[{"data":1,"prerenderedAt":938},["ShallowReactive",2],{"questions:four-operations":3},{"bank":4,"contentHash":926,"dependencyHashes":927,"releaseId":937},{"schemaVersion":5,"conceptId":6,"revision":5,"title":7,"intro":8,"sections":9,"questions":42,"sourceIds":913,"reviewStatus":922,"authoring":923},1,"four-operations","Four operations: question bank","Practise adding, subtracting, multiplying and dividing: column methods with large numbers, remainders, choosing the right operation, multi-step problems in Indian contexts, estimating and checking, money and units, and the vocabulary of the operations. Every question has a worked solution. Try it first, then compare your method. Prices, crop yields, farm costs and bills in these questions are illustrative round figures chosen to make the arithmetic clear, not real market rates.",[10,14,18,22,26,30,34,38],{"id":11,"title":12,"description":13},"add-sub","Addition and subtraction","Column addition and subtraction with carrying and borrowing, up to lakhs and crores, missing numbers and spotting mistakes.",{"id":15,"title":16,"description":17},"multiplication","Multiplication","Multiplying by 10, 100 and 1,000, long multiplication, zeros in the multiplier and mental shortcuts.",{"id":19,"title":20,"description":21},"division","Division and remainders","Long division, zeros in the quotient, remainders and dividend = divisor × quotient + remainder.",{"id":23,"title":24,"description":25},"choosing","Choosing the operation","Deciding which operation a story needs from its structure, including problems that use misleading clue words.",{"id":27,"title":28,"description":29},"multi-step","Multi-step word problems","Kirana bills, trains, school trips, cricket, crops, elections and the unitary method.",{"id":31,"title":32,"description":33},"estimation","Estimation and checking","Rounding, estimating before calculating, inverse checks, last-digit and digit-sum checks.",{"id":35,"title":36,"description":37},"money-measure","Money and measurement","Operations with rupees and paise, km and m, kg and g, L and mL, converting units first.",{"id":39,"title":40,"description":41},"vocabulary","Vocabulary","Sum, difference, product, quotient, dividend, divisor, remainder, minuend, subtrahend, multiplicand, multiplier, addend.",[43,57,68,79,89,99,111,120,130,142,152,162,172,182,192,203,226,236,246,256,266,276,286,296,305,316,337,348,357,367,386,395,405,424,444,454,465,474,484,495,505,516,525,534,543,552,562,573,583,601,619,637,655,673,691,700,720,730,740,749,759,768,777,786,795,810,822,831,840,851,869,878,886,895],{"id":44,"section":11,"level":45,"prompt":46,"check":47,"hints":51,"solution":53,"skills":54},"four-operations.q001","foundation","Add: 4,567 + 2,385",{"kind":48,"answer":49,"tolerance":50},"number",6952,0,[52],"Start with the ones column and carry when a column makes 10 or more.","Line up the columns. Ones: 7 + 5 = 12, write 2 carry 1. Tens: 6 + 8 + 1 = 15, write 5 carry 1. Hundreds: 5 + 3 + 1 = 9. Thousands: 4 + 2 = 6. Answer: **6,952**. Estimate check: 4,600 + 2,400 = 7,000. ✓",[55,56],"column addition","carrying",{"id":58,"section":11,"level":45,"prompt":59,"check":60,"hints":62,"solution":64,"skills":65},"four-operations.q002","Subtract: 8,000 − 3,456",{"kind":48,"answer":61,"tolerance":50},4544,[63],"You need to borrow across three zeros, or use 7,999 − 3,455.","Borrowing across zeros is easier with a trick: take 1 from both numbers. 8,000 − 3,456 = 7,999 − 3,455. Now no borrowing is needed: 9 − 5 = 4, 9 − 5 = 4, 9 − 4 = 5, 7 − 3 = 4. Answer: **4,544**. Check: 4,544 + 3,456 = 8,000. ✓",[66,67],"column subtraction","borrowing",{"id":69,"section":11,"level":70,"prompt":71,"check":72,"hints":74,"solution":76,"skills":77},"four-operations.q003","core","Add: 3,45,678 + 2,78,965",{"kind":48,"answer":73,"tolerance":50},624643,[75],"There is a carry in almost every column.","Ones 8 + 5 = 13 (write 3, carry 1). Tens 7 + 6 + 1 = 14 (4, carry 1). Hundreds 6 + 9 + 1 = 16 (6, carry 1). Thousands 5 + 8 + 1 = 14 (4, carry 1). Ten-thousands 4 + 7 + 1 = 12 (2, carry 1). Lakhs 3 + 2 + 1 = 6. Answer: **6,24,643** (six lakh twenty-four thousand six hundred forty-three).",[55,78],"large numbers",{"id":80,"section":11,"level":70,"prompt":81,"check":82,"hints":84,"solution":86,"skills":87},"four-operations.q004","Subtract: 6,00,000 − 2,45,789",{"kind":48,"answer":83,"tolerance":50},354211,[85],"Borrowing across five zeros is error-prone. Try making it 5,99,999 − 2,45,788.","Take 1 from both: 5,99,999 − 2,45,788. No borrowing needed now: 9−8 = 1, 9−8 = 1, 9−7 = 2, 9−5 = 4, 9−4 = 5, 5−2 = 3. Answer: **3,54,211**. Check: 3,54,211 + 2,45,789 = 6,00,000. ✓",[88],"borrowing across zeros",{"id":90,"section":11,"level":70,"prompt":91,"check":92,"hints":94,"solution":96,"skills":97},"four-operations.q005","Arjun wrote **5,003 − 1,678 = 4,675**. What is the correct answer?",{"kind":48,"answer":93,"tolerance":50},3325,[95],"Check Arjun's answer by adding it back to 1,678.","Arjun's answer is too big. Check by adding: 4,675 + 1,678 = 6,353, not 5,003. He subtracted the smaller digit from the bigger one in each column (8 − 3 = 5, 7 − 0 = 7, 6 − 0 = 6) instead of borrowing. Borrow properly: the 5 thousands become 4 thousands, 9 hundreds, 9 tens and 13 ones. Then ones 13 − 8 = 5, tens 9 − 7 = 2, hundreds 9 − 6 = 3, thousands 4 − 1 = 3. Answer: **3,325**. Check: 3,325 + 1,678 = 5,003. ✓",[98,67],"spot the