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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":34,"initialQuestionWindow":[{"id":"9a1d0d1e-5399-41d0-a670-48c40d528fa4","revisionId":"eb7b2aa7-3e11-47b2-b36e-81f5d44ea1a9","questionNumber":null,"questionType":"LA","specification":"The table shows pairs of values for $u$ and $v$\n\n| $u$ | 4 | 10 |\n| :--- | :--- | :--- |\n| $v$ | 48 | 120 |","difficulty":2,"difficultyBand":"easy","totalMarks":4,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"b867447e-3575-49eb-9dec-666fb211887d","label":"i","content":"Tick the correct statement below.\n\n- $v \\propto u$ .....\n- $v \\propto u^2$ .....\n- $v \\propto u^3$ .....","markscheme":"- Tests relationships: $48/4 = 12$ and $120/10 = 12$ (linear); $48/4^2 = 3$ vs $120/10^2 = 1.2$ (not quadratic) M1\n- Identifies $v \\propto u$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Evidence of checking the ratio $v/u$ for both pairs of values.","code":"M"},{"id":"p2","marks":1,"description":"Correct identification of the linear relationship $v \\propto u$.","code":"A"}],"finalAnswers":["$$v \\propto u$"],"generatedAt":"2026-05-06T14:01:57.019Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"compare_ratios","label":"Compare the ratios","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{48}{4^2}=\\htmlData{anchor=square_one}{\\color{@red}{3}}$"},{"latex":"$$\\dfrac{120}{10^2}=\\htmlData{anchor=square_two}{\\color{@red}{1.2}}$"}],"highlights":[{"color":"red","style":"text-color-with-background","anchor":"square_one"},{"color":"red","style":"text-color-with-background","anchor":"square_two"}]},"order":0,"title":"Eliminate the square","description":"Divide each value of v by the corresponding value of u squared. The constants are 3 and 1.2, so the square option is wrong. Simply noticing that both variables increase is not enough: in a square relationship, v changes by the square of u's scale factor."},{"type":"text","order":1,"title":"Eliminate the cube","content":"From the first pair to the second, both $u$ and $v$ are multiplied by $2.5$. A common mistake is to choose the cubic option simply because both variables increase. In a cubic relationship, multiplying $u$ by $2.5$ would multiply $v$ by $2.5^3=15.625$, not by $2.5$. Therefore the cubic option is false.","highlights":[{"text":"$$v \\propto u^3$ .....","index":2,"location":"eliminate"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\dfrac{48}{4}=\\color{@sky}{\\htmlData{anchor=ratio_one}{12}}$"},{"latex":"$$\\dfrac{120}{10}=\\color{@sky}{\\htmlData{anchor=ratio_two}{12}}$"}],"highlights":[{"color":"sky","style":"text-color-with-background","anchor":"ratio_one"},{"color":"sky","style":"text-color-with-background","anchor":"ratio_two"}]},"order":2,"title":"Compare linear ratios","description":"For direct proportion, divide each value of v by the corresponding value of u. Both ratios equal 12. Together with the unequal square ratios, these equal linear ratios earn M1."},{"type":"text","order":3,"title":"Tick direct proportion","content":"The ratio $v/u$ is constant, so tick $v\\propto u$. This correct identification earns **A1**.","highlights":[{"text":"$$v \\propto u$ .....","index":0,"location":"option"}]}]}],"generatedAt":"2026-07-19T18:50:25.874Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]},{"id":"7d9c33fa-e2f3-4cef-b1e8-ff28e168b822","label":"ii","content":"Write a formula for $v$ in terms of $u$","markscheme":"- Uses $v = ku$ and substitutes values to find $k = 12$ M1\n- $v = 12u$ A1","marks":2,"order":1,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Method to determine the constant of proportionality $k$ for direct proportion.","code":"M"},{"id":"p2","marks":1,"description":"Correct formula $v = 12u$.","code":"A"}],"finalAnswers":["$$v = 12u$"],"generatedAt":"2026-05-06T14:01:57.019Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"constant_of_proportionality","label":"Find the constant of proportionality","steps":[{"type":"text","order":0,"title":"Choose the relationship","content":"The pairs have the same ratio $v\\div u$, so $v$ is directly proportional to $u$. Start with\n\n$$v=ku$$\n\nwhere $k$ is the **constant of proportionality**, meaning the fixed number that multiplies $u$.","highlights":[{"text":"Write a formula for $v$ in terms of $u$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$v=ku\\htmlData{anchor=blue_start}{\\phantom{\\qquad}}$"},{"latex":"$$\\htmlData{anchor=amber_start}{\\phantom{\\qquad}}48=k\\times4\\htmlData{anchor=blue_end}{\\phantom{\\qquad}}$"},{"latex":"$$\\htmlData{anchor=amber_end}{\\phantom{\\qquad}}k=\\dfrac{48}{4}$"},{"latex":"$$k=12$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"blue_start"},{"color":"sky","style":"text-color","anchor":"blue_end"},{"color":"amber","style":"text-color","anchor":"amber_start"},{"color":"amber","style":"text-color","anchor":"amber_end"}],"connections":[{"to":"blue_end","from":"blue_start","color":"sky","label":"substitute"},{"to":"amber_end","from":"amber_start","color":"amber","label":"divide by 4"}]},"order":1,"title":"Find the constant","description":"Writing the general form v = ku, substituting one pair of values and obtaining k = 12 earns the method mark."