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bg-background","children":["$","$L39",null,{"backHref":"/gcse/maths/aqa/questions","basePath":"/gcse/maths/aqa/grade-7-7-3-direct-and-inverse-proportion/questions","formulaUrl":"https://www.aqa.org.uk/files/resources.mathematics.AQA-8300H-FS-INS-2025_PDF/c87a0d4512612ebfb053370b84e9339e546e9e27.pdf","initialIndex":18,"initialQuestionWindow":[{"id":"3a98a21f-5f11-4c9c-9396-b4a5651ca063","revisionId":"dabf5753-8cd0-4c47-b439-d08e63a12732","questionNumber":null,"questionType":"LA","specification":"Here are four sketches representing various proportionality relationships.\n\n![Graph P](https://assets.mathsgenie.co.uk/questions/64839398-fb14-4120-b8f2-81c87c8e2c50.png)\n**P**\n\n![Graph Q](https://assets.mathsgenie.co.uk/questions/13bf1e35-a008-4840-af6b-3f33f860a245.png)\n**Q**\n\n![Graph R](https://assets.mathsgenie.co.uk/questions/3dd812f8-ae9b-4505-9d79-5ab5cc6a67e3.png)\n**R**\n\n![Graph S](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/3a98a21f-5f11-4c9c-9396-b4a5651ca063/image-4-f79598e7021f6dc4f80d87ee.svg)\n**S**\n\n*Note: Graph P is a parabola starting at the origin, Graph Q is a hyperbola, Graph R is a straight line through the origin, and Graph S shows a curve asymptotic to both axes steeper than Q.*","difficulty":1,"difficultyBand":"easy","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"b157848c-593d-4133-acfc-db4cfe5bcc2e","label":"a","content":"Match each graph with a statement in the table below.\n\n| Proportionality relationship | Graph letter |\n| :--- | :--- |\n| $y$ is directly proportional to $x$ | |\n| $y$ is inversely proportional to $x$ | |\n| $y$ is directly proportional to $x^2$ | |\n| $y$ is inversely proportional to $x^2$ | |","markscheme":"- $y$ is directly proportional to $x$: **R**\n- $y$ is inversely proportional to $x$: **Q**\n- $y$ is directly proportional to $x^2$: **P**\n- $y$ is inversely proportional to $x^2$: **S**\n\n- Award 2 marks for all 4 correct. 2\n- Award 1 mark for 2 or 3 correct. 1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Correctly matches 2 or 3 proportionality statements.","code":"B"},{"id":"p2","marks":1,"description":"Correctly matches all 4 proportionality statements.","code":"B"}],"finalAnswers":["R, Q, P, S"],"generatedAt":"2026-05-06T14:01:42.018Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"recognise_standard_shapes","label":"Recognise standard graph shapes","steps":[{"type":"text","order":0,"title":"Spot the line","content":"Direct proportion has the form $y=kx$, where $k$ is a constant. Its graph is always a **straight line through the origin**.\n\nGraph **R** is the only straight line passing through the origin, so write **R** for “$y$ is directly proportional to $x$”.","highlights":[{"text":"$$y$ is directly proportional to $x$","location":"specification"}]},{"type":"text","order":1,"title":"Spot the parabola","content":"When $y$ is directly proportional to $x^2$, the equation has the form $y=kx^2$. Starting from the origin, the curve becomes steeper as $x$ increases: this is a parabola.\n\nGraph **P** has this shape, so write **P** for “$y$ is directly proportional to $x^2$”.","highlights":[{"text":"$$y$ is directly proportional to $x^2$","location":"specification"}]},{"type":"text","order":2,"title":"Complete the table","content":"The two curves left are inverse relationships. **Q** is the less-steep reciprocal curve, so it is $y\\propto\\frac{1}{x}$. **S** is the steeper squared-reciprocal curve, so it is $y\\propto\\frac{1}{x^2}$.