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recurring part, e.g., $1000x$ and $10x$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly subtracts to find $990x = 3222$.","code":"M"},{"id":"p3","marks":1,"description":"Simplifies the fraction to $\\frac{179}{55}$ or $3\\frac{14}{55}$.","code":"A"}],"finalAnswers":["179/55"],"generatedAt":"2026-05-07T02:03:47.477Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"shift_and_subtract","label":"Shift and subtract","steps":[{"type":"text","order":0,"title":"Define $x$","content":"Let $x=3.2\\dot{5}\\dot{4}$. The dots mean that $54$ repeats: $3.254545\\ldots$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=first}{10x=32.\\dot{5}\\dot{4}}"},{"latex":"\\htmlData{anchor=second}{1000x=3254.\\dot{5}\\dot{4}}"}],"connections":[{"to":"second","from":"first","color":"sky","label":"multiply by 100","style":"solid"}]},"order":1,"title":"Line up the repeats","description":"First shift past the non-repeating digit, then shift by the two-digit recurring block. This setup earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=equations}{1000x-10x=3254-32}"},{"latex":"\\htmlData{anchor=combined}{990x=3222}"}],"connections":[{"to":"combined","from":"equations","color":"orange","label":"subtract","style":"solid"}]},"order":2,"title":"Subtract the equations","description":"Subtract the aligned equations; the recurring tails cancel. This earns the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=combined}{990x=3222}"},{"latex":"\\htmlData{anchor=answer}{x=\\frac{3222}{990}}"}],"connections":[{"to":"answer","from":"combined","color":"sky","label":"divide by 990","style":"solid"}]},"order":3,"title":"Divide by 990","description":"Divide both sides by 990 to make x the subject."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=unsimplified}{x=\\frac{3222}{990}}"},{"latex":"\\htmlData{anchor=simplified}{x=\\frac{179}{55}}"}],"connections":[{"to":"simplified","from":"unsimplified","color":"green","label":"divide numerator and denominator by 18","style":"solid"}]},"order":4,"title":"Simplify the fraction","description":"Divide the numerator and denominator by 18. This earns the accuracy mark for the simplest form."}]}],"generatedAt":"2026-07-21T03:11:16.626Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"7850200c-6720-4092-a014-b06e5c46c745","revisionId":"b54bb027-d139-4000-8b5d-aea48d9f5408","questionNumber":"14","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Recurring Decimals to Fractions","calculatorUse":null,"isCalculation":false,"parts":[{"id":"6e569b96-6aed-4692-ab75-c3cb62c32624","label":"0","content":"Write $2.7\\dot{4}\\dot{2}$ as a fraction in its simplest form.","markscheme":"- Sets $10x = 27.\\dot{4}\\dot{2}$ and $1000x = 2742.\\dot{4}\\dot{2}$ M1\n- Subtracts to get $990x = 2715$ M1\n- Simplifies to $\\frac{181}{66}$ or $2\\frac{49}{66}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up equations to isolate the recurring part, e.g., $1000x$ and $10x$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly subtracts to find $990x = 2715$.","code":"M"},{"id":"p3","marks":1,"description":"Simplifies the fraction to $\\frac{181}{66}$ or $2\\frac{49}{66}$.","code":"A"}],"finalAnswers":["181/66"],"generatedAt":"2026-05-07T02:03:47.477Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"eliminate_repeating_block","label":"Eliminate the repeating block","steps":[{"type":"text","order":0,"title":"Name the decimal","content":"Let $$x=2.7\\dot{4}\\dot{2}=2.7424242\\ldots$$\n\nThe dots show that the block $42$ repeats forever. There is one non-repeating digit ($7$) before that block, so we will use powers of $10$ to line up the repeating parts.","highlights":[{"text":"Write $2.7\\dot{4}\\dot{2}$ as a fraction in its simplest form.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=tenx}{\\color{@sky}{10x}}=\\htmlData{anchor=tailone}{27.\\dot{4}\\dot{2}}"},{"latex":"\\htmlData{anchor=thousandx}{\\color{@orange}{1000x}}=\\htmlData{anchor=tailtwo}{2742.\\dot{4}\\dot{2}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"tenx"},{"color":"orange","style":"text-color-with-background","anchor":"thousandx"}],"connections":[{"to":"tailtwo","from":"tailone","color":"green","label":"multiply by 100","style":"solid"}]},"order":1,"title":"Shift the decimal","description":"First move past the non-repeating digit; then move three places from the original decimal so the recurring tails match. Writing both shifted equations earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=top}{\\color{@orange}{1000x}=2742.