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Writing the resulting equation earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"2ax-3a=4+2bx"},{"latex":"2ax-2bx=4+3a"}]},"order":3,"title":"Expand and rearrange","description":"Expand the bracket. Then subtract 2bx from both sides and add 3a to both sides, collecting the terms containing x on the left; this earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"x(2a-2b)=4+3a"},{"latex":"x=\\frac{4+3a}{2a-2b}"}]},"order":5,"title":"Make x the subject","description":"Factorise the left-hand side, then divide both sides by the coefficient of x. This gives the final answer and earns A1."}]}],"generatedAt":"2026-07-21T04:05:11.258Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"4cbe4739-918d-4f11-9d03-99fb594d5d29","revisionId":"be4013a9-d28f-416c-857e-9ce85bdb6ad5","questionNumber":"13","questionType":"LA","specification":"","difficulty":3,"difficultyBand":"medium","totalMarks":4,"sourceTitle":"Rearranging Harder Formulae","calculatorUse":null,"isCalculation":false,"parts":[{"id":"9d4f26d9-ad11-4b2b-b69b-e6d46dc83ce8","label":"0","content":"Make $b$ the subject of the formula $\\frac{1}{a} = \\frac{1}{b} + \\frac{1}{c}$","markscheme":"- Multiply by $abc$ (or common denominator): $bc = ac + ab$ M1\n- Rearrange to get $b$ terms on one side: $bc - ab = ac$ M1\n- Factorise $b$: $b(c - a) = ac$ M1\n- Divide: $b = \\frac{ac}{c - a}$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Multiplies through by $abc$ or finds a common denominator to eliminate fractions.","code":"M"},{"id":"p2","marks":1,"description":"Rearranges the equation to group all terms with $b$ on one side.","code":"M"},{"id":"p3","marks":1,"description":"Factorises $b$ from the expression.","code":"M"},{"id":"p4","marks":1,"description":"Divides to isolate $b$ as the subject.","code":"A"}],"finalAnswers":["b = \\frac{ac}{c - a}"],"generatedAt":"2026-05-07T02:09:01.541Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"clear_denominators","label":"Clear the fractions","steps":[{"type":"text","order":0,"title":"Choose a common multiplier","content":"To make $b$ the subject, first remove the fractions. The expression $abc$ is a common multiple of the three denominators $a$, $b$ and $c$, so multiply **every term** by $abc$. This earns the first method mark (M1) once the fractions have been cleared.","highlights":[{"text":"Make $b$ the subject of the formula $\\frac{1}{a} = \\frac{1}{b} + \\frac{1}{c}$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\frac{1}{a}=\\frac{1}{b}+\\frac{1}{c}"},{"latex":"\\htmlData{anchor=multiplied}{abc}\\left(\\frac{1}{a}\\right)=abc\\left(\\frac{1}{b}\\right)+abc\\left(\\frac{1}{c}\\right)"},{"latex":"\\color{@green}{\\htmlData{anchor=cleared}{bc=ac+ab}}"}],"highlights":[{"color":"green","style":"text-color","anchor":"cleared"}]},"order":1,"title":"Clear denominators","description":"Multiply every term by the common denominator abc, so that each denominator cancels."},{"type":"text","order":2,"title":"Collect the b terms","content":"Both $bc$ and $ab$ contain $b$. Subtract $ab$ from both sides so that all the terms involving $b$ are together. This is the second method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"bc=ac+\\htmlData{anchor=abterm}{ab}"},{"latex":"bc-ab=ac+ab-ab"},{"latex":"\\color{@green}{\\htmlData{anchor=rearranged}{bc-ab=ac}}"}],"highlights":[{"color":"green","style":"text-color","anchor":"rearranged"}]},"order":3,"title":"Rearrange the equation","description":"Subtract ab from both sides, then simplify the right-hand side."},{"type":"text","order":4,"title":"Factorise b","content":"Factorise means write a common factor outside brackets. The common factor of $bc$ and $ab$ is $b$, leaving $c-a$ inside the bracket. This earns the third method mark (M1)."},{"type":"hyper_latex","block":{"nodes":[{"latex":"bc-ab=ac"},{"latex":"\\color{@purple}{\\htmlData{anchor=factorised}{b(c-a)=ac}}"}],"highlights":[{"color":"purple","style":"text-color","anchor":"factorised"}]},"order":5,"title":"Factorise b","description":"Both terms on the left contain b, so factorise b."},{"type":"text","order":6,"title":"Divide by c minus a","content":"Now divide both sides by $c-a$ to leave $b$ on its own. This final accurate division earns the **A1** mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=start}{b(c-a)=ac}"},{"latex":"\\color{@teal}{\\htmlData{anchor=answer}{b=\\frac{ac}{c-a}}}"}],"highlights":[{"color":"teal","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"start","color":"teal","label":"divide by c minus a","style":"solid"}]},"order":7,"title":"Write b alone","description":"Divide both sides by c minus a."}]}],"generatedAt":"2026-07-21T04:06:04.469Z","modelVersion":"gpt-5.6-terra","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":null}],"subjectCode":"igcse_maths__oxford_aqa","summaries":[{"id":"2221441c-51c6-4cfd-93ef-53180ca8ac25","difficultyBand":"easy","calculatorUse":null,"totalMarks":1,"questionType":"LA","isCalculation":false},{"id":"12ad6831-a4eb-48e4-b127-d2dda549930e","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"8e9b6070-da7d-4c7d-a510-9a8450b986cd","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"c7909575-fb91-43f8-8546-2b607d2c86aa","difficultyBand":"easy","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"677e251b-a63e-4950-b2ba-89b1e78838a9","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"ddfb58b2-0ae1-442b-a426-76015be5284f","difficultyBand":"medium","calculatorUse":null,"totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"0d84ec73-daad-4d69-85c5-1c042c944964","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"eb2afcdb-1c77-41d5-8538-845fe750eb12","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"755c4481-3ea7-44ff-9272-f4357df6e354","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"f6801ff0-5743-4c8c-b825-57d53a40fe4f","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"1ce4d46d-9bd2-4a8c-92f3-7f6bfc30bef8","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"e33c522c-f4e0-4a9f-9178-2bb9f609023d","difficultyBand":"medium","calculatorUse":null,"totalMarks":3,"questionType":"LA","isCalculation":false},{"id":"4cbe4739-918d-4f11-9d03-99fb594d5d29","difficultyBand":"medium","calculatorUse":null,"totalMarks":4,"questionType":"LA","isCalculation":false}],"title":"Rearranging Harder Formulae","topicId":"17e30389-ddfd-4fad-9a02-78a7d4791133","topicSidebar":"$L3c","topicSlug":"grade-7-7-7-rearranging-harder-formulae","topicTitle":"Rearranging Harder Formulae"}]}],"$L3d","$L3e"]}]

Rearranging Harder Formulae

Question 9

Rearranging Harder Formulae Questions

Rearranging Harder Formulae

Question 9

Rearranging Harder Formulae Questions