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recurring digit, e.g., $10x = 2.\\dot{3}$ and $100x = 23.\\dot{3}$.","code":"M"},{"id":"p2","marks":1,"description":"Subtracts to get $90x = 21$ and simplifies to $\\frac{7}{30}$.","code":"A"}],"finalAnswers":["7/30"],"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 the decimal","content":"Let $x=0.2\\dot{3}=0.2333\\ldots$. 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Writing both of these equations earns the M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=upper}{\\color{@orange}{100x}}=23.\\dot{3}"},{"latex":"\\htmlData{anchor=subtract}{\\color{@orange}{100x}-\\color{@sky}{10x}}=23.\\dot{3}-2.\\dot{3}"},{"latex":"\\htmlData{anchor=result}{\\color{@green}{90x}=\\color{@green}{21}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"result"}]},"order":2,"title":"Subtract the equations","description":"Subtract the first equation from the second. The recurring decimal parts cancel because they are identical."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=start}{\\color{@sky}{90x}}=21"},{"latex":"\\htmlData{anchor=fraction}{x}=\\frac{21}{90}"},{"latex":"\\htmlData{anchor=answer}{\\color{@green}{x=\\frac{7}{30}}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"start"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":3,"title":"Solve for x","description":"Divide by $90$, then simplify the fraction by dividing its numerator and denominator by $3$. This completes the A1 mark."}]}],"generatedAt":"2026-07-21T04:06:48.232Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"2d42d68b-29c3-4e12-b137-7718f4955be2","revisionId":"dcab6d6a-4fa6-4d94-a37f-9f3524a75638","questionNumber":"9","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"medium","totalMarks":2,"sourceTitle":"Recurring Decimals to Fractions","calculatorUse":null,"isCalculation":false,"parts":[{"id":"171c9d21-b7f3-4a76-af8d-8c5e73878454","label":"0","content":"Write $0.4\\dot{3}$ as a fraction in its simplest form.","markscheme":"- Sets $10x = 4.\\dot{3}$ and $100x = 43.\\dot{3}$ M1\n- Subtracts to get $90x = 39$ and simplifies to $\\frac{13}{30}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Uses an algebraic method to eliminate the recurring part, e.g., $100x - 10x = 39$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly simplifies $\\frac{39}{90}$ to $\\frac{13}{30}$.","code":"A"}],"finalAnswers":["13/30"],"generatedAt":"2026-05-07T02:03:47.477Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"recurring_decimal","label":"Convert the recurring decimal","steps":[{"type":"text","order":0,"title":"Define the decimal","content":"Let $x=0.4\\dot{3}$. The dot means that only the $3$ repeats, so $x=0.4333\\ldots$.","highlights":[{"text":"Write $0.4\\dot{3}$ as a fraction in its simplest form.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=first}{10x=4.\\dot{3}}"},{"latex":"\\htmlData{anchor=second}{100x=43.\\dot{3}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"first"},{"color":"sky","style":"text-color-with-background","anchor":"second"}],"connections":[{"to":"second","from":"first","color":"sky","label":"multiply by 10","style":"solid"}]},"order":1,"title":"Align the repeating digits","description":"Multiply by powers of 10 so that the repeating tails in both decimals are identical. M1: Writing both equations shown below earns this mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=subtract}{100x-10x=43.\\dot{3}-4.\\dot{3}}"},{"latex":"\\htmlData{anchor=result}{90x=39}"}],"highlights":[{"color":"orange","style":"text-color","anchor":"subtract"},{"color":"sky","style":"text-color-with-background","anchor":"result"}],"connections":[{"to":"result","from":"subtract","color":"orange","label":"subtract","style":"solid"}]},"order":2,"title":"Subtract the equations","description":"Subtract the first equation from the second; the identical recurring parts cancel. A1 requires this subtraction and the final simplified fraction."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=equation}{90x=39}"},{"latex":"\\htmlData{anchor=divide}{x=\\frac{39}{90}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"divide"}],"connections":[{"to":"divide","from":"equation","color":"green","label":"divide by 90","style":"solid"}]},"order":3,"title":"Divide by 90","description":"Divide both sides by 90 to make x the subject."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=unsimplified}{x=\\frac{39}{90}}"},{"latex":"\\htmlData{anchor=simplified}{x=\\frac{13}{30}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"simplified"}],"connections":[{"to":"simplified","from":"unsimplified","color":"sky","label":"divide both by 3","style":"solid"}]},"order":4,"title":"Simplify the fraction","description":"Divide the numerator and denominator by 3. This gives the fraction in its simplest form and completes A1."}]}],"generatedAt":"2026-07-21T04:03:23.380Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0cac3bff-993c-4246-a2b4-a5ea8407f64f","revisionId":"778a8da0-aa29-43af-af2b-0bbf0c4cb345","questionNumber":"10","questionType":"LA","specification":"","difficulty":2,"difficultyBand":"medium","totalMarks":2,"sourceTitle":"Recurring Decimals to Fractions","calculatorUse":null,"isCalculation":true,"parts":[{"id":"5606aab2-be94-4bd6-b9a3-892a3ecb1dc5","label":"0","content":"Prove algebraically that the recurring decimal $0.6\\dot{8}\\dot{1}$ can be written as $\\frac{15}{22}$","markscheme":"- Sets $10x = 6.\\dot{8}\\dot{1}$ and $1000x = 681.\\dot{8}\\dot{1}$ M1\n- Subtracts to get $990x = 675$ and simplifies $\\frac{675}{990}$ to $\\frac{15}{22}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Sets up equations to align the recurring block, e.g., $1000x - 10x = 675$.","code":"M"},{"id":"p2","marks":1,"description":"Correctly simplifies the resulting fraction to $\\frac{15}{22}$.","code":"A"}],"finalAnswers":["15/22"],"generatedAt":"2026-05-07T02:03:47.477Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"align_repeating_blocks","label":"Align the repeating block","steps":[{"type":"text","order":0,"title":"Name the decimal","content":"Let $x=0.6\\dot{8}\\dot{1}$. The dots show that $81$ repeats, so we will create two versions of $x$ whose repeating tails line up.","highlights":[{"text":"recurring decimal $0.6\\dot{8}\\dot{1}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"10x=6.\\dot{8}\\dot{1}"},{"latex":"1000x=681.\\dot{8}\\dot{1}"}]},"order":1,"title":"Shift both decimals","description":"Multiply by powers of 10 so that the recurring part, 81, appears in the same position in both equations. Writing both shifted equations earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=left}{\\color{@sky}{1000x-10x}}=681.\\dot{8}\\dot{1}-6.\\dot{8}\\dot{1}"},{"latex":"\\htmlData{anchor=equation}{990x=675}"},{"text":"Divide both sides by 990."},{"latex":"x=\\htmlData{anchor=fraction}{\\frac{675}{990}}"},{"text":"Divide the numerator and denominator by 45."},{"latex":"x=\\htmlData{anchor=answer}{\\frac{15}{22}}"}]},"order":3,"title":"Subtract and simplify","description":"Subtract the first shifted equation from the second. The identical recurring decimals cancel; obtaining 990x = 675 and simplifying 675/990 to 15/22 earns A1."}]}],"generatedAt":"2026-07-21T04:02:01.587Z","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 4

Recurring Decimals to Fractions Questions

Recurring Decimals to Fractions

Question 4

Recurring Decimals to Fractions Questions