label":722},"There is no mistake",[292],[725],"What should happen to (2 + 8) first?",{"correct":727,"incorrect":728},"Right. (2 + 8) = 10 first, then 3 × 10 = 30, then 45 − 30 = **15**.","The brackets were dropped in the first step. Correct: 45 − 3 × 10 = 45 − 30 = 15.",{"id":730,"type":55,"title":731,"eyebrow":732,"navLabel":733},"ch8","Calculators, phones and spreadsheets","Chapter 08","8 Calculators",{"id":735,"type":43,"markdown":736},"calcs","Machines have to be told the rule too, and not all of them are.\n\n- **Basic (four-function) calculators** usually use *immediate execution*: each time you press an operation key, they finish the previous step. So 2 + 3 × 4 = shows **20**. They are built for running totals (adding up a bill item by item).\n- **Scientific calculators** and most **phone calculator apps** wait until you press = and then follow the order of operations: 2 + 3 × 4 = shows **14**.\n- **Spreadsheets** (like Excel or Google Sheets) use * for × and \u002F for ÷. Typing =2+3*4 gives **14**, and =(2+3)*4 gives 20.\n- **Programming languages** such as Python and Scratch follow the order of operations too: 2 + 3 * 4 is 14.\n\nTo get the right answer from a basic calculator, work out the × and ÷ chunks first, or use its memory keys: **M+** adds the display to memory, **M−** subtracts it, **MR** recalls the total.",{"id":738,"type":446,"title":739,"items":740},"steps-memory","Working out 3 × 12 + 2 × 40 on a basic calculator",[741,745,749,753],{"title":742,"tag":743,"text":744},"Clear memory","MC","Press MC (or the key that clears memory) so the memory starts at 0.",{"title":746,"tag":747,"text":748},"First chunk","3 × 12 = M+","The display shows 36; M+ adds it to memory.",{"title":750,"tag":751,"text":752},"Second chunk","2 × 40 = M+","The display shows 80; memory now holds 116.",{"title":754,"tag":755,"text":756},"Recall","MR","The display shows 116, the correct total.",{"id":758,"type":286,"prompt":759,"options":760,"explanation":769},"predict-calc","On a **basic** calculator, Tara presses **20 − 4 × 3 =**. What does it show, and what is the correct value?",[761,763,765,767],{"id":290,"label":762},"Shows 48; correct is 8",{"id":292,"label":764},"Shows 8; correct is 8",{"id":295,"label":766},"Shows 48; correct is 48",{"id":298,"label":768},"Shows 8; correct is 48","**Shows 48; correct is 8.** The basic calculator does 20 − 4 = 16, then 16 × 3 = 48. The correct value multiplies first: 20 − 4 × 3 = 20 − 12 = 8.",{"id":771,"type":49,"variant":772,"title":773,"markdown":774},"tryit-calc","try_it","Test the calculators at home","Find as many calculators as you can: a basic one, a phone app, a computer's calculator, a scientific calculator, a spreadsheet.\n\n1. Type 2 + 3 × 4 = into each. Which show 14 and which show 20?\n2. Try 10 − 3 + 2 =. Does any calculator show something other than 9?\n3. Try 8 ÷ 4 × 2 =. All should give 4.\n\nMake a table of your results. Now you know which calculators you can trust with a whole expression.",{"id":776,"type":55,"title":777,"eyebrow":778,"navLabel":779},"ch9","Checking your answer","Chapter 09","9 Checking",{"id":781,"type":43,"markdown":782},"checking","Good mathematicians check. Three quick checks for this topic:\n\n1. **Estimate first.