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\\frac{1}{4}b$ M1\n- Final answer $b = 20$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Establishes the constant of proportionality $k = 0.25$ or the relationship $b = 4a$.","code":"M"},{"id":"p2","marks":1,"description":"Substitutes $a=5$ into their proportional relationship.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of 20.","code":"A"}],"finalAnswers":["20"],"generatedAt":"2026-05-07T02:06:23.377Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"constant_of_proportionality","label":"Find the constant","steps":[{"type":"text","order":0,"title":"Write the proportion rule","content":"“Directly proportional” means one variable is a constant multiple of the other. Write\n\n$$a=kb,$$\n\nwhere $k$ is the constant of proportionality.","highlights":[{"text":"$$a$ is directly proportional to $b$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=general}{a=kb}"},{"latex":"\\htmlData{anchor=known}{7=k(28)}"},{"latex":"\\htmlData{anchor=constant}{k=\\frac{1}{4}}"}],"highlights":[{"color":"orange","style":"text-color-with-background","anchor":"constant"}],"connections":[{"to":"known","from":"general","color":"sky","label":"substitute known values"},{"to":"constant","from":"known","color":"orange","label":"divide by 28"}]},"order":1,"title":"Find the constant","description":"Use the known pair of values to calculate the constant of proportionality. This setup earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=rule}{a=\\frac{1}{4}b}"},{"latex":"\\htmlData{anchor=substitution}{5=\\frac{1}{4}b}"},{"latex":"\\htmlData{anchor=answer}{b=20}"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"substitution","from":"rule","color":"sky","label":"substitute a = 5","style":"solid"},{"to":"answer","from":"substitution","color":"green","label":"multiply by 4","style":"solid"}]},"order":2,"title":"Substitute the new value","description":"Keep b as the unknown, then solve the resulting one-step equation. This substitution earns the second method mark, and b = 20 earns the final accuracy mark."}]}],"generatedAt":"2026-07-21T03:58:06.185Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":48},"walkthroughVideo":null},{"id":"86ff1292-75a9-4e8b-9d7c-9122c7810a30","revisionId":"556022cc-8e8d-496c-bf2b-2d10731cd794","questionNumber":"3","questionType":"LA","specification":"$$c$ is inversely proportional to $d$\n\nWhen $c = 3, d = 8$","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Direct and Inverse Proportion","calculatorUse":null,"isCalculation":true,"parts":[{"id":"d03a66c2-4c3b-4683-acff-76e1a128633a","label":"0","content":"Find the value of $c$ when $d = 2$","markscheme":"- Sets up equation $c = \\frac{k}{d}$ or $cd = k$: $3 \\times 8 = 24$ M1\n- Substitutes $d = 2$ into equation: $c = \\frac{24}{2}$ M1\n- Final answer $c = 12$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Finds the constant of proportionality $k = 24$.","code":"M"},{"id":"p2","marks":1,"description":"Divides their constant $k$ by 2.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of 12.","code":"A"}],"finalAnswers":["12"],"generatedAt":"2026-05-07T02:06:23.377Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"inverse_proportion","label":"Use the inverse formula","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"“Inversely proportional” means that the product of the two quantities stays constant. Write\n$$cd=k,$$\nwhere $k$ is a constant.","highlights":[{"text":"$$c$ is inversely proportional to $d$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=formula}{cd=k}"},{"latex":"3\\times 8=\\htmlData{anchor=substitute}{k}"},{"latex":"k=24"}],"connections":[{"to":"substitute","from":"formula","color":"sky","label":"use given values","style":"solid"}]},"order":1,"title":"Find the constant","description":"Use the given pair of values in the equation. This setup earns the first method mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"c=\\frac{\\htmlData{anchor=numerator}{\\color{@sky}{24}}}{\\htmlData{anchor=denominator}{d}}"},{"latex":"c=\\frac{24}{\\htmlData{anchor=newd}{\\color{@orange}{2}}}"},{"latex":"c=12"}],"connections":[{"to":"newd","from":"denominator","color":"orange","label":"substitute d = 2","style":"solid"}]},"order":2,"title":"Substitute the new value","description":"Put the new value of d into the formula. Writing this substitution earns the second method mark."