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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":4,"initialQuestionWindow":[{"id":"57c1fe12-6e15-49de-92b2-ddae4b8b9c9a","revisionId":"4d3ab473-a1ca-4330-9486-075e7e5cf127","questionNumber":null,"questionType":"LA","specification":"The table shows pairs of values for $x$ and $y$\n\n| $x$ | 6 | 10 |\n| :--- | :--- | :--- |\n| $y$ | 72 | 120 |","difficulty":2,"difficultyBand":"easy","totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"a50014ba-03ec-452a-8a16-2f89c4f15cc6","label":"i","content":"Tick the correct statement below.\n\n- $y \\propto x$ .....\n- $y \\propto x^2$ .....\n- $y \\propto x^3$ .....","markscheme":"- Tests relationships: $72/6 = 12$ vs $120/10 = 12$ (linear); $72/6^2 = 2$ vs $120/10^2 = 1.2$ (not quadratic) M1\n- Identifies $y \\propto x$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Calculates ratios of $y$ to $x, x^2$ or $x^3$ for both pairs of values to test for proportionality.","code":"M"},{"id":"p2","marks":1,"description":"Correctly identifies that the relationship is linear direct proportion.","code":"A"}],"finalAnswers":["$$y \\propto x$"],"generatedAt":"2026-05-06T14:01:44.115Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":null,"options":[]},{"id":"3602f652-7fed-4571-98ac-69714a1efc7f","label":"ii","content":"Write a formula for $y$ in terms of $x$","markscheme":"- Uses $y = kx$ and substitutes values to find $k = 12$ M1\n- $y = 12x$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Evidence of finding the constant of proportionality $k$.","code":"M"},{"id":"p2","marks":1,"description":"Final formula $y = 12x$.","code":"A"}],"finalAnswers":["$$y = 12x$"],"generatedAt":"2026-05-06T14:01:44.115Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"table_multiplier","label":"Find the multiplier","steps":[{"type":"text","order":0,"title":"Choose the relationship","content":"Each $y$-value is found by multiplying its matching $x$-value by the same constant. Call this constant $k$, so start with\n\n$$y=kx$$","highlights":[{"text":"Write a formula for $y$ in terms of $x$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=\\htmlData{anchor=model}{\\color{@sky}{kx}}$"},{"latex":"$$72=\\htmlData{anchor=substituted}{\\color{@sky}{6k}}$"},{"latex":"$$k=\\dfrac{72}{6}$"},{"latex":"$$k=12$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"model"},{"color":"sky","style":"text-color","anchor":"substituted"}],"connections":[{"to":"substituted","from":"model","color":"sky","label":"substitute the pair"}]},"order":1,"title":"Find the multiplier","description":"Use the table pair where x is 6 and y is 72. Substituting this pair and finding the constant earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y=\\htmlData{anchor=general}{\\color{@green}{kx}}$"},{"latex":"$$y=\\htmlData{anchor=answer}{\\color{@green}{12x}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"general"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"answer","from":"general","color":"green","label":"substitute k = 12"}]},"order":2,"title":"Write the formula","description":"Replace the constant with the value found. This exact formula earns A1."}]}],"generatedAt":"2026-07-19T07:54:54.484Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"a50014ba-03ec-452a-8a16-2f89c4f15cc6:ratio_comparison","label":"(i) Comparing Ratios","steps":[{"type":"text","order":0,"title":"Understand the method","content":"To find which proportional relationship is correct, we use the **Comparing Ratios** method.