mistake",{"id":100,"section":11,"level":101,"prompt":102,"check":103,"hints":105,"solution":107,"skills":108},"four-operations.q006","stretch","Find the missing number: **? − 38,476 = 51,829**",{"kind":48,"answer":104,"tolerance":50},90305,[106],"Subtraction is undone by addition.","The missing number is the one we started with before taking away 38,476. Undo the subtraction by adding: 51,829 + 38,476 = **90,305**. Check: 90,305 − 38,476 = 51,829. ✓",[109,110],"inverse operations","missing number",{"id":112,"section":11,"level":101,"prompt":113,"check":114,"hints":116,"solution":118,"skills":119},"four-operations.q007","Add 1,23,45,678 and 87,65,432. Write the answer in the Indian system.",{"kind":48,"answer":115,"tolerance":50},21111110,[117],"Look at pairs of digits in each column: 8 + 2, 7 + 3, 6 + 4…","Every column (except the last) adds to 10, so each produces a 0 plus a carry, and each carry turns the next column's 10 into 11: 8 + 2 = 10 → 0 carry 1; 7 + 3 + 1 = 11 → 1 carry 1; and so on. The crore column is 1 + 0 + 1 = 2. Answer: **2,11,11,110** (two crore eleven lakh eleven thousand one hundred ten).",[78,56],{"id":121,"section":11,"level":70,"prompt":122,"check":123,"hints":125,"solution":127,"skills":128},"four-operations.q008","What is the difference between the largest and the smallest **5-digit numbers that use five different digits**?",{"kind":48,"answer":124,"tolerance":50},88531,[126],"The smallest cannot start with 0.","Largest 5-digit number with different digits: use the biggest digits in order, **98,765**. Smallest: the first digit cannot be 0, so start with 1, then 0, 2, 3, 4: **10,234**. Difference: 98,765 − 10,234 = **88,531**.",[129,66],"place value",{"id":131,"section":11,"level":132,"prompt":133,"check":134,"hints":136,"solution":139,"skills":140},"four-operations.q009","challenge","Use each of the digits 1, 2, 3, 4, 5, 6 exactly once to make two 3-digit numbers. What is the **smallest possible positive difference** between them?",{"kind":48,"answer":135,"tolerance":50},47,[137,138],"Make the hundreds digits differ by only 1.","Then make the bigger number's last two digits as small as possible.","To make the difference small, the hundreds digits should be neighbours (differ by 1). Then the bigger number should have the **smallest** possible tens-and-ones, and the smaller number the **largest**. For example with hundreds 4 and 3: the larger is 4 _ _ with the smallest remaining digits (1, 2) → 412, and the smaller is 3 _ _ with the largest remaining (6, 5) → 365. 412 − 365 = **47**. Checking every arrangement (a computer tried all 720) confirms that 47 is the smallest.",[141,129],"reasoning",{"id":143,"section":11,"level":101,"prompt":144,"check":145,"hints":147,"solution":149,"skills":150},"four-operations.q010","Three wards of a town have populations 1,25,640, 98,755 and 87,960. Compare the largest ward with the other two wards put together. By how many people do they differ?",{"kind":48,"answer":146,"tolerance":50},61075,[148],"First add the two smaller wards.","Other two together: 98,755 + 87,960 = 1,86,715. Largest ward: 1,25,640. Since 1,86,715 is bigger, the largest ward has **fewer** people than the other two together, by 1,86,715 − 1,25,640 = **61,075**.",[27,151],"comparing",{"id":153,"section":15,"level":45,"prompt":154,"check":155,"hints":157,"solution":159,"skills":160},"four-operations.q011","Multiply: 475 × 100",{"kind":48,"answer":156,"tolerance":50},47500,[158],"Each digit moves two places left.","Multiplying by 100 makes each digit worth 100 times as much, so every digit moves two places to the left: 4 hundreds become 4 ten-thousands, and so on. 475 × 100 = **47,500**.",[161],"multiply by 10, 100, 1000",{"id":163,"section":15,"level":45,"prompt":164,"check":165,"hints":167,"solution":169,"skills":170},"four-operations.q012","Multiply: 234 × 6",{"kind":48,"answer":166,"tolerance":50},1404,[168],"Work from the ones, carrying as you go.","6 × 4 = 24: write 4, carry 2. 6 × 3 = 18, + 2 = 20: write 0, carry 2. 6 × 2 = 12, + 2 = 14. Answer: **1,404**. Estimate: 230 × 6 ≈ 1,400. ✓",[171],"short multiplication",{"id":173,"section":15,"level":70,"prompt":174,"check":175,"hints":177,"solution":179,"skills":180},"four-operations.q013","Multiply: 368 × 47",{"kind":48,"answer":176,"tolerance":50},17296,[178],"Multiply by the 7 first, then by 40 (remember the 0 place-holder).","Partial products: 368 × 7 = 2,576; 368 × 40 = 14,720. Add: 2,576 + 14,720 = **17,296**. Estimate: 400 × 50 = 20,000, and the exact answer is a bit smaller, as expected because both numbers were rounded up.",[181],"long multiplication",{"id":183,"section":15,"level":70,"prompt":184,"check":185,"hints":187,"solution":189,"skills":190},"four-operations.q014","Multiply: 1,208 × 305",{"kind":48,"answer":186,"tolerance":50},368440,[188],"The 0 in 305 means there are no tens; the next row is × 300.","1,208 × 5 = 6,040. 1,208 × 0 tens = 0 (a row of zeros you can skip). 