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A common denominator for $3$ and $4$ is $12$.","highlights":[{"text":"$$\\frac{2}{3} - \\frac{1}{4}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{2}{3}=\\frac{2\\times4}{3\\times4}"},{"latex":"\\frac{2\\times4}{3\\times4}=\\frac{8}{12}"},{"latex":"\\frac{1}{4}=\\frac{1\\times3}{4\\times3}"},{"latex":"\\frac{1\\times3}{4\\times3}=\\frac{3}{12}"},{"latex":"\\frac{2}{3}-\\frac{1}{4}=\\frac{8}{12}-\\frac{3}{12}"}]},"order":1,"title":"Write equivalent fractions","description":"Use 12 as the common denominator. M1: Correctly rewrites the fractions as 8/12 − 3/12 using a common denominator."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{8}{12}-\\frac{3}{12}=\\frac{8-3}{12}"},{"latex":"\\frac{8-3}{12}=\\frac{5}{12}"},{"latex":"\\text{Final answer: }\\frac{5}{12}"}]},"order":2,"title":"Subtract the numerators","description":"The denominators now match, so subtract the top numbers and keep 12 underneath. A1: 8/12 − 3/12 = 5/12."}]}],"generatedAt":"2026-07-21T02:55:09.562Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]},{"id":"b25e1278-f1d4-465c-b9f7-cfc939acdd5b","label":"b","content":"Work out $\\frac{3}{4} \\times \\frac{4}{9}$\n\nGive your answer as a fraction in its simplest form.","markscheme":"- Multiplies numerators and denominators, e.g. $\\frac{12}{36}$ M1\n- Simplifies to $\\frac{1}{3}$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Multiplies numerators and denominators, e.g. $\\frac{12}{36}$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies final answer to $\\frac{1}{3}$.","code":"A"}],"finalAnswers":["1/3"],"generatedAt":"2026-05-06T15:44:07.141Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"multiply_and_simplify","label":"Multiply then simplify","steps":[{"type":"text","order":0,"title":"Multiply matching parts","content":"To multiply fractions, multiply the numerators (top numbers) together and multiply the denominators (bottom numbers) together. This is the method mark, so show this intermediate fraction clearly.","highlights":[{"text":"$$\\frac{3}{4} \\times \\frac{4}{9}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{3}{4}\\times\\frac{4}{9}=\\frac{\\htmlData{anchor=top}{\\color{@sky}{3\\times4}}}{\\htmlData{anchor=bottom}{\\color{@orange}{4\\times9}}}"},{"latex":"\\frac{\\htmlData{anchor=top_result}{\\color{@sky}{12}}}{\\htmlData{anchor=bottom_result}{\\color{@orange}{36}}}"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"top_result"},{"color":"orange","style":"text-color-with-background","anchor":"bottom_result"}],"connections":[{"to":"top_result","from":"top","color":"sky","label":"multiply"},{"to":"bottom_result","from":"bottom","color":"orange","label":"multiply"}]},"order":1,"title":"Form the product","description":"Multiply the top numbers for the new numerator and the bottom numbers for the new denominator."},{"type":"text","order":2,"title":"Simplify the fraction","content":"Both $12$ and $36$ divide by $12$. Divide the numerator and denominator by the **same** number to keep the fraction equivalent. Reaching the simplest form earns the accuracy mark.","highlights":[{"text":"Give your answer as a fraction in its simplest form.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{12}{36}=\\frac{\\htmlData{anchor=numerator}{\\color{@green}{12\\div12}}}{\\htmlData{anchor=denominator}{\\color{@green}{36\\div12}}}"},{"latex":"\\frac{\\htmlData{anchor=top_simplified}{\\color{@green}{1}}}{\\htmlData{anchor=bottom_simplified}{\\color{@green}{3}}}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"top_simplified"},{"color":"green","style":"text-color-with-background","anchor":"bottom_simplified"}],"connections":[{"to":"top_simplified","from":"numerator","color":"green","label":"divide by 12"},{"to":"bottom_simplified","from":"denominator","color":"green","label":"divide by 12"}]},"order":3,"title":"Give the simplest form","description":"Divide both parts of the fraction by $12$."