\n\nIf two variables are proportional in the form $y \\propto x^n$, then the ratio of $y$ to $x^n$ must be constant for all pairs of values:\n\n$$\\frac{y}{x^n} = k$$\n\nwhere $k$ is the constant of proportionality. We will calculate this ratio for each option using our two data points:\n* **Point 1:** $x = 6$, $y = 72$\n* **Point 2:** $x = 10$, $y = 120$"},{"type":"text","order":1,"title":"Test linear relationship","content":"Let's first test the linear relationship $y \\propto x$ (where $n = 1$). We calculate the ratio $\\frac{y}{x}$ for both data points:\n\n* **For the first point ($x = 6, y = 72$):**\n $$\\frac{72}{6} = 12$$\n\n* **For the second point ($x = 10, y = 120$):**\n $$\\frac{120}{10} = 12$$\n\nSince both ratios are equal to $12$, the ratio is constant. This strongly suggests that $y \\propto x$ is the correct relationship.","highlights":[{"text":"y \\propto x","index":0,"location":"option"}]},{"type":"text","order":2,"title":"Test quadratic relationship","content":"To be thorough, let's also test the quadratic relationship $y \\propto x^2$ (where $n = 2$) by calculating the ratio $\\frac{y}{x^2}$:\n\n* **For the first point ($x = 6, y = 72$):**\n $$\\frac{72}{6^2} = \\frac{72}{36} = 2$$\n\n* **For the second point ($x = 10, y = 120$):**\n $$\\frac{120}{10^2} = \\frac{120}{100} = 1.2$$\n\nSince the ratios ($2$ and $1.2$) are not equal, the relationship is **not** $y \\propto x^2$.","highlights":[{"text":"y \\propto x^2","index":1,"location":"option"}]},{"type":"text","order":3,"title":"Test cubic relationship","content":"Lastly, let's test the cubic relationship $y \\propto x^3$ (where $n = 3$) by calculating the ratio $\\frac{y}{x^3}$:\n\n* **For the first point ($x = 6, y = 72$):**\n $$\\frac{72}{6^3} = \\frac{72}{216} \\approx 0.333$$\n\n* **For the second point ($x = 10, y = 120$):**\n $$\\frac{120}{10^3} = \\frac{120}{1000} = 0.12$$\n\nSince these ratios are not equal, the relationship is **not** $y \\propto x^3$.","highlights":[{"text":"y \\propto x^3","index":2,"location":"option"}]},{"type":"text","order":4,"title":"Identify correct statement","content":"Comparing our results, only the relationship $y \\propto x$ maintains a constant ratio of $12$ across both data points.\n\nTherefore, the correct statement to tick is:\n* **$y \\propto x$**","highlights":[{"text":"y \\propto x","index":0,"location":"option"}]}]},{"id":"a50014ba-03ec-452a-8a16-2f89c4f15cc6:solving_for_constant","label":"(i) Finding and Testing the Constant","steps":[{"type":"text","order":0,"title":"Define the relationship","content":"To find which statement is correct, we can write down equations for the three possible proportional relationships using a constant of proportionality, $k$:\n\n1. For linear proportion ($y \\propto x$):\n$$y = kx$$\n\n2. For quadratic proportion ($y \\propto x^2$):\n$$y = kx^2$$\n\n3. For cubic proportion ($y \\propto x^3$):\n$$y = kx^3$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Let's find the value of $k$ for each relationship using our first pair of values: $x = 6$ and $y = 72$."},{"latex":"$$$\n\\begin{aligned}\n\\text{If } y &= kx: & 72 &= k(6) & \\Rightarrow \\quad \\htmlData{anchor=k-linear}{\\color{@sky}{k}} &= \\htmlData{anchor=calc-linear}{\\color{@sky}{12}} \\\\\n\\text{If } y &= kx^2: & 72 &= k(6^2) & \\Rightarrow \\quad \\htmlData{anchor=k-quad}{\\color{@amber}{k}} &= \\htmlData{anchor=calc-quad}{\\color{@amber}{2}} \\\\\n\\text{If } y &= kx^3: & 72 &= k(6^3) & \\Rightarrow \\quad \\htmlData{anchor=k-cubic}{\\color{@purple}{k}} &= \\htmlData{anchor=calc-cubic}{\\color{@purple}{\\frac{1}{3}}}\n\\end{aligned}\n$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"k-linear"},{"color":"amber","style":"text-color","anchor":"k-quad"},{"color":"purple","style":"text-color","anchor":"k-cubic"}],"connections":[]},"order":1,"title":"Solve for the constant","description":"We substitute $x=6$ and $y=72$ into each model to solve for the constant $k$."},{"type":"text","order":2,"title":"Test the second pair","content":"Now we test which of these equations correctly predicts the second pair of values from our table: $x = 10$ and $y = 120$.\n\n* **Testing $y = 12x$ (Linear):**\n$$y = 12 \\times 10 = 120$$\nThis matches the table value of $120$!