1,208 × 300 = 3,62,400. Total: 6,040 + 3,62,400 = **3,68,440**. Careful: skipping the zero row is fine, but the 300-row must still be shifted two places.",[181,191],"zeros",{"id":193,"section":15,"level":70,"prompt":194,"check":195,"hints":197,"solution":199,"skills":200},"four-operations.q015","Work out 250 × 48 in your head using halving and doubling.",{"kind":48,"answer":196,"tolerance":50},12000,[198],"250 × 4 = 1,000. How many 4s are there in 48?","Double 250 and halve 48: 500 × 24. Again: 1,000 × 12 = **12,000**. Halving one factor and doubling the other keeps the product the same.",[201,202],"mental strategies","halving and doubling",{"id":204,"section":15,"level":101,"prompt":205,"check":206,"hints":222,"solution":224,"skills":225},"four-operations.q016","Neha multiplied 456 × 23 like this: 456 × 3 = 1,368, 456 × 2 = 912, then 1,368 + 912 = 2,280. What went wrong?",{"kind":207,"options":208,"correct":221},"choice",[209,212,215,218],{"id":210,"label":211},"a","She added instead of multiplying",{"id":213,"label":214},"b","The 2 in 23 means 2 tens, so the second row should be 9,120, not 912",{"id":216,"label":217},"c","She should have multiplied 456 × 5",{"id":219,"label":220},"d","Nothing, 2,280 is correct",[213],[223],"What is the 2 in 23 really worth?","The 2 in 23 stands for 20. So the second partial product is 456 × 20 = 9,120. Correct total: 1,368 + 9,120 = **10,488**. An estimate shows the problem at once: 456 × 23 ≈ 500 × 20 = 10,000, so 2,280 is far too small.",[98,129],{"id":227,"section":15,"level":101,"prompt":228,"check":229,"hints":231,"solution":233,"skills":234},"four-operations.q017","What is the product of the largest 3-digit number and the largest 2-digit number?",{"kind":48,"answer":230,"tolerance":50},98901,[232],"99 = 100 − 1.","999 × 99. A neat way: 999 × 100 = 99,900, then take away one 999: 99,900 − 999 = **98,901**.",[201,235],"distributive",{"id":237,"section":15,"level":70,"prompt":238,"check":239,"hints":241,"solution":243,"skills":244},"four-operations.q018","Work out 25 × 37 × 4 without long multiplication.",{"kind":48,"answer":240,"tolerance":50},3700,[242],"Which two numbers multiply to make 100?","Multiplication can be done in any order, so pair 25 with 4 first: 25 × 4 = 100. Then 100 × 37 = **3,700**.",[201,245],"commutative",{"id":247,"section":15,"level":132,"prompt":248,"check":249,"hints":251,"solution":254,"skills":255},"four-operations.q019","Use each of the digits 2, 3, 4, 5, 6 exactly once to make a 3-digit number and a 2-digit number. What is the **largest** product you can make?",{"kind":48,"answer":250,"tolerance":50},34146,[252,253],"Put 6 and 5 in the two leading places.","Try a few arrangements and compare.","The two biggest digits, 6 and 5, should be the leading digits (hundreds of one number and tens of the other). Trying both ways: 6__ × 5_ and 5__ × 6_. Then place 4 and 3 as the next digits and 2 last. Testing the candidates: 642 × 53 = 34,026, 632 × 54 = 34,128, 542 × 63 = 34,146, 532 × 64 = 34,048. The largest is **542 × 63 = 34,146**. (A computer check of all 120 arrangements agrees.)",[141,181],{"id":257,"section":19,"level":45,"prompt":258,"check":259,"hints":261,"solution":263,"skills":264},"four-operations.q020","Divide: 84 ÷ 7",{"kind":48,"answer":260,"tolerance":50},12,[262],"Think of the 7 times table.","7 × 12 = 84, so 84 ÷ 7 = **12**. Division asks: how many 7s make 84?",[265],"division facts",{"id":267,"section":19,"level":45,"prompt":268,"check":269,"hints":271,"solution":273,"skills":274},"four-operations.q021","50 sweets are shared equally among 6 children. How many sweets are **left over**?",{"kind":48,"answer":270,"tolerance":50},2,[272],"Find the biggest multiple of 6 that is not more than 50.","6 × 8 = 48, which is the most that can be shared equally. 50 − 48 = **2** left over. So 50 ÷ 6 = 8 remainder 2: each child gets 8.",[275],"remainders",{"id":277,"section":19,"level":70,"prompt":278,"check":279,"hints":281,"solution":283,"skills":284},"four-operations.q022","Divide: 7,452 ÷ 6",{"kind":48,"answer":280,"tolerance":50},1242,[282],"Divide, multiply, subtract, bring down.","7 ÷ 6 = 1 r 1. Bring down 4: 14 ÷ 6 = 2 r 2. Bring down 5: 25 ÷ 6 = 4 r 1. Bring down 2: 12 ÷ 6 = 2 r 0. Answer: **1,242**. Check: 1,242 × 6 = 7,452. ✓",[285],"long division",{"id":287,"section":19,"level":70,"prompt":288,"check":289,"hints":291,"solution":293,"skills":294},"four-operations.q023","Divide: 6,324 ÷ 6",{"kind":48,"answer":290,"tolerance":50},1054,[292],"When a step does not go, write 0 in the quotient.","6 ÷ 6 = 1. Bring down 3: 3 ÷ 6 = **0** (write the 0 in the quotient!), remainder 3. Bring down 2: 32 ÷ 6 = 5 r 2. Bring down 4: 24 ÷ 6 = 4. Answer: **1,054**. Forgetting the 0 gives 154, which is far too small: 6 × 154 = 924.",[285,295],"zeros in quotient",{"id":297,"section":19,"level":70,"prompt":298,"check":299,"hints":301,"solution":303,"skills":304},"four-operations.q024","Find the **remainder** when 9,876 is divided by 23.",{"kind":48,"answer":300,"tolerance":50},9,[302],"Estimate each step with 23 ≈ 20 or 25, then adjust.","98 ÷ 23 = 4 (92), remainder 6. Bring down 7: 67 ÷ 23 = 2 (46), remainder 21. Bring down 6: 216 ÷ 23 = 9 (207), remainder **9**. So 9,876 ÷ 23 = 429 r 9. Check: 23 × 429 + 9 = 9,867 + 9 = 9,876. ✓",[285,275],{"id":306,"section":19,"level":101,"prompt":307,"check":308,"hints":310,"solution":312,"skills":313},"four-operations.q025","In a division, the divisor is 37, the quotient is 58 and the remainder is 29. What is the dividend?",{"kind":48,"answer":309,"tolerance":50},2175,[311],"Use dividend = divisor × quotient + remainder.","Dividend = divisor × quotient + remainder = 37 × 58 + 29 = 2,146 + 29 = **2,175**. The remainder 29 is less than 37, so this division is possible.",[314,315],"division algorithm","checking",{"id":317,"section":19,"level":101,"prompt":318,"check":319,"hints":333,"solution":335,"skills":336},"four-operations.q026","When a number is divided by 12, which of these could **not** be the