\n\n| Proportionality relationship | Graph letter |\n| :--- | :--- |\n| $y$ is directly proportional to $x$ | **R** |\n| $y$ is inversely proportional to $x$ | **Q** |\n| $y$ is directly proportional to $x^2$ | **P** |\n| $y$ is inversely proportional to $x^2$ | **S** |\n\nThis completes all **4** entries. All four correct earn **2 marks**; **2** or **3** correct earn **1 mark**.","highlights":[{"text":"y$ is inversely proportional to $x","location":"specification"},{"text":"y$ is inversely proportional to $x^2","location":"specification"}]}]}],"generatedAt":"2026-09-01T09:02:03.455Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"b157848c-593d-4133-acfc-db4cfe5bcc2e:recognise_standard_shapes","label":"Recognise standard graph shapes","steps":[{"type":"text","order":0,"title":"Spot the line","content":"Direct proportion has the form $y=kx$, where $k$ is a constant. Its graph is always a **straight line through the origin**.\n\nGraph **R** is the only straight line passing through the origin, so write **R** for “$y$ is directly proportional to $x$”.","highlights":[{"text":"$$y$ is directly proportional to $x$","location":"specification"}]},{"type":"text","order":1,"title":"Spot the parabola","content":"When $y$ is directly proportional to $x^2$, the equation has the form $y=kx^2$. Starting from the origin, the curve becomes steeper as $x$ increases: this is a parabola.\n\nGraph **P** has this shape, so write **P** for “$y$ is directly proportional to $x^2$”.","highlights":[{"text":"$$y$ is directly proportional to $x^2$","location":"specification"}]},{"type":"text","order":2,"title":"Complete the table","content":"The two curves left are inverse relationships. **Q** is the less-steep reciprocal curve, so it is $y\\propto\\frac{1}{x}$. **S** is the steeper squared-reciprocal curve, so it is $y\\propto\\frac{1}{x^2}$.\n\n| Proportionality relationship | Graph letter |\n| :--- | :--- |\n| $y$ is directly proportional to $x$ | **R** |\n| $y$ is inversely proportional to $x$ | **Q** |\n| $y$ is directly proportional to $x^2$ | **P** |\n| $y$ is inversely proportional to $x^2$ | **S** |\n\nThis completes all **4** entries. All four correct earn **2 marks**; **2** or **3** correct earn **1 mark**.","highlights":[{"text":"y$ is inversely proportional to $x","location":"specification"},{"text":"y$ is inversely proportional to $x^2","location":"specification"}]}]}],"generatedAt":"2026-09-01T09:02:03.772Z","modelVersion":"gpt-5.6-terra+diagrams-2","explanationVersion":"v3"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"6eb45812-8536-448d-bb74-6cfe990e889a","revisionId":"17e8d385-cb29-48fc-a339-2d5fa5849887","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"c05d92bc-4c28-4c1d-bebf-160e2f51d891","label":"a","content":"The distance $d$ is directly proportional to the time $t$\n\nWhen $d = 15, t = 5$\n\nFind the value of $t$ when $d = 48$","markscheme":"- Sets up equation $d = kt$ or $\\frac{d}{t} = k$ M1\n- Finds $k = \\frac{15}{5} = 3$ or $t = \\frac{d}{3}$ M1\n- Substitutes $d = 48$ to find $t = 16$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up a correct proportionality equation such as $d = kt$ or $k = \\frac{d}{t}$.","code":"M"},{"id":"p2","marks":1,"description":"Calculates the constant $k = 3$ or the rearranged equation $t = \\frac{d}{3}$.","code":"M"},{"id":"p3","marks":1,"description":"Calculates the correct value for $t$.","code":"A","acceptable":["16"]}],"finalAnswers":[],"generatedAt":"2026-05-06T14:01:45.262Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"constant_of_proportionality","label":"Find the constant of proportionality","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"“Directly proportional” means that the ratio between distance and time stays constant. So introduce a constant of proportionality, $k$:\n\n$$d=kt$$\n\nThis earns the first method mark, M1. The letter $k$ represents the same constant for every pair of values.","highlights":[{"text":"The distance $d$ is directly proportional to the time $t$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=rule}{d=kt}$"},{"latex":"$$\\htmlData{anchor=substitution}{15=k\\times 