\n\n* **Testing $y = 2x^2$ (Quadratic):**\n$$y = 2 \\times 10^2 = 2 \\times 100 = 200$$\nThis does not match $120$.\n\n* **Testing $y = \\frac{1}{3}x^3$ (Cubic):**\n$$y = \\frac{1}{3} \\times 10^3 = \\frac{1000}{3} \\approx 333.3$$\nThis does not match $120$.","highlights":[{"text":"y \\propto x","location":"specification"},{"text":"y \\propto x^2","location":"specification"},{"text":"y \\propto x^3","location":"specification"}]},{"type":"text","order":3,"title":"Select correct statement","content":"Because only the relationship $y = 12x$ is consistent for both data points, the correct proportion is direct linear proportion:\n\n$$y \\propto x$$\n\nSo, we tick the first option.","highlights":[{"text":"y \\propto x","index":0,"location":"option"}]}]},{"id":"3602f652-7fed-4571-98ac-69714a1efc7f:direct_variation","label":"(ii) Direct Variation Approach","steps":[{"type":"text","order":0,"title":"Set up the variation","content":"Since $y$ is directly proportional to $x$, we can represent their relationship using the direct variation equation:\n\n$$y = kx$$\n\nwhere $k$ is a constant of proportionality that we need to find."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x = \\htmlData{anchor=x-val}{\\color{@sky}{6}}, \\quad y = \\htmlData{anchor=y-val}{\\color{@amber}{72}}$"},{"latex":"$$\\htmlData{anchor=y-eq}{\\color{@amber}{y}} = k\\htmlData{anchor=x-eq}{\\color{@sky}{x}}$"},{"latex":"$$\\color{@amber}{72} = k(\\color{@sky}{6})$"},{"latex":"$$k = \\frac{72}{6} = 12$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x-val"},{"color":"amber","style":"text-color","anchor":"y-val"}],"connections":[{"to":"x-eq","from":"x-val","color":"sky","label":"substitute x","style":"solid"},{"to":"y-eq","from":"y-val","color":"amber","label":"substitute y","style":"solid"}]},"order":1,"title":"Substitute to find k","description":"We can choose any coordinate pair from the table to solve for $k$. Let's use the first pair where $x = 6$ and $y = 72$ to substitute into our equation:"},{"type":"text","order":2,"title":"Write the final formula","content":"Now that we have found the constant $k = 12$, we write the final formula by substituting $k$ back into $y = kx$:\n\n$$y = 12x$$\n\nThis formula works for all pairs in the table. For example, when $x = 10$:\n\n$$y = 12 \\times 10 = 120$$\n\nwhich matches the second column of our table."}]},{"id":"3602f652-7fed-4571-98ac-69714a1efc7f:linear_equation","label":"(ii) Linear Equation Method","steps":[{"type":"text","order":0,"title":"Identify coordinate points","content":"To find a formula for $y$ in terms of $x$ using the linear equation method, we can treat the given pairs of $(x, y)$ values as points on a coordinate plane.\n\nFrom the table, we have two points:\n- Point 1: $(x_1, y_1) = (6, 72)$\n- Point 2: $(x_2, y_2) = (10, 120)$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$m = \\frac{\\htmlData{anchor=y2}{\\color{@pink}{y_2}} - \\htmlData{anchor=y1}{\\color{@sky}{y_1}}}{\\htmlData{anchor=x2}{\\color{@purple}{x_2}} - \\htmlData{anchor=x1}{\\color{@amber}{x_1}}}$"},{"latex":"$$m = \\frac{\\htmlData{anchor=y2-val}{\\color{@pink}{120}} - \\htmlData{anchor=y1-val}{\\color{@sky}{72}}}{\\htmlData{anchor=x2-val}{\\color{@purple}{10}} - \\htmlData{anchor=x1-val}{\\color{@amber}{6}}}$"},{"latex":"$$m = \\frac{48}{4} = 12$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"y2-val"},{"color":"sky","style":"text-color","anchor":"y1-val"},{"color":"purple","style":"text-color","anchor":"x2-val"},{"color":"amber","style":"text-color","anchor":"x1-val"}],"connections":[{"to":"y2","from":"y2-val","color":"pink","label":"y₂ value","style":"solid"},{"to":"y1","from":"y1-val","color":"sky","label":"y₁ value","style":"solid"},{"to":"x2","from":"x2-val","color":"purple","label":"x₂ value","style":"solid"},{"to":"x1","from":"x1-val","color":"amber","label":"x₁ value","style":"solid"}]},"order":1,"title":"Calculate the gradient","description":"We calculate the gradient (rate of change), $m$, using the gradient formula:"},{"type":"text","order":2,"title":"Determine the y-intercept","content":"The equation of a straight line is written in the form:\n\n$$y = mx + c$$\n\nSubstituting our gradient $m = 12$ and the coordinates of the first point $(6, 72)$ into this equation:\n\n$$\\begin{aligned} 72 &= 12(6) + c \\\\ 72 &= 72 + c \\\\ c &= 0 \\end{aligned}$$\n\nThis means the line passes through the origin $(0,0)$, which confirms that $y$ is directly proportional to $x$."},{"type":"text","order":3,"title":"Write the final formula","content":"Now we can write the final equation by substituting $m = 12$ and $c = 0$ back into our linear equation form:\n\n$$y = 12x$$\n\nThis earns the final **A1** accuracy mark."}]}],"generatedAt":"2026-06-12T16:57:26.071Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":40},"walkthroughVideo":null},{"id":"5a0ad258-3fe5-418a-a138-bb1d4d87e87e","revisionId":"f08f5673-fd74-4715-b7e4-2e1fa70441f0","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"f06a97c1-4268-4486-b9ed-60e628a9a837","label":"a","content":"$$y$ is directly proportional to $x$\n\nWhen $y = 6, x = 48$\n\nFind the value of $x$ when $y = 10$","markscheme":"- Sets up equation $y = kx$ or $\\frac{y}{x} = k$ M1\n- Finds $k = \\frac{6}{48} = \\frac{1}{8}$ or $x = 8y$ M1\n- Substitutes $y = 10$ to find $x = 80$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up a correct proportionality equation such as $y = kx$ or $\\frac{y}{x} = k$.","code":"M"},{"id":"p2","marks":1,"description":"Calculates the constant of proportionality $k = 0.125$ or finds the relationship $x = 8y$.","code":"M"},{"id":"p3","marks":1,"description":"Calculates the correct value for $x$.","code":"A","acceptable":["80"]}],"finalAnswers":[],"generatedAt":"2026-05-06T14:01:43.501Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"direct_proportion","label":"Find the constant","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"“Directly proportional” means that the ratio of $y$ to $x$ stays constant. Call this constant $k$ and write\n\n$$y=kx$$\n\nThis correct direct-proportion equation earns the first **M1** mark.","highlights":[{"text":"$$y$ is directly proportional to $x$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=given_equation}{6=48k}$"},{"latex":"$$\\htmlData{anchor=fraction_equation}{k=\\dfrac{6}{48}}$"},{"latex":"$$k=\\dfrac{1}{8}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"given_equation"},{"color":"sky","style":"text-color","anchor":"fraction_equation"}],"connections":[{"to":"fraction_equation","from":"given_equation","color":"sky","label":"÷ 48"}]},"order":1,"title":"Find the constant","description":"Substitute the given pair of values into the direct-proportion equation, then divide both sides by 48. Finding the correct constant earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=source_equation}{10=\\dfrac{x}{8}}$"},{"latex":"$$\\boxed{\\htmlData{anchor=result_equation}{x=80}}$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"source_equation"},{"color":"green","style":"text-color-with-background","anchor":"result_equation"}],"connections":[{"to":"result_equation","from":"source_equation","color":"green","label":"× 8"}]},"order":2,"title":"Substitute the new value","description":"Now substitute y = 10 into the same relationship and solve for x. The correct boxed value earns A1."