remainder?",{"kind":207,"options":320,"correct":332},[321,323,325,327,329],{"id":210,"label":322},"0",{"id":213,"label":324},"7",{"id":216,"label":326},"11",{"id":219,"label":328},"12",{"id":330,"label":331},"e","13",[219,330],[334],"More than one answer is correct.","The remainder must always be **less than the divisor**, so dividing by 12 can leave only 0, 1, 2, …, 11. A remainder of 12 or 13 means another group of 12 could still be made.",[275,141],{"id":338,"section":19,"level":101,"prompt":339,"check":340,"hints":342,"solution":345,"skills":346},"four-operations.q027","What is the smallest whole number that must be **added** to 1,000 to make it exactly divisible by 17?",{"kind":48,"answer":341,"tolerance":50},3,[343,344],"Find the remainder first.","How far is 1,000 from the next multiple of 17?","1,000 ÷ 17 = 58 remainder 14 (17 × 58 = 986). The next multiple of 17 is 17 × 59 = 1003. So add 1003 − 1,000 = **3**.",[275,347],"multiples",{"id":349,"section":19,"level":132,"prompt":350,"check":351,"hints":353,"solution":355,"skills":356},"four-operations.q028","What is the largest 4-digit number that is exactly divisible by 47?",{"kind":48,"answer":352,"tolerance":50},9964,[354],"Start from 9,999 and remove the remainder.","Divide the largest 4-digit number, 9,999, by 47: 9,999 = 47 × 212 + 35. Subtract the remainder: 9,999 − 35 = **9,964**, which is 47 × 212.",[314,141],{"id":358,"section":19,"level":132,"prompt":359,"check":360,"hints":362,"solution":364,"skills":365},"four-operations.q029","A number leaves remainder 3 when divided by 5 and remainder 2 when divided by 7. What is the smallest such positive number?",{"kind":48,"answer":361,"tolerance":50},23,[363],"List numbers that are 2 more than a multiple of 7.","Numbers leaving remainder 2 on division by 7: 2, 9, 16, **23**, 30… Check each for remainder 3 on division by 5: 2 (r 2), 9 (r 4), 16 (r 1), **23 (r 3)** ✓. So the answer is **23**. The next one is 23 + 35 = 58, because 35 is the smallest number divisible by both 5 and 7.",[275,366],"puzzles",{"id":368,"section":23,"level":45,"prompt":369,"check":370,"hints":381,"solution":383,"skills":384},"four-operations.q030","36 children sit in 4 equal rows. Which calculation tells you how many are in each row?",{"kind":207,"options":371,"correct":380},[372,374,376,378],{"id":210,"label":373},"36 + 4",{"id":213,"label":375},"36 − 4",{"id":216,"label":377},"36 × 4",{"id":219,"label":379},"36 ÷ 4",[219],[382],"Is the total known? Are the groups equal?","Sharing a total into a known number of equal groups is division: 36 ÷ 4 = **9** in each row. Check: 4 × 9 = 36. ✓",[385],"choosing the operation",{"id":387,"section":23,"level":45,"prompt":388,"check":389,"hints":391,"solution":393,"skills":394},"four-operations.q031","A pencil costs ₹8. A pen costs ₹15 more than the pencil. How much does the pen cost?",{"kind":48,"answer":361,"tolerance":50,"unit":390},"₹",[392],"Which costs more?","The pen costs **more**, so add: 8 + 15 = **₹23**.",[385,151],{"id":396,"section":23,"level":70,"prompt":397,"check":398,"hints":400,"solution":402,"skills":403},"four-operations.q032","Asha has 45 stamps. That is 12 more than Bina has. How many stamps does Bina have?",{"kind":48,"answer":399,"tolerance":50},33,[401],"Who has more, Asha or Bina?","Careful: the word 'more' describes **Asha**. Bina has fewer, so subtract: 45 − 12 = **33**. Check: 33 + 12 = 45. ✓",[404,151],"keyword traps",{"id":406,"section":23,"level":70,"prompt":407,"check":408,"hints":419,"solution":421,"skills":422},"four-operations.q033","9 identical books cost ₹540 altogether. Which calculation gives the cost of **one** book?",{"kind":207,"options":409,"correct":418},[410,412,414,416],{"id":210,"label":411},"540 × 9",{"id":213,"label":413},"540 ÷ 9",{"id":216,"label":415},"540 − 9",{"id":219,"label":417},"540 + 9",[213],[420],"The total is known; you want the size of one equal part.","The total is shared equally over 9 books, so divide: 540 ÷ 9 = **₹60**. The word 'altogether' did not mean add.",[385,423],"unitary method",{"id":425,"section":23,"level":70,"prompt":426,"check":427,"hints":440,"solution":442,"skills":443},"four-operations.q034","Which of these problems are solved by **multiplication**? (Choose all that apply.)",{"kind":207,"options":428,"correct":439},[429,431,433,435,437],{"id":210,"label":430},"A box holds 24 bangles. How many bangles in 15 boxes?",{"id":213,"label":432},"360 mangoes are packed 24 to a crate. How many crates?",{"id":216,"label":434},"A ticket costs ₹35. What do 18 tickets cost?",{"id":219,"label":436},"Ravi scored 48 runs and Sunil 67. How many more did Sunil score?",{"id":330,"label":438},"A train covers 72 km every hour. How far in 5 hours?",[210,216,330],[441],"Look for equal groups where the total is missing.","(a) 24 × 15 = 360 bangles; (c) 35 × 18 = ₹630; (e) 72 × 5 = 360 km: each has equal groups with the **total** unknown. (b) is division: 360 ÷ 24 = 15 crates (the number of groups is unknown). (d) is subtraction: 67 − 48 = 19 (comparing).",[385],{"id":445,"section":23,"level":101,"prompt":446,"check":447,"hints":449,"solution":451,"skills":452},"four-operations.q035","After giving 125 marbles to her brother, Rani has 348 marbles left. How many did she have at first?",{"kind":48,"answer":448,"tolerance":50},473,[450],"Draw a bar: one part given away, one part kept.","The word 'left' tempts you to subtract, but the **start** is unknown. Start = what she gave + what she kept: 125 + 348 = **473**. Check: 473 − 125 = 348. ✓",[404,453],"working