5}$"},{"latex":"$$\\htmlData{anchor=division}{k=\\dfrac{15}{5}}$"},{"latex":"$$\\htmlData{anchor=constant}{k=3}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substitution"},{"color":"orange","style":"text-color","anchor":"division"},{"color":"green","style":"text-color","anchor":"constant"}],"connections":[{"to":"substitution","from":"rule","color":"sky","label":"substitute d = 15 and t = 5","style":"solid"},{"to":"division","from":"substitution","color":"orange","label":"divide by 5","style":"solid"},{"to":"constant","from":"division","color":"green","label":"calculate","style":"solid"}]},"order":1,"title":"Find the constant","description":"Use the known pair of values, d = 15 and t = 5, in the proportionality equation. Dividing the distance by the time finds k and earns the second method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=known_relation}{d=3t}$"},{"latex":"$$\\color{@sky}{\\htmlData{anchor=new_distance}{48=3t}}$"},{"latex":"$$\\color{@orange}{\\htmlData{anchor=divide_three}{t=\\dfrac{48}{3}}}$"},{"latex":"$$\\color{@green}{\\htmlData{anchor=final_time}{t=16}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"new_distance"},{"color":"orange","style":"text-color","anchor":"divide_three"},{"color":"green","style":"text-color-with-background","anchor":"final_time"}],"connections":[{"to":"new_distance","from":"known_relation","color":"sky","label":"substitute d = 48","style":"solid"},{"to":"divide_three","from":"new_distance","color":"orange","label":"divide by 3","style":"solid"},{"to":"final_time","from":"divide_three","color":"green","label":"calculate","style":"solid"}]},"order":2,"title":"Find the new time","description":"Substitute d = 48 into the relationship and obtain t = 16 to earn A1. A quick check is 3 × 16 = 48, so the result fits the relationship."}]}],"generatedAt":"2026-07-19T18:27:20.441Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"c05d92bc-4c28-4c1d-bebf-160e2f51d891:proportionality-constant","label":"Using Proportionality Constant","steps":[{"type":"text","order":0,"title":"Set up the proportion equation","content":"Since we are told that the distance $d$ is directly proportional to the time $t$, we can express this relationship mathematically using a constant of proportionality, $k$:\n\n$$d = kt$$\n\nSetting up this equation correctly earns you your first method mark (**M1**).","highlights":[{"text":"directly proportional","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"We are given that when "},{"latex":"$$\\htmlData{anchor=d-val}{\\color{@sky}{d = 15}}$"},{"text":", "},{"latex":"$$\\htmlData{anchor=t-val}{\\color{@amber}{t = 5}}$"},{"text":". Substitute these initial values into our formula to solve for $k$:"},{"latex":"$$$\\htmlData{anchor=d-sub}{\\color{@sky}{15}} = k \\times \\htmlData{anchor=t-sub}{\\color{@amber}{5}}$$"},{"latex":"$$$k = \\frac{15}{5} = 3$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"d-val"},{"color":"amber","style":"text-color","anchor":"t-val"}],"connections":[{"to":"d-sub","from":"d-val","color":"sky","label":"substitute d","style":"solid"},{"to":"t-sub","from":"t-val","color":"amber","label":"substitute t","style":"solid"}]},"order":1,"title":"Find the constant $k$","description":"Finding that the constant of proportionality $k = 3$ earns the second method mark (**M1**)."},{"type":"text","order":2,"title":"Calculate the final time","content":"Now that we know $k = 3$, we can write our complete formula relating $d$ and $t$:\n\n$$d = 3t$$\n\nTo find the value of $t$ when $d = 48$, substitute this value into the formula and solve:\n\n$$\\begin{aligned}\n48 &= 3t \\\\\nt &= \\frac{48}{3} \\\\\nt &= 16\n\\end{aligned}$$\n\nThis final answer completes the calculation and earns the accuracy mark (**A1**).","highlights":[{"text":"value of $t$ when $d = 48$","location":"specification"}]}]},{"id":"c05d92bc-4c28-4c1d-bebf-160e2f51d891:ratio-method","label":"Using Direct Ratios","steps":[{"type":"text","order":0,"title":"Identify proportional variables","content":"We are given that the distance $d$ is directly proportional to the time $t$. This means that the ratio of distance to time remains constant throughout:\n\n$$\\frac{d_1}{t_1} = \\frac{d_2}{t_2}$$","highlights":[{"text":"The distance $d$ is directly proportional to the time $t$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Given values: } \\htmlData{anchor=d1}{\\color{@sky}{d_1 = 15}}, \\; \\htmlData{anchor=t1}{\\color{@amber}{t_1 = 5}} \\quad \\text{and} \\quad \\htmlData{anchor=d2}{\\color{@purple}{d_2 = 48}}$"},{"latex":"$$$\\frac{\\htmlData{anchor=d1-sub}{\\color{@sky}{15}}}{\\htmlData{anchor=t1-sub}{\\color{@amber}{5}}} = \\frac{\\htmlData{anchor=d2-sub}{\\color{@purple}{48}}}{\\htmlData{anchor=t2}{\\color{@teal}{t}}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"d1"},{"color":"amber","style":"text-color","anchor":"t1"},{"color":"purple","style":"text-color","anchor":"d2"}],"connections":[{"to":"d1-sub","from":"d1","color":"sky","label":null,"style":"solid"},{"to":"t1-sub","from":"t1","color":"amber","label":null,"style":"solid"},{"to":"d2-sub","from":"d2","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up the ratio","description":"We substitute our known values into the ratio equivalence formula."},{"type":"text","order":2,"title":"Solve for t","content":"Simplify the fraction on the left side of our equation:\n\n$$\\frac{15}{5} = 3$$\n\nNow, rewrite the equation with this simplified value:\n\n$$3 = \\frac{48}{t}$$\n\nMultiply both sides by $t$ and then divide by $3$ to isolate $t$:\n\n$$\\begin{aligned} 3t &= 48 \\\\ t &= \\frac{48}{3} \\\\ t &= 16 \\end{aligned}$$"}]}],"generatedAt":"2026-06-12T19:29:23.665Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"0b64bec9-f207-4f1a-be86-92ca919fd208","revisionId":"08da8af6-c9ef-420d-8594-868039d6081c","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"94bb779b-099f-441f-84b2-eecef1f43103","label":"a","content":"$$p$ is directly proportional to $q$\n\nWhen $p = 6$, $q = 18$\n\nFind the value of $q$ when $p = 11$","markscheme":"- Sets up equation $p = kq$ or $\\frac{p}{q} = k$ M1\n- Finds $k = \\frac{6}{18} = \\frac{1}{3}$ or $q = 3p$ M1\n- Substitutes $p = 11$ to find $q = 33$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up equation $p = kq$ or $\\frac{p}{q} = k$","code":"M"},{"id":"p2","marks":1,"description":"Finds $k = \\frac{6}{18} = \\frac{1}{3}$ or $q = 3p$","code":"M"},{"id":"p3","marks":1,"description":"Substitutes $p = 11$ to find $q = 33$","code":"A"}],"finalAnswers":["33"],"generatedAt":"2026-05-06T14:02:00.736Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"unitary_ratio","label":"Use the unitary ratio","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"Direct proportion means the ratio between the two quantities is constant, so write\n\n$$\\frac{p}{q}=k$$\n\nWriting this proportionality equation earns the first **M1** mark.","highlights":[{"text":"$$p$ is directly proportional to $q$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=RATIO}{18 \\div 6 = 3}$"},{"latex":"$$\\htmlData{anchor=RULE}{q = 3p}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"RATIO"},{"color":"orange","style":"text-color","anchor":"RULE"}],"connections":[{"to":"RULE","from":"RATIO","color":"orange","label":"q is 3 times p"}]},"order":1,"title":"Find q per p","description":"For each 1 in p, there are 3 in q, so q = 3p. This earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=SUBSTITUTION}{q = 3 \\times 11}$"},{"latex":"$$\\htmlData{anchor=ANSWER}{q = 33}$"}],"highlights":[{"color":"green","style":"text-color-with-background","anchor":"ANSWER"}],"connections":[{"to":"ANSWER","from":"SUBSTITUTION","color":"green","label":"calculate"}]},"order":2,"title":"Substitute the new p","description":"Substitute p = 11 into q = 3p. Reaching q = 33 earns the A1 mark."