}]}],"generatedAt":"2026-07-19T15:54:47.898Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"f06a97c1-4268-4486-b9ed-60e628a9a837:constant_of_proportionality","label":"Constant of Proportionality Method","steps":[{"type":"text","order":0,"title":"Set up the equation","content":"Since $y$ is directly proportional to $x$, we can write this relationship as:\n\n$$x = ky$$\n\nwhere $k$ is the constant of proportionality. \n\nSetting up this equation correctly earns the first mark.","highlights":[{"text":"y is directly proportional to x","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\text{Substitute } \\htmlData{anchor=y-val}{\\color{@sky}{y = 6}} \\text{ and } \\htmlData{anchor=x-val}{\\color{@amber}{x = 48}} \\text{ into the equation:}$$"},{"latex":"$$$\\htmlData{anchor=sub-eq}{\\color{@amber}{48}} = k \\times \\htmlData{anchor=sub-y}{\\color{@sky}{6}}$$"},{"latex":"$$$k = \\frac{48}{6} = 8$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"y-val"},{"color":"amber","style":"text-color","anchor":"x-val"}],"connections":[{"to":"sub-y","from":"y-val","color":"sky","label":null,"style":"solid"},{"to":"sub-eq","from":"x-val","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Find the constant $k$","description":"We substitute the given values to solve for our constant $k$. This step earns the second mark."},{"type":"text","order":2,"title":"Update the formula","content":"Now that we know $k = 8$, we can write our formula for $x$ in terms of $y$:\n\n$$x = 8y$$"},{"type":"text","order":3,"title":"Find the value of $x$","content":"To find $x$ when $y = 10$, substitute $y = 10$ into our formula:\n\n$$\\begin{aligned} x &= 8 \\times 10 \\\\ x &= 80 \\end{aligned}$$\n\nThis final calculation earns the third and final mark.","highlights":[{"text":"Find the value of x when y = 10","location":"specification"}]}]},{"id":"f06a97c1-4268-4486-b9ed-60e628a9a837:ratio_equivalence","label":"Ratio Equivalence Method","steps":[{"type":"text","order":0,"title":"Understand direct proportion","content":"When two variables $y$ and $x$ are **directly proportional**, the ratio between them remains constant. \n\nThis means that the ratio of $x$ to $y$ for one set of values is equal to the ratio of $x$ to $y$ for any other set of values:\n\n$$\\frac{x_1}{y_1} = \\frac{x_2}{y_2}$$","highlights":[{"text":"y is directly proportional to x","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"We are given the initial pair of values:"},{"latex":"$$\\htmlData{anchor=y1}{\\color{@sky}{y_1 = 6}} \\quad \\text{and} \\quad \\htmlData{anchor=x1}{\\color{@amber}{x_1 = 48}}$"},{"text":"And we want to find $x_2$ when:"},{"latex":"$$\\htmlData{anchor=y2}{\\color{@purple}{y_2 = 10}}$"},{"text":"Setting up our equivalence formula:"},{"latex":"$$\\frac{\\htmlData{anchor=x2-var}{x_2}}{\\htmlData{anchor=y2-sub}{\\color{@purple}{10}}} = \\frac{\\htmlData{anchor=x1-sub}{\\color{@amber}{48}}}{\\htmlData{anchor=y1-sub}{\\color{@sky}{6}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"y1"},{"color":"amber","style":"text-color","anchor":"x1"},{"color":"purple","style":"text-color","anchor":"y2"}],"connections":[{"to":"y1-sub","from":"y1","color":"sky","label":null,"style":"solid"},{"to":"x1-sub","from":"x1","color":"amber","label":null,"style":"solid"},{"to":"y2-sub","from":"y2","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up the ratio equivalence","description":"We substitute our known values into the ratio equation."},{"type":"text","order":2,"title":"Simplify the known ratio","content":"First, simplify the fraction on the right-hand side of the equation:\n\n$$\\frac{48}{6} = 8$$\n\nNow our equation is:\n\n$$\\frac{x_2}{10} = 8$$"},{"type":"text","order":3,"title":"Solve for the unknown x","content":"To solve for $x_2$, multiply both sides of the equation by $10$:\n\n$$\\begin{aligned} x_2 &= 8 \\times 10 \\\\ x_2 &= 80 \\end{aligned}$$\n\nSo, when $y = 10$, the value of $x$ is $80$."