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bg-background","children":["$","$L3b",null,{"backHref":"/igcse/maths/edexcel/questions","basePath":"/igcse/maths/edexcel/grade-6-6-1-recurring-decimals-to-fractions/questions","formulaUrl":null,"initialIndex":15,"initialQuestionWindow":[{"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},{"id":"a1b80afa-0eb8-4b3e-a262-0304c7e348af","revisionId":"5d048b18-5cd9-4d9c-bb7d-528fbadcc0c2","questionNumber":"16","questionType":"LA","specification":"$$x$ is an integer such that $1 \\leq x \\leq 9$","difficulty":3,"difficultyBand":"medium","totalMarks":2,"sourceTitle":"Recurring Decimals to Fractions","calculatorUse":null,"isCalculation":true,"parts":[{"id":"f463ee6b-15d2-48c5-ae53-938eaf500c6f","label":"0","content":"Prove that $0.\\dot{0}x = \\frac{x}{99}$","markscheme":"- Sets $y = 0.\\dot{0}x$ and $100y = x.\\dot{0}x$ M1\n- Subtracts to get $99y = x$ and concludes $y = \\frac{x}{99}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Sets up equations to align the 2-digit recurring block, e.g., $100y = x.\\dot{0}x$.","code":"M"},{"id":"p2","marks":1,"description":"Subtracts to eliminate the recurring part and reaches the required proof.","code":"A"}],"finalAnswers":[],"generatedAt":"2026-05-07T02:03:47.477Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"3f46a231-2a60-4f5b-8229-d5a3954ef397","revisionId":"10ef547e-1ee5-4304-95a0-f4714233b10b","questionNumber":"17","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"medium","totalMarks":4,"sourceTitle":"Recurring Decimals to Fractions","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d1945aa1-3e72-47ba-9d10-f82b235dcbe5","label":"0","content":"Work out: $0.\\dot{5}\\dot{4} \\times 0.\\dot{5}$","markscheme":"- Converts $0.\\dot{5}\\dot{4}$ to $\\frac{54}{99}$ or $\\frac{6}{11}$ M1\n- Converts $0.\\dot{5}$ to $\\frac{5}{9}$ M1\n- Multiplies fractions: $\\frac{6}{11} \\times \\frac{5}{9} = \\frac{30}{99}$ M1\n- Simplifies to $\\frac{10}{33}$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Correctly converts $0.\\dot{5}\\dot{4}$ to a fraction.","code":"M"},{"id":"p2","marks":1,"description":"Correctly converts $0.\\dot{5}$ to a fraction.","code":"M"},{"id":"p3","marks":1,"description":"Multiplies the two fractions correctly.","code":"M"},{"id":"p4","marks":1,"description":"Final answer $\\frac{10}{33}$ in simplest form.","code":"A"}],"finalAnswers":["10/33"],"generatedAt":"2026-05-07T02:03:47.477Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"recurring_decimals_to_fractions","label":"Convert to fractions","steps":[{"type":"text","order":0,"title":"Spot the repeats","content":"The dots mean the digits underneath repeat forever: $0.\\dot{5}\\dot{4}=0.545454\\ldots$ and $0.\\dot{5}=0.555\\ldots$. Converting both numbers to fractions makes the multiplication exact.","highlights":[{"text":"$$0.\\dot{5}\\dot{4} \\times 0.\\dot{5}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"0.\\dot{5}\\dot{4}=\\frac{54}{99}"},{"latex":"\\frac{54}{99}=\\frac{6}{11}"}]},"order":1,"title":"Convert the first decimal","description":"A two-digit recurring block is written over 99. This conversion earns the first method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"0.\\dot{5}=\\frac{5}{9}"},{"text":"This fraction is already in its simplest form."}]},"order":2,"title":"Convert the second decimal","description":"A single recurring digit is written over 9. This earns the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{6}{11}\\times\\frac{5}{9}=\\frac{6\\times5}{11\\times9}"},{"latex":"\\frac{6\\times5}{11\\times9}=\\frac{30}{99}"}]},"order":3,"title":"Multiply the fractions","description":"Multiply numerators together and denominators together. Reaching 30 over 99 earns the third method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{30}{99}=\\frac{30\\div3}{99\\div3}"},{"latex":"\\frac{30\\div3}{99\\div3}=\\frac{10}{33}"}]},"order":4,"title":"Simplify the answer","description":"Both parts of the fraction divide by 3. The simplified fraction earns the final answer mark (A1)."}]}],"generatedAt":"2026-07-21T03:36:09.148Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"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 11

Recurring Decimals to Fractions Questions

Recurring Decimals to Fractions

Question 11

Recurring Decimals to Fractions Questions