}]}],"generatedAt":"2026-07-21T02:58:41.825Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":67},"walkthroughVideo":null},{"id":"ea6a61c1-5d8d-4cb5-a0f2-c865119f56db","revisionId":"8a7e3ea6-1f08-48ed-ae26-b6cfae75d56a","questionNumber":"3","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":"Fractions","calculatorUse":null,"isCalculation":true,"parts":[{"id":"14eeb63a-3337-4506-88dd-516715deb7c4","label":"0","content":"Work out $\\frac{3}{4} \\times \\frac{5}{6}$","markscheme":"- Multiplies numerators and denominators, e.g. $\\frac{15}{24}$ M1\n- Simplifies to $\\frac{5}{8}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Multiplies numerators and denominators, e.g. $\\frac{15}{24}$.","code":"M"},{"id":"p2","marks":1,"description":"Simplifies final answer to $\\frac{5}{8}$.","code":"A"}],"finalAnswers":["5/8"],"generatedAt":"2026-05-06T15:44:07.141Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"multiply_fractions","label":"Multiply then simplify","steps":[{"type":"text","order":0,"title":"Multiply matching parts","content":"For fractions, multiply the top numbers (the **numerators**) together and multiply the bottom numbers (the **denominators**) together."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{\\frac{\\color{@sky}{3}}{\\color{@orange}{4}}\\times\\frac{\\color{@sky}{5}}{\\color{@orange}{6}}}"},{"latex":"\\htmlData{anchor=middle}{\\frac{\\htmlData{anchor=topproduct}{\\color{@sky}{3\\times5}}}{\\htmlData{anchor=bottomproduct}{\\color{@orange}{4\\times6}}}}"},{"latex":"\\frac{\\htmlData{anchor=topanswer}{\\color{@sky}{15}}}{\\htmlData{anchor=bottomanswer}{\\color{@orange}{24}}}"}],"connections":[{"to":"middle","from":"original","color":"green","label":"multiply matching parts","style":"solid"},{"to":"topanswer","from":"topproduct","color":"sky","label":"calculate","style":"solid"},{"to":"bottomanswer","from":"bottomproduct","color":"orange","label":"calculate","style":"solid"}]},"order":1,"title":"Form the product","description":"Multiply the numerators, then multiply the denominators. Reaching this unsimplified fraction earns the M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{\\htmlData{anchor=starttop}{15}}{\\htmlData{anchor=startbottom}{24}}"},{"latex":"\\htmlData{anchor=middle}{\\frac{\\htmlData{anchor=divisiontop}{\\color{@sky}{15}\\div3}}{\\htmlData{anchor=divisionbottom}{\\color{@orange}{24}\\div3}}}"},{"latex":"\\htmlData{anchor=answer}{\\frac{\\color{@green}{5}}{\\color{@green}{8}}}"}],"connections":[{"to":"divisiontop","from":"starttop","color":"sky","label":"divide by 3","style":"solid"},{"to":"divisionbottom","from":"startbottom","color":"orange","label":"divide by 3","style":"solid"},{"to":"answer","from":"middle","color":"green","label":"calculate","style":"solid"}]},"order":2,"title":"Simplify the fraction","description":"Divide the numerator and denominator by 3. The final fraction, 5/8, earns the A1 mark."}]}],"generatedAt":"2026-07-21T02:59:32.701Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":75},"walkthroughVideo":null},{"id":"30050684-d8c4-41cc-a12f-76f9e9a7532c","revisionId":"14fbed99-ed86-42b7-a47d-499138378769","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":4,"sourceTitle":"Fractions","calculatorUse":null,"isCalculation":true,"parts":[{"id":"6c02b292-18d5-4475-97e5-8eb986e43ff6","label":"a","content":"Work out $\\frac{1}{5} + \\frac{3}{4}$","markscheme":"- Finds a common denominator, e.g. $\\frac{4}{20} + \\frac{15}{20}$ M1\n- Final answer $\\frac{19}{20}$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds a common denominator, e.g. $\\frac{4}{20} + \\frac{15}{20}$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $\\frac{19}{20}$.","code":"A"}],"finalAnswers":["19/20"],"generatedAt":"2026-05-06T15:44:07.141Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"common_denominator","label":"Use a common denominator","steps":[{"type":"text","order":0,"title":"Choose a denominator","content":"To add fractions, their denominators must be the same. A common denominator of $5$ and $4$ is $20$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\color{@sky}{\\frac{1}{5}}+\\color{@orange}{\\frac{3}{4}}$"},{"latex":"$$\\color{@sky}{\\frac{1\\times4}{5\\times4}}+\\color{@orange}{\\frac{3\\times5}{4\\times5}}$"},{"latex":"$$\\color{@green}{\\frac{4}{20}+\\frac{15}{20}}$"}]},"order":1,"title":"Rewrite both fractions","description":"Multiply each fraction by a form of 1 so that both denominators become 20. Writing the two equivalent fractions earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{4}{20}+\\dfrac{15}{20}$"},{"latex":"$$\\dfrac{\\htmlData{anchor=sum}{\\color{@green}{4+15}}}{20}$"},{"latex":"$$\\dfrac{\\htmlData{anchor=answer}{\\color{@green}{19}}}{20}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"sum"},{"color":"green","style":"text-color","anchor":"answer"}],"connections":[{"to":"answer","from":"sum","color":"green","label":"add numerators","style":"solid"}]},"order":2,"title":"Add the numerators","description":"Keep the common denominator and add only the numerators. The final answer earns A1."