}]},{"id":"f06a97c1-4268-4486-b9ed-60e628a9a837:unitary_scaling","label":"Unitary Scaling Method","steps":[{"type":"text","order":0,"title":"Identify the approach","content":"Since $y$ is directly proportional to $x$, we can use the **Unitary Scaling Method**. This method works by first finding the value of $x$ that corresponds to a single unit of $y$ ($y = 1$), and then multiplying this scale factor by our target value of $y = 10$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$y = 6 \\implies x = 48$"},{"latex":"$$\\text{For } y = 1 \\implies x = \\frac{48}{6} = \\htmlData{anchor=scale-factor}{\\color{@amber}{8}}$"},{"latex":"$$\\text{For } y = 10 \\implies x = \\htmlData{anchor=scaled-x}{\\color{@amber}{8}} \\times 10 = 80$"}],"highlights":[{"color":"amber","style":"box","anchor":"scale-factor"},{"color":"amber","style":"box","anchor":"scaled-x"}],"connections":[{"to":"scaled-x","from":"scale-factor","color":"amber","label":"multiply by 10","style":"solid"}]},"order":1,"title":"Calculate the scale factor","description":"We divide the given value of $x = 48$ by the corresponding $y = 6$ to find that $1$ unit of $y$ corresponds to $8$ units of $x$. Then, we scale this unit value up by multiplying by $10$ to find the value of $x$ when $y = 10$."},{"type":"text","order":2,"title":"State the final answer","content":"Therefore, when $y = 10$, we have $x = 80$.\\n\\n**Markscheme breakdown:**\\n- Sets up the proportion relationship: **M1**\\n- Finds the scale factor $x = 8y$ (or $y = \\frac{1}{8}x$): **M1**\\n- Substitutes $y = 10$ to get $x = 80$: **A1**"}]}],"generatedAt":"2026-06-12T17:06:24.750Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":62},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/5a0ad258-3fe5-418a-a138-bb1d4d87e87e.mp4","durationSeconds":80,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/5a0ad258-3fe5-418a-a138-bb1d4d87e87e.jpg"}},{"id":"757331c8-a6fa-4b6d-97ba-d245f0d45b28","revisionId":"8e455933-9b7b-4c33-8d40-9db636c3abc2","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"c347ae6e-fa8c-4643-83c2-2a31da19986a","label":"a","content":"$$y$ is directly proportional to $x$\n\nWhen $y = 9$, $x = 45$\n\nFind the value of $x$ when $y = 7$","markscheme":"- Sets up equation $y = kx$ or $\\frac{y}{x} = k$ M1\n- Finds $k = \\frac{9}{45} = \\frac{1}{5}$ or $x = 5y$ M1\n- Substitutes $y = 7$ to find $x = 35$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up equation $y = kx$ or $\\frac{y}{x} = k$","code":"M"},{"id":"p2","marks":1,"description":"Finds $k = \\frac{9}{45} = \\frac{1}{5}$ or $x = 5y$","code":"M"},{"id":"p3","marks":1,"description":"Substitutes $y = 7$ to find $x = 35$","code":"A"}],"finalAnswers":["35"],"generatedAt":"2026-05-06T14:01:57.137Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"unitary_method","label":"Unitary method","steps":[{"type":"text","order":0,"title":"Write the proportion equation","content":"For direct proportion, start with $y=kx$, where $k$ is the constant of proportionality. Writing this equation earns the first M1 mark.","highlights":[{"text":"$$y$ is directly proportional to $x$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=ratio}{\\color{@sky}{45 \\div 9}}=5$"},{"latex":"$$x=\\htmlData{anchor=multiplier}{\\color{@sky}{5}}y$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"ratio"},{"color":"sky","style":"text-color-with-background","anchor":"multiplier"}]},"order":1,"title":"Find the multiplier","description":"Calculate how many times larger x is than y. This earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x=5\\times \\htmlData{anchor=new_y}{\\color{@orange}{7}}$"},{"latex":"$$\\htmlData{anchor=answer}{\\boxed{x=35}}$"}],"highlights":[{"color":"orange","style":"text-color-with-background","anchor":"new_y"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":2,"title":"Apply the multiplier","description":"Substitute y = 7, then calculate. The correct value earns the A1 mark."