backwards",{"id":455,"section":23,"level":70,"prompt":456,"check":457,"hints":460,"solution":462,"skills":463},"four-operations.q036","Mount Everest is 8,849 m high and Kanchenjunga is 8,586 m high. How much higher is Everest?",{"kind":48,"answer":458,"tolerance":50,"unit":459},263,"m",[461],"Comparing asks for a difference.","Comparing two heights is subtraction: 8,849 − 8,586 = **263 m**. No 'minus' word appears, but 'how much higher' asks for the difference.",[151,464],"subtraction",{"id":466,"section":23,"level":101,"prompt":467,"check":468,"hints":470,"solution":472,"skills":473},"four-operations.q037","A mango costs ₹15. A papaya costs 4 times as much as a mango, and a watermelon costs ₹20 more than the papaya. What does the watermelon cost?",{"kind":48,"answer":469,"tolerance":50,"unit":390},80,[471],"Find the papaya's price first.","Two steps with two different operations. Papaya: '4 times as much' is scaling, so multiply: 15 × 4 = ₹60. Watermelon: '₹20 more' means add: 60 + 20 = **₹80**.",[385,27],{"id":475,"section":27,"level":70,"prompt":476,"check":477,"hints":479,"solution":481,"skills":482},"four-operations.q038","At a kirana shop, Farida buys 3 kg of rice at ₹56 per kg, 2 L of oil at ₹145 per litre and 5 soaps at ₹38 each. She pays with a ₹1,000 note. How much change does she get?",{"kind":48,"answer":478,"tolerance":50,"unit":390},352,[480],"Find the cost of each item first, then the total.","Rice: 3 × 56 = ₹168. Oil: 2 × 145 = ₹290. Soap: 5 × 38 = ₹190. Bill: 168 + 290 + 190 = ₹648. Change: 1,000 − 648 = **₹352**.",[27,483],"money",{"id":485,"section":27,"level":70,"prompt":486,"check":487,"hints":490,"solution":492,"skills":493},"four-operations.q039","A goods train travels at 78 km per hour for 6 hours, then at 64 km per hour for 3 hours. How far does it travel altogether?",{"kind":48,"answer":488,"tolerance":50,"unit":489},660,"km",[491],"Work out each part of the journey separately.","First part: 78 × 6 = 468 km. Second part: 64 × 3 = 192 km. Total: 468 + 192 = **660 km**.",[27,494],"distance",{"id":496,"section":27,"level":101,"prompt":497,"check":498,"hints":500,"solution":502,"skills":503},"four-operations.q040","48 students go on a trip. Museum tickets cost ₹150 each and the bus costs ₹6,000 in total. If the whole cost is shared equally by the students, how much does each student pay?",{"kind":48,"answer":499,"tolerance":50,"unit":390},275,[501],"Find the total cost, then share it among 48.","Tickets: 48 × 150 = ₹7,200. Total: 7,200 + 6,000 = ₹13,200. Each student: 13,200 ÷ 48 = **₹275**. Another way: each pays ₹150 for the ticket plus 6,000 ÷ 48 = ₹125 for the bus, and 150 + 125 = 275. ✓",[27,504],"budget",{"id":506,"section":27,"level":101,"prompt":507,"check":508,"hints":511,"solution":513,"skills":514},"four-operations.q041","In a 50-over match a team needs 300 runs to win. After 30 overs it has scored 160. How many runs per over must it score in the remaining overs?",{"kind":48,"answer":509,"tolerance":50,"unit":510},7,"runs per over",[512],"Two things are needed: runs left and overs left.","Runs still needed: 300 − 160 = 140. Overs left: 50 − 30 = 20. Required rate: 140 ÷ 20 = **7 runs per over**.",[27,515],"cricket",{"id":517,"section":27,"level":101,"prompt":518,"check":519,"hints":521,"solution":523,"skills":524},"four-operations.q042","A farmer grows wheat on 12 hectares. Each hectare gives 45 quintals, and wheat sells at ₹2,275 per quintal. How much money does the whole crop fetch?",{"kind":48,"answer":520,"tolerance":50,"unit":390},1228500,[522],"First find the total quintals.","Total wheat: 12 × 45 = 540 quintals. Money: 540 × 2,275 = **₹12,28,500** (twelve lakh twenty-eight thousand five hundred). Estimate: 500 × 2,000 = 10,00,000, and the answer is a little more. ✓",[27,78],{"id":526,"section":27,"level":70,"prompt":527,"check":528,"hints":530,"solution":532,"skills":533},"four-operations.q043","8 notebooks cost ₹360. At the same price, how much do 13 notebooks cost?",{"kind":48,"answer":529,"tolerance":50,"unit":390},585,[531],"First find the cost of one notebook.","Unitary method: one notebook costs 360 ÷ 8 = ₹45. Thirteen cost 45 × 13 = **₹585**.",[423],{"id":535,"section":27,"level":101,"prompt":536,"check":537,"hints":539,"solution":541,"skills":542},"four-operations.q044","A car travels 150 km on 10 litres of petrol. How far can it go on 26 litres?",{"kind":48,"answer":538,"tolerance":50,"unit":489},390,[540],"How far on one litre?","On 1 litre it goes 150 ÷ 10 = 15 km. On 26 litres: 15 × 26 = **390 km**.",[423],{"id":544,"section":27,"level":132,"prompt":545,"check":546,"hints":548,"solution":550,"skills":551},"four-operations.q045","A trader doubles his money, then spends ₹500. He doubles what is left, then spends ₹500 again. Now he has ₹700. How much did he start with?",{"kind":48,"answer":547,"tolerance":50,"unit":390},550,[549],"Start from ₹700 and undo each step in reverse order.","Work backwards, undoing each step with the inverse operation. End: ₹700. Undo 'spend ₹500': 700 + 500 = 1,200. Undo 'double': 1,200 ÷ 2 = 600. Undo 'spend ₹500': 600 + 500 = 1,100. Undo 'double': 1,100 ÷ 2 = **₹550**. Check forwards: 550 → 1,100 → 600 → 1,200 → 700. ✓",[453,109],{"id":553,"section":27,"level":132,"prompt":554,"check":555,"hints":557,"solution":559,"skills":560},"four-operations.q046","In a village election between two candidates, 1,25,640 valid votes were cast. The winner got 18,420 more votes than the loser. How many votes did