}]}],"generatedAt":"2026-07-21T04:00:25.492Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":59},"walkthroughVideo":null},{"id":"ba748b17-55bc-4631-a821-e3e32204b750","revisionId":"0d70f1ad-71ef-4ea5-8206-a135b0070bc2","questionNumber":"4","questionType":"LA","specification":"$$e$ is directly proportional to $f$\n\nWhen $e = 3, f = 36$","difficulty":1,"difficultyBand":"medium","totalMarks":3,"sourceTitle":"Direct and Inverse Proportion","calculatorUse":null,"isCalculation":true,"parts":[{"id":"31c9c621-e52b-4877-a1f3-e74bd2a963c6","label":"0","content":"Find the value of $f$ when $e = 4$","markscheme":"- Finds $k$ in $e = kf$: $3 = k(36) \\Rightarrow k = \\frac{1}{12}$ M1\n- Substitutes $e = 4$: $4 = \\frac{1}{12}f$ M1\n- Final answer $f = 48$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Finds the constant of proportionality $k = 1/12$ or the ratio $f = 12e$.","code":"M"},{"id":"p2","marks":1,"description":"Applies the relationship to $e = 4$.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of 48.","code":"A"}],"finalAnswers":["48"],"generatedAt":"2026-05-07T02:06:23.377Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"constant_of_proportionality","label":"Constant of proportionality","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"“Directly proportional” means one variable is a constant multiple of the other. Following the form required here, write\n\n$$e=kf$$\n\nwhere $k$ is the **constant of proportionality**.","highlights":[{"text":"$$e$ is directly proportional to $f$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$3=k(36)\\qquad \\htmlData{anchor=divide_from}{\\phantom{\\qquad}}$"},{"latex":"$$\\dfrac{3}{36}=k\\qquad \\htmlData{anchor=divide_to}{\\phantom{\\qquad}}\\qquad \\htmlData{anchor=simplify_from}{\\phantom{\\qquad}}$"},{"latex":"$$k=\\dfrac{1}{12}\\qquad \\htmlData{anchor=simplify_to}{\\phantom{\\qquad}}$"}],"connections":[{"to":"divide_to","from":"divide_from","color":"sky","label":"divide by 36"},{"to":"simplify_to","from":"simplify_from","color":"orange","label":"simplify fraction"}]},"order":1,"title":"Find the constant","description":"Substitute the given pair of values into e = kf, then divide both sides by 36. This earns the first M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$e=kf\\qquad \\htmlData{anchor=sub_from}{\\phantom{\\qquad\\qquad}}$"},{"latex":"$$4=\\dfrac{1}{12}f\\qquad \\htmlData{anchor=sub_to}{\\phantom{\\qquad\\qquad}}$"}],"connections":[{"to":"sub_to","from":"sub_from","color":"green","label":"substitute values"}]},"order":2,"title":"Substitute the new value","description":"Now put the new value of e and the value of k into e = kf. Writing this substitution earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$4=\\frac{1}{12}f\\htmlData{anchor=multiply_from}{\\phantom{\\qquad}}$"},{"latex":"$$48=f\\htmlData{anchor=multiply_to}{\\phantom{\\qquad}}$"},{"latex":"$$f=48$"}],"connections":[{"to":"multiply_to","from":"multiply_from","color":"sky","label":"multiply both sides by 12"}]},"order":3,"title":"Solve for f","description":"Multiply both sides by 12. This gives 48 = f, which can be written as f = 48. The correct final value earns the A1 mark."}]}],"generatedAt":"2026-07-20T16:53:21.842Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":{"percent":60},"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"1a19fa08-ddec-481d-a988-a6c35637d89f","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"86ff1292-75a9-4e8b-9d7c-9122c7810a30","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ba748b17-55bc-4631-a821-e3e32204b750","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bf9eceb9-7e0a-4d2e-9614-dc9ab5c72b4a","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"ef17e09f-9c60-47f7-a276-3a00da3b690b","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1ff0110f-4140-4ec2-b5f7-025901953d96","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"35c749ad-8106-4880-a6ee-69b017086d76","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3a8cf4ff-f66e-4e18-95de-767b1dda9fb2","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9ee55ef1-318a-44fe-8e8b-4c06f741cbe9","difficultyBand":"easy","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"e73e44e5-dce9-43ef-8582-fd3f810b4a4d","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false}],"title":"Direct and Inverse Proportion","topicId":"f8c7a70b-a833-42ef-8ccc-6d04f9a4e1cc","topicSidebar":"$L3c","topicSlug":"grade-7-7-3-direct-and-inverse-proportion","topicTitle":"Direct and Inverse Proportion"}]}],"$L3d","$L3e"]}]