** 38 × 5 + 97 is about 40 × 5 + 100 = 300. The exact answer, 287, is close. If you had got 3,876 (by adding 5 + 97 first), the estimate would tell you at once that something went wrong.\n2. **Put the brackets in yourself.** Rewrite the expression with brackets around every × and ÷ chunk: 38 × 5 + 97 becomes (38 × 5) + 97. Then there is only one way to read it.\n3. **Use meaning.** If the expression came from a story, does the answer fit the story? Change cannot be more than the note you paid with; a share cannot be bigger than the whole.",{"id":784,"type":49,"variant":785,"title":786,"markdown":787},"example-estimate","example","Estimating catches the letter trap","Someone claims 250 − 50 + 100 = 100. Estimate: start with 250, take away a small amount, add a bigger amount. The answer must be **more** than 250! The correct value is 250 − 50 + 100 = 200 + 100 = 300. The claimed 100 came from doing 50 + 100 first.",{"id":789,"type":55,"title":790,"eyebrow":791,"navLabel":792},"ch-sheets","Spreadsheet formulas","Chapter 10","10 Spreadsheets",{"id":794,"type":43,"markdown":795},"sheets","Shops, schools and offices keep bills in **spreadsheets** such as Excel, Google Sheets or LibreOffice Calc. A formula starts with **=**, uses * for × and \u002F for ÷, and can refer to other cells by their address (like B2, meaning column B, row 2).\n\nSpreadsheets follow the order of operations, so a formula means exactly what the same expression means on paper:\n\n- =2+3*4 gives **14**.\n- =(2+3)*4 gives **20**.\n\nImagine a kirana bill where column B holds the quantity and column C the price. Row 2 is rice (5 kg at ₹60), row 3 is dal (2 kg at ₹110) and row 4 is oil (1 bottle at ₹180). The total is\n\n**=B2*C2+B3*C3+B4*C4**, which is 5 × 60 + 2 × 110 + 1 × 180 = ₹700.\n\nNo brackets are needed, because every product is done before the adding, exactly as on paper.",{"id":797,"type":71,"caption":798,"columns":799,"rows":804},"tbl-sheets","Spreadsheet formulas and what they compute",[800,801,802,803],"Formula","Means","With B2 = 5, C2 = 60, B3 = 2, C3 = 110","Correct?",[805,810,815,820,825,830],[806,807,808,809],"=B2*C2+B3*C3","5 × 60 + 2 × 110","520","Yes: total of two lines",[811,812,813,814],"=(B2+B3)*C2","(5 + 2) × 60","420","Only if both items cost ₹60",[816,817,818,819],"=(C2+C3)\u002F2","(60 + 110) ÷ 2","85","Yes: average price",[821,822,823,824],"=C2+C3\u002F2","60 + 110 ÷ 2","115","No: halves only the dal price",[826,827,828,829],"=500-(B2*C2+B3*C3)","500 − (300 + 220)","-20","Yes: change (negative means ₹20 more is needed)",[831,832,813,833],"=500-B2*C2+B3*C3","500 − 300 + 220","No: adds the dal instead of subtracting",{"id":835,"type":49,"variant":364,"title":836,"markdown":837},"careful-average","The most common spreadsheet slip","To average two cells you must write **=(C2+C3)\u002F2**. Writing =C2+C3\u002F2 divides only C3 by 2 and then adds C2, because \u002F is done before +. With C2 = 60 and C3 = 110 you get 115 instead of the correct 85.