the winner get?",{"kind":48,"answer":556,"tolerance":50},72030,[558],"Take away the lead, then halve what remains.","Remove the winner's lead of 18,420 votes and the two candidates would be equal: (1,25,640 − 18,420) ÷ 2 = 53,610 each. So the winner has 53,610 + 18,420 = **72,030**. Check: loser 53,610; sum 1,25,640 ✓; difference 18,420 ✓.",[561,78],"sum and difference",{"id":563,"section":27,"level":70,"prompt":564,"check":565,"hints":568,"solution":570,"skills":571},"four-operations.q047","A shared water tank holds 12,000 litres. Three families use 180 L, 240 L and 330 L a day. How many days will a full tank last?",{"kind":48,"answer":566,"tolerance":50,"unit":567},16,"days",[569],"First find how much water is used each day.","Daily use: 180 + 240 + 330 = 750 L. Days: 12,000 ÷ 750 = **16 days**. Check: 750 × 16 = 12,000. ✓",[27,572],"measurement",{"id":574,"section":31,"level":45,"prompt":575,"check":576,"hints":578,"solution":580,"skills":581},"four-operations.q048","Round 4,76,382 to the nearest thousand.",{"kind":48,"answer":577,"tolerance":50},476000,[579],"Look at the hundreds digit.","The thousands digit is 6. Look at the digit to its right, the hundreds digit: 3. That is less than 5, so round down: **4,76,000**.",[582],"rounding",{"id":584,"section":31,"level":45,"prompt":585,"check":586,"hints":597,"solution":599,"skills":600},"four-operations.q049","Which is the best estimate of 398 + 604?",{"kind":207,"options":587,"correct":596},[588,590,592,594],{"id":210,"label":589},"About 100",{"id":213,"label":591},"About 1,000",{"id":216,"label":593},"About 10,000",{"id":219,"label":595},"About 200",[213],[598],"Round each number to the nearest hundred.","398 is about 400 and 604 is about 600, so the sum is about 1,000. Exact: 1002.",[31],{"id":602,"section":31,"level":70,"prompt":603,"check":604,"hints":615,"solution":617,"skills":618},"four-operations.q050","Which is the best estimate of 47 × 62?",{"kind":207,"options":605,"correct":614},[606,608,610,612],{"id":210,"label":607},"About 300",{"id":213,"label":609},"About 3,000",{"id":216,"label":611},"About 30,000",{"id":219,"label":613},"About 109",[213],[616],"Round each to the nearest ten.","47 ≈ 50 and 62 ≈ 60, so 50 × 60 = 3,000. Exact: 2,914. Option (d) adds the numbers instead of multiplying.",[31],{"id":620,"section":31,"level":70,"prompt":621,"check":622,"hints":633,"solution":635,"skills":636},"four-operations.q051","Which calculation checks that 7,425 ÷ 25 = 297?",{"kind":207,"options":623,"correct":632},[624,626,628,630],{"id":210,"label":625},"297 × 25",{"id":213,"label":627},"7,425 − 297",{"id":216,"label":629},"7,425 × 25",{"id":219,"label":631},"297 + 25",[210],[634],"Which operation undoes division?","Multiplication undoes division: 297 × 25 = 7,425, which gives back the dividend. ✓",[109,315],{"id":638,"section":31,"level":70,"prompt":639,"check":640,"hints":651,"solution":653,"skills":654},"four-operations.q052","Which answer to 689 × 21 is most likely correct? Decide by estimating, not by multiplying.",{"kind":207,"options":641,"correct":650},[642,644,646,648],{"id":210,"label":643},"1,449",{"id":213,"label":645},"14,469",{"id":216,"label":647},"1,44,690",{"id":219,"label":649},"3,445",[213],[652],"Round to 700 × 20.","689 × 21 ≈ 700 × 20 = 14,000. Only 14,469 is close. (Exact: 14,469.) The others are off by a factor of 10 or come from multiplying by the digits of 21 without place value.",[31,315],{"id":656,"section":31,"level":101,"prompt":657,"check":658,"hints":669,"solution":671,"skills":672},"four-operations.q053","A school buys 38 chairs at ₹198 each. Without calculating exactly, which is the best estimate of the cost?",{"kind":207,"options":659,"correct":668},[660,662,664,666],{"id":210,"label":661},"₹800",{"id":213,"label":663},"₹8,000",{"id":216,"label":665},"₹80,000",{"id":219,"label":667},"₹236",[213],[670],"Round both numbers up to friendly values.","38 ≈ 40 and 198 ≈ 200. 40 × 200 = ₹8,000. Exact: 38 × 198 = ₹7,524, a little less, because both numbers were rounded up.",[31,483],{"id":674,"section":31,"level":101,"prompt":675,"check":676,"hints":687,"solution":689,"skills":690},"four-operations.q054","Two numbers are 5,049 and 4,951. Which is the most sensible way to estimate their difference?",{"kind":207,"options":677,"correct":686},[678,680,682,684],{"id":210,"label":679},"Round both to the nearest thousand: 5,000 − 5,000 = 0",{"id":213,"label":681},"Round both to the nearest ten: 5,050 − 4,950 = 100",{"id":216,"label":683},"Add them: about 10,000",{"id":219,"label":685},"The difference cannot be estimated",[213],[688],"What happens to a small gap when you round to thousands?","When two numbers are close together, coarse rounding can wipe out the difference. Rounding to the nearest ten keeps it: 5,050 − 4,950 = 100. The exact difference is 98.",[31,141],{"id":692,"section":31,"level":101,"prompt":693,"check":694,"hints":696,"solution":698,"skills":699},"four-operations.q055","Without multiplying the whole thing, what is the **last digit** of 3,457 × 2,368?",{"kind":48,"answer":695,"tolerance":50},6,[697],"Which digits decide the ones digit?","Only the ones digits affect the ones digit of a product: 7 × 8 = 56, so the answer ends in **6**. (The full product is 81,86,176.) Checking the last digit is a quick way to catch errors.",[315,141],{"id":701,"section":31,"level":132,"prompt":702,"check":703,"hints":714,"solution":717,"skills":718},"four-operations.q056","Digit-sum check (casting out nines): reduce each number to a single digit by adding its digits repeatedly. For a correct product, the digit sums must match. Which product must be **wrong**?",{"kind":207,"options":704,"correct":713},[705,707,709,711],{"id":210,"label":706},"4,321 × 56 = 2,41,976",{"id":213,"label":708},"1,234 × 78 = 96,252",{"id":216,"label":710},"5,555 × 25 = 1,38,875",{"id":219,"label":712},"789 × 123 = 96,047",[219],[715,716],"Find the digit sum of each factor and of the product.","Multiply the two factor digit sums and reduce again.","(d): 789 → 7+8+9 = 24 → 6. 