}]}],"generatedAt":"2026-07-20T16:32:41.663Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"4cbac5db-5132-4d89-9f06-a74dcb6b9b26","label":"b","content":"Work out $\\frac{4}{5} - \\frac{1}{3}$","markscheme":"- Finds a common denominator, e.g. $\\frac{12}{15} - \\frac{5}{15}$ M1\n- Final answer $\\frac{7}{15}$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Finds a common denominator, e.g. $\\frac{12}{15} - \\frac{5}{15}$.","code":"M"},{"id":"p2","marks":1,"description":"Final answer $\\frac{7}{15}$.","code":"A"}],"finalAnswers":["7/15"],"generatedAt":"2026-05-06T15:44:07.141Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"common_denominator","label":"Use a common denominator","steps":[{"type":"text","order":0,"title":"Choose a denominator","content":"To subtract fractions, their denominators must be the same. The smallest common multiple of $5$ and $3$ is $15$, so use $15$ as the common denominator.","highlights":[{"text":"$$\\frac{4}{5} - \\frac{1}{3}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{4}{5}=\\dfrac{4\\times \\htmlData{anchor=firstfactor}{\\color{@sky}{3}}}{5\\times 3}$"},{"latex":"$$\\dfrac{4}{5}=\\htmlData{anchor=firstfraction}{\\color{@sky}{\\dfrac{12}{15}}}$"},{"latex":"$$\\dfrac{1}{3}=\\dfrac{1\\times \\color{@orange}{5}}{3\\times 5}$"},{"latex":"$$\\dfrac{1}{3}=\\color{@orange}{\\dfrac{5}{15}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"firstfactor"},{"color":"sky","style":"text-color","anchor":"firstfraction"}],"connections":[{"to":"firstfraction","from":"firstfactor","color":"sky","label":"multiply top and bottom"}]},"order":1,"title":"Make equivalent fractions","description":"Multiply each numerator and denominator by the same number. Writing both fractions with a common denominator earns the method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{4}{5}-\\dfrac{1}{3}=\\dfrac{12}{15}-\\dfrac{5}{15}$"},{"latex":"$$\\dfrac{4}{5}-\\dfrac{1}{3}=\\dfrac{\\htmlData{anchor=subtract}{\\color{@green}{12-5}}}{15}$"},{"latex":"$$\\dfrac{4}{5}-\\dfrac{1}{3}=\\htmlData{anchor=answer}{\\color{@green}{\\dfrac{7}{15}}}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"subtract","color":"green","label":"subtract"}]},"order":2,"title":"Subtract the numerators","description":"Keep the common denominator and subtract only the numerators. The correct final answer earns A1."}]}],"generatedAt":"2026-07-20T16:35:12.890Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":72},"walkthroughVideo":null}],"subjectCode":"igcse_maths","summaries":[{"id":"0e46151b-227c-46e1-a3f3-907e0a964da7","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"48c71fdc-924c-4a11-b8cd-eb38f0e91ec6","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"ea6a61c1-5d8d-4cb5-a0f2-c865119f56db","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"30050684-d8c4-41cc-a12f-76f9e9a7532c","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"e77c48ce-d11f-48e5-8729-b4c618951058","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"33dae1d1-25d4-4cb7-8022-d9b3eb21117f","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"5f60042b-0941-45ef-a915-deb13be7db84","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"5bfb14f1-bea0-4004-9cb5-9c234ad4c013","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"f6332863-1ad8-40e3-915f-008acde620fe","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"52c0eb29-a7b3-4a9a-ae8b-3c83623920f9","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"39f8557b-be11-4cea-aedc-bb2eb718a84c","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"87bcdce0-09da-4a80-96fb-85d83f58e286","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true},{"id":"3de4fab5-fe21-4c32-9a26-1597f7ffde6b","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"d2312790-ebe1-4ad1-ac17-55cd464e7720","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":true}],"title":"Fractions","topicId":"ec05d77d-5c98-4537-90d9-bfde953a2efa","topicSidebar":"$L3a","topicSlug":"grade-3-3-1-fractions","topicTitle":"Fractions"}]}],"$L3b","$L3c"]}]