\n\nThis mistake has turned up in real business spreadsheets. The fix is always the same: when a whole total is divided, put the total in brackets.",{"id":839,"type":104,"title":840,"problem":841,"steps":842},"we-sheet-avg","Average marks in a spreadsheet","A teacher has marks 72, 85 and 91 in cells B2, B3 and B4. Write a correct formula for the average and evaluate it. What would =B2+B3+B4\u002F3 give?",[843,844,845,846],"The average is the total divided by the number of values: (72 + 85 + 91) ÷ 3.","Formula: **=(B2+B3+B4)\u002F3**.","Brackets: 72 + 85 + 91 = 248. Then 248 ÷ 3 = 82⅔, which the spreadsheet shows as about 82.67.","Without brackets: =B2+B3+B4\u002F3 means 72 + 85 + 91 ÷ 3 = 72 + 85 + 30⅓ = 187⅓. That is bigger than every single mark, which an average can never be. The size check catches the error.",{"id":848,"type":116,"itemId":849,"prompt":850,"check":851,"hints":862,"feedback":864},"pr-sheet","order-of-operations.understand-spreadsheet","Cells B2 and B3 hold the prices of two notebooks. Which formula gives the price of **3 of each**?",{"kind":713,"options":852,"correct":861},[853,855,857,859],{"id":290,"label":854},"=3*B2+B3",{"id":292,"label":856},"=3*(B2+B3)",{"id":295,"label":858},"=3+B2*B3",{"id":298,"label":860},"=B2+B3*3",[292],[863],"3 of each means 3 × (one of each).",{"correct":865,"incorrect":866},"Yes: =3*(B2+B3), or equally =3*B2+3*B3.","=3*B2+B3 buys 3 of the first but only 1 of the second. You need the bracket: =3*(B2+B3), which equals 3 × B2 + 3 × B3.",{"id":868,"type":55,"title":869,"eyebrow":870,"navLabel":871},"ch-errors","Mistakes real students make","Chapter 11","11 Real mistakes",{"id":873,"type":43,"markdown":874},"real-errors","Teachers who mark thousands of answers see the same order-of-operations errors again and again. Knowing them by name is the fastest way to stop making them. Each callout below is a real, common error, with the thinking behind it and the fix.",{"id":876,"type":49,"variant":178,"title":877,"markdown":878},"misc-running-equals","The “running equals” chain","A student works out 3 + 4 × 2 on paper like this: **3 + 4 = 7 × 2 = 14**. Each small step looks fine, but the line claims that 3 + 4 equals 7 × 2, which is false (7 ≠ 14).\n\nThis habit comes from calculators, where you keep pressing keys. On paper, the = sign means **the two sides have the same value**. Write the whole expression on every line instead: 3 + 4 × 2 = 3 + 8 = 11. Notice that the running chain also got the wrong answer (14), because it added first.",{"id":880,"type":49,"variant":178,"title":881,"markdown":882},"misc-right-to-left","Working from the right","Some students, told that × comes before +, start at the **right-hand end** and work backwards. For 20 − 6 ÷ 2 + 1 they do 2 + 1 = 3 first, then 6 ÷ 3 = 2, then 20 − 2 = 18. The correct value is 20 − 6 ÷ 2 + 1 = 20 − 3 + 1 = 17 + 1 = 18.\n\nThe fix: × and ÷ are found wherever they are, but partners are always taken **from the left**.",{"id":884,"type":49,"variant":178,"title":885,"markdown":886},"misc-first-term","Multiplying only the first number in the bracket","In 5 × (4 + 3) the 5 multiplies the **whole** bracket. A common slip is 5 × 4 + 3 = 23. The bracket first gives 5 × 7 = **35**; or, using the distributive property, 5 × 4 + 5 × 3 = 20 + 15 = 35.