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Set\n\n$$x=0.\\dot{1}\\dot{2}\\dot{6}=0.126126126\\ldots$$\n\nGiving the decimal a letter lets us turn the recurring part into an equation.","highlights":[{"text":"$$0.\\dot{1}\\dot{2}\\dot{6}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original-x}{x}=0.126126126\\ldots"},{"latex":"\\htmlData{anchor=shifted-x}{1000x}=126.126126126\\ldots"}],"connections":[{"to":"shifted-x","from":"original-x","color":"sky","label":"multiply by $1000$"}]},"order":1,"title":"Align the repeats","description":"There are three digits in the repeating block, so multiplying by $1000$ shifts the decimal point three places. The same recurring tail now appears in both equations."},{"type":"hyper_latex","block":{"nodes":[{"latex":"1000x=126.126126126\\ldots"},{"latex":"-x=-0.126126126\\ldots"},{"latex":"\\htmlData{anchor=difference}{999}x=126"},{"latex":"x=\\frac{126}{999}"}],"highlights":[{"color":"green","style":"box","anchor":"difference"}]},"order":2,"title":"Subtract the equations","description":"Subtract the first equation from the second. The recurring decimals are identical, so they cancel; setting up these aligned equations earns the first mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x=\\frac{\\htmlData{anchor=num}{126}}{\\htmlData{anchor=den}{999}}"},{"latex":"x=\\frac{\\htmlData{anchor=new-num}{14}}{\\htmlData{anchor=new-den}{111}}"},{"latex":"\\boxed{0.\\dot{1}\\dot{2}\\dot{6}=\\frac{14}{111}}"}],"connections":[{"to":"new-num","from":"num","color":"purple","label":"divide by $9$"},{"to":"new-den","from":"den","color":"purple","label":"divide by $9$"}]},"order":3,"title":"Simplify the fraction","description":"Both numerator and denominator divide by $9$. This correct simplification earns the second mark."}]}],"generatedAt":"2026-09-01T09:03:47.815Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"a0e6e5a3-065d-41db-8da1-04b64df99e80:shift_subtract","label":"Shift and subtract","steps":[{"type":"text","order":0,"title":"Name the decimal","content":"The dots show that the block $126$ repeats forever. Set\n\n$$x=0.\\dot{1}\\dot{2}\\dot{6}=0.126126126\\ldots$$\n\nGiving the decimal a letter lets us turn the recurring part into an equation.","highlights":[{"text":"$$0.\\dot{1}\\dot{2}\\dot{6}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original-x}{x}=0.126126126\\ldots"},{"latex":"\\htmlData{anchor=shifted-x}{1000x}=126.126126126\\ldots"}],"connections":[{"to":"shifted-x","from":"original-x","color":"sky","label":"multiply by $1000$"}]},"order":1,"title":"Align the repeats","description":"There are three digits in the repeating block, so multiplying by $1000$ shifts the decimal point three places. The same recurring tail now appears in both equations."},{"type":"hyper_latex","block":{"nodes":[{"latex":"1000x=126.126126126\\ldots"},{"latex":"-x=-0.126126126\\ldots"},{"latex":"\\htmlData{anchor=difference}{999}x=126"},{"latex":"x=\\frac{126}{999}"}],"highlights":[{"color":"green","style":"box","anchor":"difference"}]},"order":2,"title":"Subtract the equations","description":"Subtract the first equation from the second. The recurring decimals are identical, so they cancel; setting up these aligned equations earns the first mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x=\\frac{\\htmlData{anchor=num}{126}}{\\htmlData{anchor=den}{999}}"},{"latex":"x=\\frac{\\htmlData{anchor=new-num}{14}}{\\htmlData{anchor=new-den}{111}}"},{"latex":"\\boxed{0.\\dot{1}\\dot{2}\\dot{6}=\\frac{14}{111}}"}],"connections":[{"to":"new-num","from":"num","color":"purple","label":"divide by $9$"},{"to":"new-den","from":"den","color":"purple","label":"divide by $9$"}]},"order":3,"title":"Simplify the fraction","description":"Both numerator and denominator divide by $9$. This correct simplification earns the second mark."}]}],"generatedAt":"2026-09-01T09:03:48.135Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"b6b8f5dd-5a0b-4503-8aa1-9412bcfa867a","revisionId":"5edd2227-c58d-4450-befd-8198743e31af","questionNumber":"13","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Recurring Decimals to Fractions","calculatorUse":null,"isCalculation":false,"parts":[{"id":"49fa0239-0e47-49e3-b76b-7ee4407baafd","label":"0","content":"Write $3.2\\dot{5}\\dot{4}$ as a fraction in its simplest form.","markscheme":"- Sets $10x = 32.\\dot{5}\\dot{4}$ and $1000x = 3254.\\dot{5}\\dot{4}$ M1\n- Subtracts to get $990x = 3222$ M1\n- Simplifies to $\\frac{179}{55}$ or $3\\frac{14}{55}$ 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 $990x = 3222$.","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}],"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 8

Recurring Decimals to Fractions Questions

Recurring Decimals to Fractions

Question 8

Recurring Decimals to Fractions Questions