}]},{"id":"constant_of_proportionality","label":"Constant of proportionality","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"Direct proportionality means one variable is a constant multiple of the other, so write $p=kq$. This earns the first M1 mark.","highlights":[{"text":"$$p$ is directly proportional to $q$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=SOURCE}{6 = k \\times 18}$"},{"latex":"$$\\htmlData{anchor=DIVIDE}{k = \\dfrac{6}{18}}$"},{"latex":"$$\\htmlData{anchor=SIMPLIFIED}{k = \\dfrac{1}{3}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"SOURCE"},{"color":"sky","style":"text-color","anchor":"DIVIDE"},{"color":"green","style":"text-color-with-background","anchor":"SIMPLIFIED"}],"connections":[{"to":"DIVIDE","from":"SOURCE","color":"sky","label":"divide both sides by 18"}]},"order":1,"title":"Find the constant","description":"Use p = 6 and q = 18. Finding k = 1/3 earns the second M1 mark. Simplify 6/18 to 1/3."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=EQUATION}{11 = \\dfrac{1}{3}q}$"},{"latex":"$$\\htmlData{anchor=MULTIPLY}{q = 11 \\times 3}$"},{"latex":"$$\\htmlData{anchor=ANSWER}{q = 33}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"EQUATION"},{"color":"sky","style":"text-color","anchor":"MULTIPLY"},{"color":"green","style":"text-color-with-background","anchor":"ANSWER"}],"connections":[{"to":"MULTIPLY","from":"EQUATION","color":"sky","label":"multiply both sides by 3"},{"to":"ANSWER","from":"MULTIPLY","color":"green","label":"calculate"}]},"order":2,"title":"Substitute the new value","description":"Use k = 1/3 and p = 11, then solve for q. Reaching q = 33 earns the A1 mark."}]}],"generatedAt":"2026-07-20T11:16:35.296Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"94bb779b-099f-441f-84b2-eecef1f43103:algebraic-constant","label":"Constant of Proportionality","steps":[{"type":"text","order":0,"title":"Set up the equation","content":"Since $p$ and $q$ are directly proportional, we can represent their relationship using a constant of proportionality, $k$. We write this as: $$q = kp$$","highlights":[{"text":"p is directly proportional to q","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$p = \\htmlData{anchor=p-val}{\\color{@sky}{6}}, q = \\htmlData{anchor=q-val}{\\color{@amber}{18}}$"},{"latex":"$$\\htmlData{anchor=q-sub}{\\color{@amber}{18}} = k \\times \\htmlData{anchor=p-sub}{\\color{@sky}{6}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"p-val"},{"color":"amber","style":"text-color","anchor":"q-val"}],"connections":[{"to":"p-sub","from":"p-val","color":"sky","label":null,"style":"solid"},{"to":"q-sub","from":"q-val","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Substitute the known values","description":"Use the initial values provided in the question to find the specific value for $k$."},{"type":"text","order":2,"title":"Find the constant k","content":"To solve for $k$, divide both sides of the equation by 6: $$\\begin{aligned} k &= \\frac{18}{6} \\\\ k &= 3 \\end{aligned}$$ Now we know the exact relationship is $$q = 3p$$","highlights":[{"text":"When p = 6, q = 18","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$p = \\htmlData{anchor=p-new}{\\color{@purple}{11}}, k = \\htmlData{anchor=k-val}{\\color{@green}{3}}$"},{"latex":"$$q = \\htmlData{anchor=k-sub}{\\color{@green}{3}} \\times \\htmlData{anchor=p-sub2}{\\color{@purple}{11}}$"},{"latex":"$$q = 33$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"p-new"},{"color":"green","style":"text-color","anchor":"k-val"}],"connections":[{"to":"p-sub2","from":"p-new","color":"purple","label":null,"style":"solid"},{"to":"k-sub","from":"k-val","color":"green","label":null,"style":"solid"}]},"order":3,"title":"Calculate the final value","description":"Substitute the new value of $p$ into our equation to find $q$."},{"type":"text","order":4,"title":"State the final answer","content":"When $p = 11$, the value of $q$ is 33.","calculator":[{"gdc":{"expression":"3 * 11","instructions":"Type 3 multiplied by 11 in the calculation menu."},"ti84":{"expression":"3 * 11","instructions":"Enter 3 times 11 on the home screen and press ENTER."