}]},{"id":"constant_of_proportionality","label":"Constant of proportionality","steps":[{"type":"text","order":0,"title":"Write the proportion equation","content":"Direct proportion means that one variable is a constant multiple of the other. Write $y=kx$, where $k$ is the constant of proportionality. This earns the first M1 mark.","highlights":[{"text":"$$y$ is directly proportional to $x$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$9=45k$"},{"latex":"$$k=\\dfrac{9}{45}$"},{"latex":"$$\\htmlData{anchor=constant}{k=\\dfrac{1}{5}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"constant"}]},"order":1,"title":"Find the constant","description":"Substitute the given pair of values, then divide both sides by 45. This earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=new_y}{\\color{@orange}{7}}=\\dfrac{1}{5}x$"},{"latex":"$$\\htmlData{anchor=answer}{\\boxed{x=35}}$"}],"highlights":[{"color":"orange","style":"text-color-with-background","anchor":"new_y"},{"color":"green","style":"text-color-with-background","anchor":"answer"}]},"order":2,"title":"Substitute the new value","description":"Use y = 7 in the proportional relationship. Multiply both sides by 5 to solve for x. The correct value earns the A1 mark."}]}],"generatedAt":"2026-07-19T07:04:21.143Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"c347ae6e-fa8c-4643-83c2-2a31da19986a:proportionality-constant","label":"Using Proportionality Constant","steps":[{"type":"text","order":0,"title":"Set up the equation","content":"Since we are told that $y$ is directly proportional to $x$, we can write this relationship using a constant of proportionality, $k$:\n\n$$y = kx$$","highlights":[{"text":"y is directly proportional to x","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"We substitute the known values into our equation to solve for $k$:"},{"latex":"$$y = \\htmlData{anchor=val-y}{\\color{@sky}{9}}, \\quad x = \\htmlData{anchor=val-x}{\\color{@amber}{45}}$"},{"latex":"$$y = kx \\implies \\htmlData{anchor=sub-y}{\\color{@sky}{9}} = k \\times \\htmlData{anchor=sub-x}{\\color{@amber}{45}}$"},{"latex":"$$k = \\frac{9}{45} = \\htmlData{anchor=val-k}{\\color{@purple}{\\frac{1}{5}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"val-y"},{"color":"amber","style":"text-color","anchor":"val-x"},{"color":"purple","style":"box","anchor":"val-k"}],"connections":[{"to":"sub-y","from":"val-y","color":"sky","label":null,"style":"solid"},{"to":"sub-x","from":"val-x","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Find the constant $k$","description":"We plug the given pair of values into $y = kx$ to calculate $k$."},{"type":"text","order":2,"title":"Form the specific formula","content":"Now we can write the formula linking $y$ and $x$ with our found value of $k = \\frac{1}{5}$:\n\n$$y = \\frac{1}{5}x$$\n\nTo make it easier to find $x$, we can rearrange this formula by multiplying both sides by $5$:\n\n$$x = 5y$$"},{"type":"hyper_latex","block":{"nodes":[{"text":"Now, we substitute the new value of $y$ into our formula to find $x$:"},{"latex":"$$y = \\htmlData{anchor=new-y}{\\color{@sky}{7}}$"},{"latex":"$$x = 5 \\times \\htmlData{anchor=sub-new-y}{\\color{@sky}{7}}$"},{"latex":"$$x = \\htmlData{anchor=ans}{\\color{@green}{35}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"new-y"},{"color":"green","style":"box","anchor":"ans"}],"connections":[{"to":"sub-new-y","from":"new-y","color":"sky","label":null,"style":"solid"}]},"order":3,"title":"Solve for $x$","description":"Substitute $y = 7$ into $x = 5y$ to calculate the final value of $x$."