/grade-7-7-6-algebraic-fractions/questions","formulaUrl":"https://assets.mathsgenie.co.uk/board-resources/gcse-maths/gcse-formulae-to-remember.pdf","initialIndex":40,"initialQuestionWindow":[{"id":"2918a5f8-facd-400a-ab36-666ec2a2e50b","revisionId":"3f1fd35b-1b20-4f13-97dc-6b703136b0f3","questionNumber":null,"questionType":"LA","specification":"Given that\n\n$$2x + 1 : x + 1 = x + 5 : x + 3$$","difficulty":3,"difficultyBand":"hard","totalMarks":4,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"eeaccc7c-720f-4cab-b5ae-a339b704ee53","label":"a","content":"Find the possible values of $x$.","markscheme":"- Set up equation: $\\frac{2x+1}{x+1} = \\frac{x+5}{x+3}$ M1\n- Cross multiply: $(2x+1)(x+3) = (x+1)(x+5)$ M1\n- Expand and simplify to $x^2 + x - 2 = 0$ M1\n- Solve to get $x = 1$ or $x = -2$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Sets up the initial equation from the ratio: $\\frac{2x+1}{x+1} = \\frac{x+5}{x+3}$","code":"M"},{"id":"p2","marks":1,"description":"Correctly cross multiplies the algebraic fractions: $(2x+1)(x+3) = (x+1)(x+5)$","code":"M"},{"id":"p3","marks":1,"description":"Expands brackets and simplifies the expression to reach a standard quadratic equation $x^2 + x - 2 = 0$","code":"M"},{"id":"p4","marks":1,"description":"Solves the quadratic equation to find two correct values for $x$.","code":"A"}],"finalAnswers":["x = 1, x = -2","1, -2"],"generatedAt":"2026-08-10T10:38:49.379Z","modelVersion":"gemini-3-flash-preview"},"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"2a3b1c20-f569-49d0-b0d4-f588a06c6b01","revisionId":"8e348491-2e9f-4784-9b80-3c17fb02a073","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","totalMarks":3,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"2f14d166-0395-4a28-9211-f5bda69193de","label":"","content":"Write $\\frac{4x^2 + 19x - 5}{x^2 + 6x + 5}$ in the form $\\frac{ax + b}{x + c}$ where $a$, $b$, and $c$ are integers.","markscheme":"- Factorise numerator: $(4x-1)(x+5)$ M1\n- Factorise denominator: $(x+5)(x+1)$ M1\n- Cancel common factor $(x+5)$ to get $\\frac{4x-1}{x+1}$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Correct factorisation of the numerator $4x^2 + 19x - 5$ into $(4x-1)(x+5)$.","code":"M"},{"id":"p2","marks":1,"description":"Correct factorisation of the denominator $x^2 + 6x + 5$ into $(x+5)(x+1)$.","code":"M"},{"id":"p3","marks":1,"description":"Cancellation of common factor $(x+5)$ to reach the final simplified form.","code":"A"}],"finalAnswers":["\\frac{4x-1}{x+1}"],"generatedAt":"2026-08-10T10:38:39.660Z","modelVersion":"gemini-3-flash-preview"},"explanation":null,"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null},{"id":"55550632-c5bc-488c-8d35-360da049e2e4","revisionId":"cdd806f2-b9e6-4f46-ac80-161e5ebfbf93","questionNumber":null,"questionType":"LA","specification":"","difficulty":3,"difficultyBand":"hard","totalMarks":4,"sourceTitle":null,"calculatorUse":null,"parts":[{"id":"560a9b16-0dbc-42e8-9967-2f292fcdec5e","label":"","content":"Solve $\\frac{1}{x} + \\frac{1}{x + 2} = \\frac{3}{4}$","markscheme":"- Multiply by common denominator $4x(x + 2)$: $4(x + 2) + 4x = 3x(x + 2)$ M1\n- Expand and simplify: $8x + 8 = 3x^2 + 6x$ M1\n- Rearrange to $3x^2 - 2x - 8 = 0$ M1\n- Solve $(3x + 4)(x - 2) = 0$ to get $x = -\\frac{4}{3}$ or $x = 2$ A1","marks":4,"order":0,"markingRubric":{"totalMarks":4,"points":[{"id":"p1","marks":1,"description":"Clearance of all fractions by multiplying by a common denominator (e.g., $4x(x+2)$).","code":"M"},{"id":"p2","marks":1,"description":"Expansion of brackets to reach a linear expression vs a quadratic expression.","code":"M"},{"id":"p3","marks":1,"description":"Correctly reaching the quadratic $3x^2 - 2x - 8 = 0$.","

Algebraic Fractions

Question 39

Algebraic Fractions Questions

Algebraic Fractions

Question 39

Algebraic Fractions Questions