},"desmos":{"expression":"3 * 11","instructions":"Type 3 * 11 into the expression bar."},"description":"You can quickly verify the multiplication using a calculator."}],"highlights":[{"text":"Find the value of q when p = 11","location":"specification"}]}]},{"id":"94bb779b-099f-441f-84b2-eecef1f43103:ratio-method","label":"Ratio Method","steps":[{"type":"text","order":0,"title":"Identify the relationship","content":"Since $p$ is directly proportional to $q$, the ratio between them is constant. This means that as one value changes, the other changes at the same rate. We can represent this relationship using the proportion formula:\n\n$$\\frac{q_1}{p_1} = \\frac{q_2}{p_2}$$","highlights":[{"text":"p is directly proportional to q","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$p_1 = \\htmlData{anchor=p1}{\\color{@sky}{6}}, \\quad q_1 = \\htmlData{anchor=q1}{\\color{@amber}{18}}, \\quad p_2 = \\htmlData{anchor=p2}{\\color{@purple}{11}}$"},{"latex":"$$\\frac{\\htmlData{anchor=q1-sub}{\\color{@amber}{18}}}{\\htmlData{anchor=p1-sub}{\\color{@sky}{6}}} = \\frac{q}{\\htmlData{anchor=p2-sub}{\\color{@purple}{11}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"p1"},{"color":"amber","style":"text-color","anchor":"q1"},{"color":"purple","style":"text-color","anchor":"p2"}],"connections":[{"to":"q1-sub","from":"q1","color":"amber","label":null,"style":"solid"},{"to":"p1-sub","from":"p1","color":"sky","label":null,"style":"solid"},{"to":"p2-sub","from":"p2","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up the proportion","description":"We plug the known values into our ratio formula to find the missing value of $q$."},{"type":"text","order":2,"title":"Solve for q","content":"First, simplify the ratio on the left side: \n$$\\frac{18}{6} = 3$$\n\nNow, our equation looks like this:\n$$3 = \\frac{q}{11}$$\n\nMultiply both sides by $11$ to isolate $q$:\n$$q = 3 \\times 11 = 33$$","calculator":[{"gdc":{"expression":"(18 / 6) * 11","instructions":"Calculate the ratio first, then multiply by the new value."},"ti84":{"expression":"(18 / 6) * 11","instructions":"Type 18 divided by 6, then multiply the result by 11."},"desmos":{"expression":"\\frac{18}{6} \\cdot 11","instructions":"Calculate the result of the proportion."},"description":"Perform the calculation directly on a calculator."}]},{"type":"text","order":3,"title":"Final answer","content":"The value of $q$ when $p = 11$ is $33$."}]},{"id":"94bb779b-099f-441f-84b2-eecef1f43103:multiplier-method","label":"Multiplier Method","steps":[{"type":"text","order":0,"title":"Identify the relationship","content":"Since $p$ is directly proportional to $q$, we know that $q$ will always be a fixed number of times larger (or smaller) than $p$. Our goal is to find this multiplier using the values we already have.","highlights":[{"text":"p is directly proportional to q","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Multiplier} = \\frac{\\htmlData{anchor=q-known}{\\color{@purple}{18}}}{\\htmlData{anchor=p-known}{\\color{@sky}{6}}} = \\htmlData{anchor=k-val}{\\color{@amber}{3}}$"},{"text":"This means that for any pair, $q$ is 3 times the value of $p$."}],"highlights":[{"color":"purple","style":"text-color","anchor":"q-known"},{"color":"sky","style":"text-color","anchor":"p-known"},{"color":"amber","style":"box","anchor":"k-val"}],"connections":[{"to":"k-val","from":"q-known","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Calculate the multiplier","description":"We look at the first pair of values to see how many times larger $q$ is than $p$."},{"type":"text","order":2,"title":"Find the missing value","content":"Now we apply that same multiplier to our new value of $p$. Since $p = 11$, we multiply it by $3$ to find $q$.\n\n$$q = 11 \\times 3 = 33$$\n\nYou would earn one mark for finding the multiplier and the final accuracy mark for the answer 33.","calculator":[{"gdc":{"expression":"11 * 3","instructions":"Type the product into the calculation menu."