123 → 6. 6 × 6 = 36 → 9. But 96,047 → 9+6+0+4+7 = 26 → 8. 9 ≠ 8, so it must be wrong. (The true product is 97,047.) The others pass: e.g. (a) 4,321 → 1, 56 → 2, 1 × 2 = 2, and 2,41,976 → 29 → 11 → 2. ✓ Passing the check does not prove an answer right, but failing it proves it wrong.",[719,315],"casting out nines",{"id":721,"section":35,"level":45,"prompt":722,"check":723,"hints":725,"solution":727,"skills":728},"four-operations.q057","Add ₹45.50 and ₹32.75.",{"kind":48,"answer":724,"tolerance":50,"unit":390},78.25,[726],"100 paise = ₹1.","Line up the decimal points (rupees under rupees, paise under paise). Paise: 50 + 75 = 125 paise = ₹1.25, so write 25 and carry ₹1. Rupees: 45 + 32 + 1 = 78. Total **₹78.25**.",[483,729],"addition",{"id":731,"section":35,"level":45,"prompt":732,"check":733,"hints":735,"solution":737,"skills":738},"four-operations.q058","How many metres are there in 3 km 250 m?",{"kind":48,"answer":734,"tolerance":50,"unit":459},3250,[736],"1 km = 1,000 m.","1 km = 1,000 m, so 3 km = 3,000 m. Add the 250 m: 3,000 + 250 = **3,250 m**.",[572,739],"conversion",{"id":741,"section":35,"level":70,"prompt":742,"check":743,"hints":745,"solution":747,"skills":748},"four-operations.q059","Kiran pays for a ₹187.50 book with a ₹500 note. How much change does she get?",{"kind":48,"answer":744,"tolerance":50,"unit":390},312.5,[746],"Count up from ₹187.50 to ₹500.","Work in paise to avoid slips: 50,000 − 18,750 = 31,250 paise = **₹312.50**. Or count up: 187.50 + 12.50 = 200, then + 300 = 500, so 12.50 + 300 = ₹312.50.",[483,464],{"id":750,"section":35,"level":70,"prompt":751,"check":752,"hints":755,"solution":757,"skills":758},"four-operations.q060","Add 2 kg 450 g and 1 kg 875 g. Give the answer in grams.",{"kind":48,"answer":753,"tolerance":50,"unit":754},4325,"g",[756],"1 kg = 1,000 g.","Grams: 450 + 875 = 1,325 g = 1 kg 325 g. Kilograms: 2 + 1 + 1 = 4. Total 4 kg 325 g = **4,325 g**.",[572,56],{"id":760,"section":35,"level":70,"prompt":761,"check":762,"hints":764,"solution":766,"skills":767},"four-operations.q061","A crate holds 12 bottles of juice, each 750 mL. How many litres of juice are in the crate?",{"kind":48,"answer":300,"tolerance":50,"unit":763},"L",[765],"Multiply first, then convert mL to L.","12 × 750 = 9,000 mL. Since 1 L = 1,000 mL, 9,000 mL = **9 L**.",[572,15],{"id":769,"section":35,"level":101,"prompt":770,"check":771,"hints":773,"solution":775,"skills":776},"four-operations.q062","A tailor has 25 m of cloth. Each kurta needs 3 m 40 cm. How many complete kurtas can she make, and how much cloth is left? Give the number of kurtas.",{"kind":48,"answer":509,"tolerance":50,"unit":772},"kurtas",[774],"Change both lengths into centimetres.","Convert to one unit: 25 m = 2,500 cm; 3 m 40 cm = 340 cm. 2,500 ÷ 340 = 7 remainder 120. She can make **7 kurtas**, with 120 cm (1 m 20 cm) left over. Here we **round down**: a part-kurta is no use.",[572,275],{"id":778,"section":35,"level":101,"prompt":779,"check":780,"hints":782,"solution":784,"skills":785},"four-operations.q063","2 kg 500 g of cashews cost ₹1,750. What is the price per kilogram?",{"kind":48,"answer":781,"tolerance":50,"unit":390},700,[783],"How many 500 g lots are there in 2 kg 500 g?","2 kg 500 g = 2,500 g. Price per 500 g: 1,750 ÷ 5 = ₹350 (because 2,500 g is 5 lots of 500 g). Per kilogram = 2 × 350 = **₹700**. Check: 700 × 2.5 = 1,750. ✓",[423,483],{"id":787,"section":35,"level":101,"prompt":788,"check":789,"hints":791,"solution":793,"skills":794},"four-operations.q064","A 5 L can of lassi is poured into glasses that hold 300 mL each. How many glasses can be filled completely?",{"kind":48,"answer":566,"tolerance":50,"unit":790},"glasses",[792],"Convert litres to millilitres first.","5 L = 5,000 mL. 5,000 ÷ 300 = 16 remainder 200. So **16 glasses** are full and 200 mL is left for a part-glass.",[572,275],{"id":796,"section":39,"level":45,"prompt":797,"check":798,"hints":806,"solution":808,"skills":809},"four-operations.q065","In 56 ÷ 8 = 7, which number is the **divisor**?",{"kind":207,"options":799,"correct":805},[800,802,804],{"id":210,"label":801},"56",{"id":213,"label":803},"8",{"id":216,"label":324},[213],[807],"It is the number you divide by.","The **dividend** (56) is the number being divided, the **divisor** (8) is the number you divide by, and the **quotient** (7) is the answer.",[39],{"id":811,"section":39,"level":45,"prompt":812,"check":813,"hints":818,"solution":820,"skills":821},"four-operations.q066","What is the answer to a subtraction called?",{"kind":814,"accept":815},"text",[816,817],"difference","the difference",[819],"It tells you how far apart two numbers are.","The answer to a subtraction is the **difference**. In 845 − 367 = 478, the difference is 478.",[39],{"id":823,"section":39,"level":45,"prompt":824,"check":825,"hints":827,"solution":829,"skills":830},"four-operations.q067","What is the **product** of 6 and 