\n\nIf you want to open the bracket, every number inside must be multiplied.",{"id":888,"type":49,"variant":178,"title":889,"markdown":890},"misc-minus-bracket","Losing a sign after a minus bracket","12 − (5 − 2) is 12 − 3 = **9**. A student who removes the bracket and writes 12 − 5 − 2 = 5 has forgotten that taking away (5 − 2) means taking away 5 and giving back 2: 12 − 5 + 2 = 9. When in doubt, work out the bracket first instead of removing it.",{"id":892,"type":49,"variant":364,"title":893,"markdown":894},"misc-two-steps","Doing two steps at once","Strong students sometimes lose marks by doing several steps in their head: 36 ÷ 4 × 3 − 2 × 5 → \"27 − 10\" is fine, but under pressure \"36 ÷ 12 − 10\" slips out. Write **one step per line** until the method is automatic, and in exams always show the line where each × or ÷ chunk is worked out.",{"id":896,"type":897,"title":898,"prompt":899,"options":900},"explorer-first-step","explorer","What do I do first?","Pick the kind of expression you are looking at.",[901,912,919,929,941],{"id":902,"label":903,"chain":904,"badge":908,"note":911},"only-plus-minus","Only + and −",[905,906,907],"Start at the left","Do each + or − in turn","One number left",{"text":909,"tone":910},"Left to right","yes","Example: 45 − 18 + 7 − 10 = 27 + 7 − 10 = 34 − 10 = 24. Do not add 18 + 7 first.",{"id":913,"label":914,"chain":915,"badge":917,"note":918},"only-times-divide","Only × and ÷",[905,916,907],"Do each × or ÷ in turn",{"text":909,"tone":910},"Example: 90 ÷ 3 × 2 ÷ 5 = 30 × 2 ÷ 5 = 60 ÷ 5 = 12.",{"id":920,"label":921,"chain":922,"badge":926,"note":928},"mixed","Mixed, no brackets",[923,924,925],"Find the terms","Do × and ÷ in each term","Then + and −, left to right",{"text":927,"tone":910},"Terms first","Example: 8 + 6 × 5 − 12 ÷ 4 has terms 8, 6 × 5 and 12 ÷ 4. Term values 8, 30 and 3; then 8 + 30 − 3 = 35.",{"id":930,"label":931,"chain":932,"badge":938,"note":940},"brackets","With brackets",[933,934,935,936,937],"Find the innermost bracket","Use the usual order inside it","Replace it by its value","Repeat","Finish outside",{"text":939,"tone":910},"Innermost first","Example: 3 × [20 − (4 + 2) × 3] → (4 + 2) = 6 → [20 − 18] = 2 → 3 × 2 = 6.",{"id":407,"label":942,"chain":943,"badge":947,"note":949},"With “of”",[944,945,946],"Do each “of” as a chunk","Then ÷ and × left to right","Then + and −",{"text":948,"tone":910},"BODMAS O","Example: 50 − ½ of 30 × 2 → ½ of 30 = 15 → 15 × 2 = 30 → 50 − 30 = 20. Brackets would make it clearer: 50 − (½ × 30) × 2.",{"id":951,"type":116,"itemId":952,"prompt":953,"check":954,"hints":955,"feedback":958},"pr-running","order-of-operations.understand-running-equals","Karan wrote **10 + 5 = 15 × 2 = 30** for 10 + 5 × 2. What is the correct value of **10 + 5 × 2**?",{"kind":120,"answer":326,"tolerance":122},[956,957],"× first.","5 × 2 = 10.",{"correct":959,"incorrect":960},"Yes: 10 + 5 × 2 = 10 + 10 = 20. Karan both added first and misused the = sign.","Multiply first: 5 × 2 = 10, then 10 + 10 = 20. Also, \"10 + 5 = 15 × 2\" is a false statement: 15 is not 30.",{"id":962,"type":55,"title":963,"eyebrow":964,"navLabel":965},"ch-recipe","Recipes, costs and sharing","Chapter 12","12 Recipe costs",{"id":967,"type":43,"markdown":968},"recipe","A class is cooking **suji halwa** for a school fair. The recipe for one batch needs 1 kg of suji (₹50), 1 kg of sugar (₹45) and 500 g of ghee (₹300). They make **4 batches** and sell the halwa in **20 cups**. What must each cup cost just to cover the ingredients?