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The dot means that the digit $3$ repeats forever.","highlights":[{"text":"recurring decimal $0.2\\dot{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=first}{\\color{@sky}{10x}}=2.\\dot{3}"},{"latex":"\\htmlData{anchor=second}{\\color{@orange}{100x}}=23.\\dot{3}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"first"},{"color":"orange","style":"text-color-with-background","anchor":"second"}]},"order":1,"title":"Align the recurring digits","description":"Multiply by powers of $10$ so that the repeating tails are in the same place. 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. 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Recurring Decimals to Fractions

Question 3

Recurring Decimals to Fractions Questions

Recurring Decimals to Fractions

Question 3

Recurring Decimals to Fractions Questions