}]},{"id":"c347ae6e-fa8c-4643-83c2-2a31da19986a:equivalent-ratios","label":"Using Equivalent Ratios","steps":[{"type":"text","order":0,"title":"Understand the proportional relationship","content":"Since $y$ is directly proportional to $x$, the ratio between them remains constant. This means the ratio of $x$ to $y$ for our first set of values will be exactly equal to the ratio of $x$ to $y$ for our second set of values:\n\n$$\\frac{x_1}{y_1} = \\frac{x_2}{y_2}$$","highlights":[{"text":"$$y$ is directly proportional to $x$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$x_1 = \\htmlData{anchor=x1}{\\color{@sky}{45}}, \\quad y_1 = \\htmlData{anchor=y1}{\\color{@amber}{9}}$"},{"latex":"$$x_2 = \\htmlData{anchor=x2}{\\color{@purple}{x}}, \\quad y_2 = \\htmlData{anchor=y2}{\\color{@teal}{7}}$"},{"latex":"$${\\frac{\\htmlData{anchor=x1-sub}{\\color{@sky}{45}}}{\\htmlData{anchor=y1-sub}{\\color{@amber}{9}}}} = {\\frac{\\htmlData{anchor=x2-sub}{\\color{@purple}{x}}}{\\htmlData{anchor=y2-sub}{\\color{@teal}{7}}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"x1"},{"color":"amber","style":"text-color","anchor":"y1"},{"color":"purple","style":"text-color","anchor":"x2"},{"color":"teal","style":"text-color","anchor":"y2"}],"connections":[{"to":"x1-sub","from":"x1","color":"sky","label":null,"style":"solid"},{"to":"y1-sub","from":"y1","color":"amber","label":null,"style":"solid"},{"to":"x2-sub","from":"x2","color":"purple","label":null,"style":"solid"},{"to":"y2-sub","from":"y2","color":"teal","label":null,"style":"solid"}]},"order":1,"title":"Set up the ratio equation","description":"We identify our given pairs of values and substitute them directly into our ratio formula. Setting up this ratio equivalent earns you the first method mark **(M1)**."},{"type":"text","order":2,"title":"Simplify and solve for x","content":"First, simplify the fraction on the left-hand side of the equation:\n\n$$\\frac{45}{9} = 5$$\n\nThis shows that $x$ is always $5$ times larger than $y$ (earning the second method mark **(M1)** for finding this constant relationship).\n\nNow substitute this back into our equation:\n\n$$5 = \\frac{x}{7}$$\n\nMultiply both sides by $7$ to solve for $x$:\n\n$$x = 5 \\times 7$$\n\n$$x = 35$$\n\nThis final substitution and calculation earns you the accuracy mark **(A1)**.","highlights":[{"text":"Find the value of $x$ when $y = 7$","location":"specification"}]}]}],"generatedAt":"2026-06-12T20:09:03.077Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":75},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/757331c8-a6fa-4b6d-97ba-d245f0d45b28.mp4","durationSeconds":82,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/757331c8-a6fa-4b6d-97ba-d245f0d45b28.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","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"767c3633-35ff-4573-b000-df0757112c14","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"7e12a003-738f-4975-bdc6-e688771878a1","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9a1d0d1e-5399-41d0-a670-48c40d528fa4","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"c6fee684-1139-4301-a73c-de405cf52ef6","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"c70e9a7b-87d8-4c7c-ae0a-837f0b1c7820","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"9dd9548c-a4cc-4572-ac05-70bfe6fe25e7","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"88ccb6e0-8295-48b1-b8fb-179264604cd2","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"b3690a16-6c44-4de1-9451-b7a90abbf5ac","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"60d73d12-84cf-43ce-80e9-c762d60a34f9","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"618dbf54-86cf-4c77-8aa1-da5fe9c8c9ed","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"a4b3ee91-ed2d-4e19-bcf9-4eeef64cb9b1","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":3,"questionType":"LA","isCalculation":true},{"id":"39f75a5f-cd31-48d5-999a-30219fa6de01","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false},{"id":"57ff4081-fe0e-4926-adab-5c52b2097788","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":4,"questionType":"LA","isCalculation":false}],"title":"Direct 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 2

Direct and Inverse Proportion Questions

Direct and Inverse Proportion

Question 2

Direct and Inverse Proportion Questions