\n\n- Cost of one batch: 50 + 45 + 300.\n- Cost of 4 batches: 4 × (50 + 45 + 300).\n- Cost per cup: **4 × (50 + 45 + 300) ÷ 20**.\n\n4 × (50 + 45 + 300) ÷ 20 = 4 × (95 + 300) ÷ 20 = 4 × 395 ÷ 20 = 1,580 ÷ 20 = 79. So each cup must be sold for at least **₹79** to break even.",{"id":970,"type":104,"title":971,"problem":972,"steps":973},"we-recipe-scale","Scaling a recipe for guests","A dal recipe for **4 people** uses 200 g of dal, 600 ml of water and 10 g of salt. How much of each is needed for **10 people**, and what is the total weight of dal and salt?",[974,975,976,977,978,979],"Scale each amount by 10 people ÷ 4 people. It is safest to multiply first, then divide, so every step stays whole: amount × 10 ÷ 4.","Dal: 200 × 10 ÷ 4 = 2,000 ÷ 4 = 500 g.","Water: 600 × 10 ÷ 4 = 6,000 ÷ 4 = 1,500 ml.","Salt: 10 × 10 ÷ 4 = 100 ÷ 4 = 25 g.","Total dal and salt: (200 + 10) × 10 ÷ 4 = 2,100 ÷ 4 = 525 g, the same as 500 + 25. ✓","Note: 10 ÷ 4 × 200 also equals 500, but its first step gives 2½. Multiplying first avoids the fraction; the answer is the same because × and ÷ can be regrouped as long as each ÷ stays attached to its own number.",{"id":981,"type":71,"caption":982,"columns":983,"rows":985},"tbl-recipe-costs","Costing a class cooking day",[984,74,77],"Question",[986,990,994,998,1002],[987,988,989],"Cost of one batch of halwa","50 + 45 + 300","₹395",[991,992,993],"Cost of 4 batches","4 × (50 + 45 + 300)","₹1,580",[995,996,997],"Cost per cup (20 cups)","4 × (50 + 45 + 300) ÷ 20","₹79",[999,1000,1001],"Profit if 20 cups sell at ₹100","20 × 100 − 4 × (50 + 45 + 300)","₹420",[1003,1004,1005],"Profit shared by 6 cooks","(20 × 100 − 4 × (50 + 45 + 300)) ÷ 6","₹70 each",{"id":1007,"type":116,"itemId":1008,"prompt":1009,"check":1010,"hints":1012,"feedback":1015},"pr-profit","order-of-operations.understand-profit","A stall sells **30 cups of chai at ₹15**. Ingredients cost **₹150** and the gas cylinder share is **₹60**. Write the profit as **30 × 15 − (150 + 60)** and evaluate it.",{"kind":120,"answer":1011,"tolerance":122,"unit":548},240,[1013,1014],"Sales: 30 × 15 = 450.","Costs in brackets: 150 + 60 = 210.",{"correct":1016,"incorrect":1017},"Yes: 450 − 210 = ₹240.","Brackets first: 150 + 60 = 210. Then 30 × 15 = 450, and 450 − 210 = ₹240. Without brackets, 450 − 150 + 60 = 360 would add the gas cost instead of subtracting it.",{"id":1019,"type":55,"title":1020,"eyebrow":1021,"navLabel":1022},"ch10","Words, check-up and summary","Chapter 13","13 Wrap-up",{"id":1024,"type":1025,"title":1026,"terms":1027},"glossary-understand","glossary","Vocabulary for order of operations",[1028,1031,1034,1037,1040,1043,1047,1050,1053,1057,1061,1064,1067,1070,1074],{"term":1029,"meaning":1030,"example":85},"Arithmetic expression","Numbers combined with operation signs (+, −, ×, ÷) and possibly brackets. It has one value.",{"term":67,"meaning":1032,"example":1033},"A part of an expression separated from the rest by a + or − that is not inside brackets.","In 5 × 6 − 3 the terms are 5 × 6 and 3.",{"term":462,"meaning":1035,"example":1036},"Find the value of an expression.","Evaluating 6 + 8 ÷ 2 gives 10.",{"term":1038,"meaning":1039},"Simplify","Rewrite an expression in a shorter form; for a number