7?",{"kind":48,"answer":826,"tolerance":50},42,[828],"Product is the multiplication word.","**Product** means the answer to a multiplication: 6 × 7 = **42**. (The *sum* of 6 and 7 would be 13.)",[39],{"id":832,"section":39,"level":70,"prompt":833,"check":834,"hints":836,"solution":838,"skills":839},"four-operations.q068","In 845 − 367 = 478, which number is the **minuend**?",{"kind":48,"answer":835,"tolerance":50},845,[837],"It comes first in the subtraction.","The **minuend** is the number you subtract *from* (845). The **subtrahend** is the number being subtracted (367). The **difference** is 478.",[39],{"id":841,"section":39,"level":70,"prompt":842,"check":843,"hints":847,"solution":849,"skills":850},"four-operations.q069","In 17 × 4 = 68, 17 is called the multiplicand. What is 4 called?",{"kind":814,"accept":844},[845,846],"multiplier","the multiplier",[848],"It 'multiplies' the other number.","The **multiplier** tells you how many times to take the multiplicand: 17 × 4 means four lots of 17. The answer, 68, is the **product**.",[39],{"id":852,"section":39,"level":70,"prompt":853,"check":854,"hints":865,"solution":867,"skills":868},"four-operations.q070","Which statement is always true for a division with a remainder?",{"kind":207,"options":855,"correct":864},[856,858,860,862],{"id":210,"label":857},"Dividend = divisor × quotient + remainder",{"id":213,"label":859},"Dividend = quotient + divisor + remainder",{"id":216,"label":861},"Divisor = dividend × quotient − remainder",{"id":219,"label":863},"Remainder is always bigger than the divisor",[210],[866],"Test each statement with 50 ÷ 6 = 8 r 2.","For example 50 ÷ 6 = 8 r 2 and 6 × 8 + 2 = 50. ✓ This is how we check every division. Also, the remainder is always **smaller** than the divisor.",[39,314],{"id":870,"section":39,"level":101,"prompt":871,"check":872,"hints":874,"solution":876,"skills":877},"four-operations.q071","The **sum** of two numbers is 1,000 and their **difference** is 246. What is the larger number?",{"kind":48,"answer":873,"tolerance":50},623,[875],"Sum + difference = 2 × larger number.","If you add the sum and the difference, you get two copies of the larger number: 1,000 + 246 = 1,246, and 1,246 ÷ 2 = **623**. The smaller is 1,000 − 623 = 377. Check: 623 − 377 = 246. ✓",[39,141],{"id":879,"section":39,"level":70,"prompt":880,"check":881,"hints":882,"solution":884,"skills":885},"four-operations.q072","The **product** of two numbers is 1,764. One of them is 42. What is the other?",{"kind":48,"answer":826,"tolerance":50},[883],"Which operation undoes multiplication?","Division undoes multiplication: 1,764 ÷ 42 = **42**. Check: 42 × 42 = 1,764 ✓, so both numbers are 42.",[39,109],{"id":887,"section":39,"level":132,"prompt":888,"check":889,"hints":891,"solution":893,"skills":894},"four-operations.q073","In a division, the remainder is 4. The divisor is 7 times the remainder, and the quotient is 5 more than the divisor. What is the dividend?",{"kind":48,"answer":890,"tolerance":50},928,[892],"Find the divisor first, then the quotient.","Divisor = 7 × 4 = 28. Quotient = 28 + 5 = 33. Dividend = divisor × quotient + remainder = 28 × 33 + 4 = 924 + 4 = **928**. Check: 928 ÷ 28 = 33 r 4. ✓",[39,314],{"id":896,"section":39,"level":101,"prompt":897,"check":898,"hints":909,"solution":911,"skills":912},"four-operations.q074","Match the word to its meaning: which word means **a number being added**?",{"kind":207,"options":899,"correct":908},[900,902,904,906],{"id":210,"label":901},"Addend",{"id":213,"label":903},"Minuend",{"id":216,"label":905},"Quotient",{"id":219,"label":907},"Multiplicand",[210],[910],"'Add' is hidden inside the word.","In 25 + 17 = 42, both 25 and 17 are **addends** and 42 is the sum. A minuend is the number subtracted from; a quotient is the answer to a division; a multiplicand is the number being multiplied.",[39],[914,915,916,917,918,919,920,921],"four-operations-ncert-math-magic-5","four-operations-ncert-math-6","four-operations-khan-arithmetic","four-operations-mathsisfun-long-division","four-operations-bbc-bitesize-ks2","four-operations-wiki-arithmetic","four-operations-mathsisfun-long-multiplication","four-operations-mathsisfun-estimation","needs_review",{"generatedBy":924,"notes":925},"claude-code","Draft generated locally; every numeric answer computed and asserted in Python. Pending owner review.","b05ba6d77b9234d83f912814e27bdcdec3c870c581fb25f99e0ee31ef8396082",{"logic:questions":928,"source:four-operations-bbc-bitesize-ks2":929,"source:four-operations-khan-arithmetic":930,"source:four-operations-mathsisfun-estimation":931,"source:four-operations-mathsisfun-long-division":932,"source:four-operations-mathsisfun-long-multiplication":933,"source:four-operations-ncert-math-6":934,"source:four-operations-ncert-math-magic-5":935,"source:four-operations-wiki-arithmetic":936},"e7fd7c240a65bea1bff6277f7af7cca65e2c7fcb13c6d756cb943d53f3cbc948","07eb90693160dd6149271155357cff693c59a2006a36d3cc67c133d9f2900790","ef6b590392c4299a9ce1be700f25412c2a52dea7c431f6dd28b6d7c1ce15374d","4fa46f082bd73cea931140c432a48f7ee35a7f7d6bbecb43d270505bf6c94d00","794b6ef61d62320200593d8724a8525d6a81168da99708651826563b823043a5","f673458eca6d084a854c7e5fd91ec78c1ed561c54faf5e27538711d81efda7a5","c66e4c408e97e6c5758c1b1160148cfabaf1f5e9ca4caba87383f8ca4e389c95","1856ee000472d9371b3855e38b691fbd35b221cc502b33bc887ffd8f161b636a","3fd5b7fcbfec31c612f8356e67126230dae21df1a69f1c1a06216d510015a231","preview-7e1cbbcc4f",1789899599891]