},"ti84":{"expression":"11 * 3","instructions":"Enter the multiplication on the main screen."},"desmos":{"expression":"11 \\cdot 3","instructions":"Type the expression into a new cell."},"description":"You can quickly verify the multiplication on any scientific or graphing calculator."}],"highlights":[{"text":"p = 11","location":"specification"}]}]}],"generatedAt":"2026-06-08T04:31:39.443Z","modelVersion":"gemini-3-flash-preview","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/0b64bec9-f207-4f1a-be86-92ca919fd208.mp4","durationSeconds":59,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/0b64bec9-f207-4f1a-be86-92ca919fd208.jpg"}}],"subjectCode":"gcse_maths__aqa","summaries":[{"id":"0370e440-684e-46ea-a3fa-6937e3db9a05","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0b21e4ce-2aa7-45b2-b464-98e339901c85","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"27589c1f-32da-4695-b8bd-3faa83c8d3dd","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"57c1fe12-6e15-49de-92b2-ddae4b8b9c9a","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"5a0ad258-3fe5-418a-a138-bb1d4d87e87e","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6eec5e8a-9c5b-4455-bda0-6945db079f37","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"7477ec4c-2dc1-4082-bb08-fcb3b0fd13b8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"757331c8-a6fa-4b6d-97ba-d245f0d45b28","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b979479a-05b0-49be-baf4-442063764342","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"bea86dca-43cb-4ea8-af8e-d74311b4ccb2","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"f6c936c1-63d4-4009-8918-b23c74434a5a","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4de70480-5e75-4cb2-ae7b-803db94bb6d0","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"1b43779a-f694-429a-abd3-d44499725e78","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"3a98a21f-5f11-4c9c-9396-b4a5651ca063","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"4645bd38-2552-429c-86ea-daf83fa2c4c5","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"48bf839f-860d-44d2-8a0f-4d380dccd331","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"5e5d89ac-7db4-46e5-9476-70c7ec0c36c7","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"6dbc6f93-a342-4918-817a-0df714b922a3","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"6eb45812-8536-448d-bb74-6cfe990e889a","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"975a6a49-d5bd-47a6-bb7f-8ff4445983dd","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"adc3389a-a1e5-4708-a4ac-58c3db86c759","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"d99c6e49-3805-4e02-b244-0467494abaa8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"e235a0c1-11fd-489e-a657-031eb01bc62b","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"389c9673-ed59-4511-87ba-8425f78da15c","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0b64bec9-f207-4f1a-be86-92ca919fd208","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"0e3f7d1b-c30e-452e-8f31-149710f1876e","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"20adab41-24e4-44e3-afee-e274b53b8055","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"2807369b-9d8a-4bcc-9edc-3ec3e08f5aef","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"4cf39bef-a9fd-4155-a904-09335774cb96","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"661bad9c-f711-483d-a36d-e74d145b48cc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and Inverse Proportion","topicId":"f58ae332-d381-454c-b364-27ee3900c771","topicSidebar":"$L3a","topicSlug":"grade-7-7-3-direct-and-inverse-proportion","topicTitle":"Direct and Inverse Proportion"}]}],"$L3b","$L3c"]}]

Direct and Inverse Proportion

Question 7

Direct and Inverse Proportion Questions

Direct and Inverse Proportion

Question 7

Direct and Inverse Proportion Questions