expression, reduce it to one number.",{"term":1041,"meaning":1042},"Precedence","How strongly an operation binds, i.e. which level it belongs to. × and ÷ have higher precedence than + and −.",{"term":1044,"meaning":1045,"example":1046},"Left-to-right rule","Operations of equal precedence are done in the order they appear, reading from the left.","8 ÷ 4 × 2 = 4",{"term":1048,"meaning":1049},"Parentheses","Round brackets ( ). Also called small brackets.",{"term":1051,"meaning":1052},"Braces","Curly brackets { }.",{"term":1054,"meaning":1055,"example":1056},"Square brackets","Box brackets [ ], often used outside round and curly brackets.","[20 − (3 + 5)] ÷ 4 = 3",{"term":1058,"meaning":1059,"example":1060},"Nested brackets","Brackets inside other brackets. Work from the innermost outwards.","[50 − {12 + (18 − 6) ÷ 4}]",{"term":1062,"meaning":1063},"Of","Means multiply, as in ½ of 10 = 5. In BODMAS it is done before ordinary × and ÷.",{"term":1065,"meaning":1066,"example":131},"Mnemonic","A memory aid such as a word made of first letters.",{"term":1068,"meaning":1069},"Indices \u002F exponents","Powers, like the 2 in 5² = 25. Done after brackets and before × and ÷.",{"term":1071,"meaning":1072,"example":1073},"Immediate execution","How basic calculators work: each step is finished as soon as the next key is pressed, ignoring the order of operations.","2 + 3 × 4 → 20 on a basic calculator",{"term":1075,"meaning":1076,"example":1077},"Distributive property","a × (b + c) = a × b + a × c. A property of numbers, not one of the DMAS letters.","3 × (4 + 5) = 12 + 15 = 27",{"id":1079,"type":1080,"title":1081,"questions":1082},"quiz-understand","quiz","Check your understanding",[1083,1095,1106,1118,1131,1141,1154,1165,1178,1191],{"itemId":1084,"prompt":1085,"options":1086,"correct":290,"why":1094},"order-of-operations.uq-1","Evaluate 18 − 12 ÷ 4 × 2.",[1087,1088,1090,1092],{"id":290,"label":264},{"id":292,"label":1089},"16.5",{"id":295,"label":1091},"3",{"id":298,"label":1093},"0","÷ and × left to right: 12 ÷ 4 = 3, 3 × 2 = 6. Then 18 − 6 = 12.",{"itemId":1096,"prompt":1097,"options":1098,"correct":292,"why":1105},"order-of-operations.uq-2","How many terms does 4 × 5 − 6 ÷ 2 + (3 + 1) have?",[1099,1101,1102,1104],{"id":290,"label":1100},"2",{"id":292,"label":1091},{"id":295,"label":1103},"4",{"id":298,"label":254},"The top-level + and − split it into 4 × 5, 6 ÷ 2 and (3 + 1). The + inside the bracket does not count.",{"itemId":1107,"prompt":1108,"options":1109,"correct":292,"why":1117},"order-of-operations.uq-3","What does DMAS stand for?",[1110,1112,1113,1115],{"id":290,"label":1111},"Distributive, Multiplicative, Additive, Subtractive",{"id":292,"label":149},{"id":295,"label":1114},"Digits, Multiples, Add, Subtract",{"id":298,"label":1116},"Divide, Minus, Add, Sum","DMAS = Division, Multiplication, Addition, Subtraction. The distributive property is a separate idea.",{"itemId":1119,"prompt":1120,"options":1121,"correct":295,"why":1130},"order-of-operations.uq-4","Which is true?",[1122,1124,1126,1128],{"id":290,"label":1123},"In BODMAS, all divisions are done before any multiplication",{"id":292,"label":1125},"In PEMDAS, all multiplications are done before any division",{"id":295,"label":1127},"× and ÷ have equal rank and are done left to right",{"id":298,"label":1129},"+ is always done before −","Both words describe the same rule: × and ÷ share a level, and so do + and −.",{"itemId":1132,"prompt":1133,"options":1134,"correct":290,"why":1140},"order-of-operations.uq-5","Simplify [30 − (4 + 2) × 3] ÷ 4.",[1135,1136,1137,1139],{"id":290,"label":1091},{"id":292,"label":269},{"id":295,"label":1138},"6",{"id":298,"label":259},"(4 + 2) = 6; inside the square bracket 30 − 6 × 3 = 30 − 18 = 12; 12 ÷ 4 = 3.",{"itemId":1142,"prompt":1143,"options":1144,"correct":290,"why":1153},"order-of-operations.uq-6","Which expression gives the change from ₹200 after buying 4 notebooks at ₹35?",[1145,1147,1149,1151],{"id":290,"label":1146},"200 − 4 × 35",{"id":292,"label":1148},"(200 − 4) × 35",{"id":295,"label":1150},"200 − 4 + 35",{"id":298,"label":1152},"4 × 35 − 200","The notebooks cost 4 × 35 = 140, and × is done before −, so 200 − 4 × 35 = 60 already means the right thing.",{"itemId":1155,"prompt":1156,"options":1157,"correct":290,"why":1164},"order-of-operations.uq-7","In BODMAS-style working, what is 24 ÷ ½ of 12?",[1158,1159,1161,1162],{"id":290,"label":1103},{"id":292,"label":1160},"144",{"id":295,"label":259},{"id":298,"label":1163},"48","\"Of\" first: ½ of 12 = 6. Then 24 ÷ 6 = 4.",{"itemId":1166,"prompt":1167,"options":1168,"correct":292,"why":1177},"order-of-operations.uq-8","A basic calculator shows 48 for 20 − 4 × 3. Why?",[1169,1171,1173,1175],{"id":290,"label":1170},"It rounds",{"id":292,"label":1172},"It works out each step as keys are pressed",{"id":295,"label":1174},"It follows PEMDAS",{"id":298,"label":1176},"It does ÷ first","Basic calculators use immediate execution: 20 − 4 = 16, then 16 × 3 = 48. The correct value is 8.",{"itemId":1179,"prompt":1180,"options":1181,"correct":295,"why":1190},"order-of-operations.uq-9","Which working has a mistake?",[1182,1184,1186,1188],{"id":290,"label":1183},"9 + 6 ÷ 3 = 9 + 2 = 11",{"id":292,"label":1185},"9 − 6 + 3 = 3 + 3 = 6",{"id":295,"label":1187},"9 − 6 + 3 = 9 − 9 = 0",{"id":298,"label":1189},"(9 + 6) ÷ 3 = 15 ÷ 3 = 5","9 − 6 + 3 must go left to right: 9 − 6 = 3, then 3 + 3 = 6. Doing the + first is the letter trap.",{"itemId":1192,"prompt":1193,"options":1194,"correct":290,"why":1202},"order-of-operations.uq-10","Simplify 2 × {15 − (3 + 2) × 2}.",[1195,1197,1199,1200],{"id":290,"label":1196},"10",{"id":292,"label":1198},"20",{"id":295,"label":91},{"id":298,"label":1201},"26","(3 + 2) = 5; inside the curly bracket 15 − 5 × 2 = 15 − 10 = 5; 2 × 5 = 10.",{"id":1204,"type":1205,"title":1206,"points":1207},"cheat-understand","summary","Cheat sheet",[1208,1209,1210,1211,1212,1213,1214,1215,1216,1217],"**Levels:** brackets (innermost first) → \"of\" → × and ÷ (left to right) → + and − (left to right).","**Terms:** the top-level + and − split an expression into terms. Evaluate each term, then combine left to right.","**Memory words:** DMAS, BODMAS, BIDMAS, PEMDAS, BEDMAS all mean the same rule. D\u002FM share a level; A\u002FS share a level.","**DMAS** = Division, Multiplication, Addition, Subtraction. The distributive property is a different topic.","**Why left to right:** − and ÷ belong to the number just after them. 10 − 3 + 2 = 9; 8 ÷ 4 × 2 = 4; 36 ÷ 