\\dot{4}\\dot{2}}"},{"latex":"\\htmlData{anchor=bottom}{\\color{@sky}{10x}=27.\\dot{4}\\dot{2}}"},{"latex":"\\htmlData{anchor=result}{\\color{@green}{990x=2715}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"result","from":"top","color":"green","label":"subtract","style":"solid"},{"to":"result","from":"bottom","color":"green","label":"subtract","style":"solid"}]},"order":3,"title":"Subtract the equations","description":"Subtract the 10x equation from the 1000x equation. The matching recurring parts cancel, giving 990x=2715 and earning the second method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=equation}{\\color{@sky}{990x}=2715}"},{"latex":"\\htmlData{anchor=fraction}{x=\\frac{2715}{990}}"},{"latex":"\\htmlData{anchor=answer}{\\color{@green}{x=\\frac{181}{66}}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"fraction","from":"equation","color":"sky","label":"divide by 990","style":"solid"},{"to":"answer","from":"fraction","color":"green","label":"divide both by 15","style":"solid"}]},"order":5,"title":"Simplify the fraction","description":"Divide by 990, then divide the numerator and denominator by their common factor, 15. The fully simplified answer x=181/66 earns the final accuracy mark."}]}],"generatedAt":"2026-07-21T04:11:48.316Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"f6e4045a-6c0a-4d8d-a3ab-600a398ac9db","revisionId":"ca1a160f-e9cd-49df-8c7c-a38fdd510542","questionNumber":"15","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Recurring Decimals to Fractions","calculatorUse":null,"isCalculation":false,"parts":[{"id":"a25bfa19-0138-4987-82ff-26d1ccb03895","label":"0","content":"Write $3.5\\dot{9}\\dot{4}$ as a fraction in its simplest form.","markscheme":"- Sets $x = 3.5\\dot{9}\\dot{4}$ and multiplies to align the recurring tails: $10x = 35.\\dot{9}\\dot{4}$ and $1000x = 3594.\\dot{9}\\dot{4}$ M1\n- Subtracts to get $990x = 3559$ M1\n- $x = \\frac{3559}{990}$ (already in simplest form), or $3\\frac{589}{990}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Multiplies by powers of 10 to align the recurring tails, e.g. $10x = 35.\\dot{9}\\dot{4}$ and $1000x = 3594.\\dot{9}\\dot{4}$.","code":"M","acceptable":["10x = 35.\\dot{9}\\dot{4}","1000x = 3594.\\dot{9}\\dot{4}"]},{"id":"p2","marks":1,"description":"Subtracts correctly to reach $990x = 3559$.","code":"M","acceptable":["990x = 3559","990x=3559"]},{"id":"p3","marks":1,"description":"Final answer $\\frac{3559}{990}$ (or $3\\frac{589}{990}$) in simplest form.","code":"A","acceptable":["\\frac{3559}{990}","3559/990","3\\frac{589}{990}"]}],"finalAnswers":["\\frac{3559}{990}","3559/990"]},"tableImages":[],"explanation":{"methods":[{"id":"shift-and-subtract","label":"Shift and subtract","steps":[{"type":"text","order":0,"title":"Spot the recurring block","content":"The dots sit on the $9$ and the $4$, so only those two digits repeat and the $5$ does not:\n\n$$3.5\\dot{9}\\dot{4}=3.5949494\\ldots$$\n\nLet\n\n$$x=3.5\\dot{9}\\dot{4}$$\n\nOne digit sits between the decimal point and the block, and the block is two digits long, so we shift by $10$ and by $1000$.","highlights":[{"text":"$$3.5\\dot{9}\\dot{4}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"10x=35.949494\\ldots"},{"latex":"\\htmlData{anchor=shifted}{1000x=3594.949494\\ldots}"}],"highlights":[{"color":"sky","style":"box","anchor":"shifted"}]},"order":1,"title":"Align the repeats","description":"Multiply by $10$ first so the non-recurring $5$ moves in front of the point, then by $1000$ so both lines have the same recurring part after the point. This setup earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"1000x=3594.\\htmlData{anchor=tail_a}{\\dot{9}\\dot{4}}"},{"latex":"10x=35.\\htmlData{anchor=tail_b}{\\dot{9}\\dot{4}}"},{"latex":"990x=3559"}],"connections":[{"to":"tail_b","from":"tail_a","color":"teal","label":"same recurring tail"}]},"order":2,"title":"Subtract the equations","description":"Subtract the $10x$ equation from the $1000x$ equation. The recurring tail is identical in both, so it cancels."