6 ÷ 2 = 3.","**Brackets:** ( ) round, { } curly, [ ] square. Shape does not rank; innermost goes first.","**\"Of\":** means ×. By the BODMAS textbook convention it is done before ordinary ÷ and ×; that is an agreement, not a law. Clearer: use brackets.","**Word problems:** find the chunks (how many × each), join them, bracket any total that is shared or taken away.","**Calculators:** basic ones use immediate execution (2 + 3 × 4 → 20); use chunks or M+ \u002F MR. Scientific, phones and spreadsheets follow the rule.","**Check:** estimate, add your own brackets around chunks, and ask whether the answer fits the story.",{"id":1219,"type":1220,"conceptId":1221,"relation":1222,"explanation":1223},"conn-properties","connection","properties-of-numbers","helps_understand","The distributive property, 3 × (4 + 5) = 3 × 4 + 3 × 5, is a separate topic from DMAS but explains why × is done before +.",{"id":1225,"type":1220,"conceptId":1226,"relation":1222,"explanation":1227},"conn-four-ops","four-operations","Word problems here are built from the four operations; each chunk is a multiplication or division from that topic.",{"id":1229,"type":1220,"conceptId":1230,"relation":1231,"explanation":1232},"conn-hcf","hcf-and-lcm","applied_in","Long HCF and LCM calculations, like 2 × 2 × 3 × 5, and checks such as HCF × LCM = product, are expressions that rely on the order of operations.",{"id":1234,"type":1235,"sourceIds":1236},"sources-understand","sources",[1237,1238,1239,1240,1241,1242],"order-of-operations-ncert-class7","order-of-operations-wikipedia","order-of-operations-mathsisfun-bodmas","order-of-operations-mathsisfun-pemdas","order-of-operations-khan-intro","order-of-operations-openstax-prealgebra",[1237,1238,1239,1240,1241,1242],"needs_review",{"generatedBy":1246,"notes":1247},"claude-code","Draft generated with Python-checked arithmetic (× ÷ left to right, then + − left to right); pending owner review.","673794d3f205ac0bee859d95cc165caac9d27ad4581717e14ded52fab59abc07",{"logic:practice":1250,"component:match-pairs@1":1251,"component:order-ops@1":1252,"component:arith-sprint@1":1253,"component:sort-game@1":1254,"source:order-of-operations-khan-intro":1255,"source:order-of-operations-mathsisfun-bodmas":1256,"source:order-of-operations-mathsisfun-pemdas":1257,"source:order-of-operations-ncert-class7":1258,"source:order-of-operations-openstax-prealgebra":1259,"source:order-of-operations-wikipedia":1260},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","3d630ebc066352df791b020d545e1876ce4620d55c8c1c082414d6b4717255c5","c0c63ed40e1bca5ba6d446d43886b887a3a7709e5cd6679419f64fd3ba62afe6","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","44599222ec19c395d411822012f005a3d35a5fcd89e9d519498d4099163b78b4","92959283b293565e5dbcac7d99680a63df55eef50520c997ab808634afaa0062","7d20cb0b6dc5cf2de499e3df9053481fefcc10be6aebbdbd74e72664e0b2a45a","e7cadfedc70b54014195d6d8bc2740fb10a7e8aed3509c11f1efbde6e7698ade","a23188943c5655abd93ffecc7ea2a47277d0ffeae0cf3637ec684845500e891a","88ccf3653f7d9723bbb33a67a6548ccff64980f752cba0cd94a32bc60f72bd44",{"state":1262,"reviewer":1263,"selfReview":313,"reviewedAt":1264,"method":1265},"approved","The library owner","2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597821]