},{"type":"hyper_latex","block":{"nodes":[{"latex":"990x=3559"},{"latex":"x=\\htmlData{anchor=answer}{\\frac{3559}{990}}=3\\frac{589}{990}"}],"highlights":[{"color":"green","style":"box","anchor":"answer"}]},"order":3,"title":"Divide and check the form","description":"Divide by $990$. Since $990=2\\times3^2\\times5\\times11$ and $3559$ shares none of those factors, $\\frac{3559}{990}$ is already in its simplest form, earning the accuracy mark."}]}],"generatedAt":"2026-09-01T09:10:48.997Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"a25bfa19-0138-4987-82ff-26d1ccb03895:shift-and-subtract","label":"Shift and subtract","steps":[{"type":"text","order":0,"title":"Spot the recurring block","content":"The dots show that the three digits $594$ repeat together:\n\n$$3.5\\dot{9}\\dot{4}=3.594594594\\ldots$$\n\nLet\n\n$$x=3.5\\dot{9}\\dot{4}$$\n\nThere are three digits in the repeating block, so we will shift it by three places.","highlights":[{"text":"$$3.5\\dot{9}\\dot{4}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"x=3.594594594\\ldots"},{"latex":"\\htmlData{anchor=shifted}{1000x=3594.594594594\\ldots}"}],"highlights":[{"color":"sky","style":"box","anchor":"shifted"}]},"order":1,"title":"Align the repeats","description":"Multiply both sides by $1000$ so the same recurring decimal part appears after the decimal point. This setup earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"1000x=3594.\\htmlData{anchor=tail_a}{\\dot{5}\\dot{9}\\dot{4}}"},{"latex":"x=3.\\htmlData{anchor=tail_b}{\\dot{5}\\dot{9}\\dot{4}}"},{"latex":"999x=3591"}],"connections":[{"to":"tail_b","from":"tail_a","color":"teal","label":"same recurring tail"}]},"order":2,"title":"Subtract the equations","description":"Subtract the original equation from the shifted equation. The recurring tail is identical in both equations, so it cancels."},{"type":"hyper_latex","block":{"nodes":[{"latex":"999x=3591"},{"latex":"x=\\frac{3591}{999}"},{"latex":"x=\\htmlData{anchor=answer}{\\frac{133}{37}}=3\\frac{22}{37}"}],"highlights":[{"color":"green","style":"box","anchor":"answer"}]},"order":3,"title":"Divide and simplify","description":"Divide by $999$, then cancel the common factor $27$. The fraction $\\frac{133}{37}$ is in its simplest form, earning the accuracy mark."}]}],"generatedAt":"2026-09-01T09:10:49.254Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths","summaries":[{"id":"f244db23-e5a0-4fea-bce3-9cc47d4528fa","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"cc7687ad-9cb6-497f-9d3f-35f23fb0a784","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"5de21ecf-f1e1-494e-b81d-e9448e880aaa","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"8bf5cb40-2db3-4209-aea4-e3bd7372219b","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"06b8a369-5db6-4e3d-8aa4-2627fd31f406","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"fc3b00b3-f880-43a5-b4ac-6ae6e3680d87","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"dc9fb712-fb15-4091-be45-3ec0937cb8a3","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"625ff601-005f-46fb-b8f8-17f74c115006","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"2d42d68b-29c3-4e12-b137-7718f4955be2","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"0cac3bff-993c-4246-a2b4-a5ea8407f64f","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"073279bf-edc0-4566-a8e5-e0827cea5752","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"c071e4ba-d823-4b59-b349-ce650bc443b9","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"b6b8f5dd-5a0b-4503-8aa1-9412bcfa867a","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"7850200c-6720-4092-a014-b06e5c46c745","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"f6e4045a-6c0a-4d8d-a3ab-600a398ac9db","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"a1b80afa-0eb8-4b3e-a262-0304c7e348af","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"3f46a231-2a60-4f5b-8229-d5a3954ef397","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5506caba-7956-414e-a035-19eece63babb","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e9eb3104-baee-4b17-b046-36d47d888b1d","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Recurring Decimals to Fractions","topicId":"0cfcd5c5-df4f-4c16-9f87-47e9a85ff433","topicSidebar":"$L3c","topicSlug":"grade-6-6-1-recurring-decimals-to-fractions","topicTitle":"Recurring Decimals to Fractions"}]}],"$L3d","$L3e"]}]

Recurring Decimals to Fractions

Question 9

Recurring Decimals to Fractions Questions

Recurring Decimals to Fractions

Question 9

Recurring Decimals to Fractions Questions