},{"type":"text","order":2,"title":"Write the formula","content":"Replace $k$ with $12$:\n\n$$\\boxed{v=12u}$$\n\nThis final formula earns the accuracy mark. It also fits both pairs: multiplying $4$ by $12$ gives $48$, and multiplying $10$ by $12$ gives $120$.","highlights":[{"text":"Write a formula for $v$ in terms of $u$","location":"specification"}]}]}],"generatedAt":"2026-07-19T18:57:33.546Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"b867447e-3575-49eb-9dec-666fb211887d:constant_of_proportionality","label":"(i) Constant of Proportionality","steps":[{"type":"text","order":0,"title":"Understand the goal","content":"We need to find the correct proportionality relationship between $u$ and $v$ from the given options:\n\n* $v \\propto u$\n* $v \\propto u^2$\n* $v \\propto u^3$\n\nTo do this, we will find the ratio of $v$ to the corresponding power of $u$ for both pairs of values in our table. The correct relationship is the one that gives a consistent constant of proportionality, $k$."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$v = k \\cdot \\htmlData{anchor=u-lin}{u} \\implies k = \\frac{v}{\\htmlData{anchor=u-lin-denom}{u}}$"},{"latex":"$$\\text{For } (4, 48): \\; k_1 = \\frac{\\htmlData{anchor=v1}{48}}{\\htmlData{anchor=u1}{4}} = \\htmlData{anchor=k1}{12}$"},{"latex":"$$\\text{For } (10, 120): \\; k_2 = \\frac{\\htmlData{anchor=v2}{120}}{\\htmlData{anchor=u2}{10}} = \\htmlData{anchor=k2}{12}$"}],"highlights":[{"color":"green","style":"box","anchor":"k1"},{"color":"green","style":"box","anchor":"k2"}],"connections":[{"to":"u-lin-denom","from":"u-lin","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Test linear proportion","description":"Let's first test the relationship $v \\propto u$, which means $v = ku$ for some constant $k$. We calculate $k = \\frac{v}{u}$ for both data points."},{"type":"text","order":2,"title":"Verify other options","content":"To be absolutely sure, let's check if the quadratic relationship $v \\propto u^2$ (meaning $v = k u^2$) is constant:\n\n* For $(4, 48)$: \n $$k = \\frac{48}{4^2} = \\frac{48}{16} = 3$$\n* For $(10, 120)$: \n $$k = \\frac{120}{10^2} = \\frac{120}{100} = 1.2$$\n\nSince $3 \\neq 1.2$, the relationship is not quadratic. \n\nSimilarly, for $v \\propto u^3$:\n* For $(4, 48)$:\n $$k = \\frac{48}{4^3} = \\frac{48}{64} = 0.75$$\n* For $(10, 120)$:\n $$k = \\frac{120}{10^3} = \\frac{120}{1000} = 0.12$$\n\nSince $0.75 \\neq 0.12$, it is not a cubic relationship."},{"type":"text","order":3,"title":"Select correct statement","content":"Since only the ratio $\\frac{v}{u} = 12$ remains constant for both pairs of values, we can confidently conclude that $v$ is directly proportional to $u$.\n\nTherefore, the correct statement to tick is:\n* **$v \\propto u$**","highlights":[{"text":"$$v \\propto u$ .....","index":0,"location":"option"}]}]},{"id":"b867447e-3575-49eb-9dec-666fb211887d:scaling_factors","label":"(i) Scaling Factors Comparison","steps":[{"type":"text","order":0,"title":"Analyze the scaling of $u$","content":"To determine the relationship between $u$ and $v$ using scaling factors, let's first find the multiplier (or scaling factor) that takes us from the first value of $u$ to the second value of $u$.\n\nFrom the table, our $u$-values are:\n- $u_1 = 4$\n- $u_2 = 10$\n\nLet's calculate the multiplier for $u$:\n\n$$\\text{Multiplier for } u = \\frac{u_2}{u_1} = \\frac{10}{4} = 2.5$$","highlights":[{"text":"4","location":"specification"},{"text":"10","location":"specification"}]},{"type":"text","order":1,"title":"Analyze the scaling of $v$","content":"Next, let's find the multiplier that takes us from the first value of $v$ to the second value of $v$.\n\nFrom the table, our $v$-values are:\n- $v_1 = 48$\n- $v_2 = 120$\n\nLet's calculate the multiplier for $v$:\n\n$$\\text{Multiplier for } v = \\frac{v_2}{v_1} = \\frac{120}{48} = 2.5$$","highlights":[{"text":"48","location":"specification"},{"text":"120","location":"specification"}]},{"type":"text","order":2,"title":"Compare the scaling factors","content":"Now, let's compare the two scaling factors:\n\n- The multiplier for $u$ is $2.5$\n- The multiplier for $v$ is $2.5$\n\nSince both variables scale by the exact same factor, we can see that $v$ is directly proportional to $u$. In other words:\n\n$$v \\propto u$$\n\nIf the relationship were quadratic ($v \\propto u^2$), the $v$-multiplier would have been $2.5^2 = 6.25$. Since the multipliers are identical, we have a linear relationship.","highlights":[{"text":"v \\propto u","index":0,"location":"option"}]},{"type":"text","order":3,"title":"Select the correct option","content":"The correct statement is the first one:\n\n- **$\\boldsymbol{v \\propto u}$**","highlights":[{"text":"v \\propto u","index":0,"location":"option"}]}]},{"id":"7d9c33fa-e2f3-4cef-b1e8-ff28e168b822:direct_variation","label":"(ii) Direct Variation Approach","steps":[{"type":"text","order":0,"title":"Assume direct variation","content":"Since $v$ is directly proportional to $u$, we can write a direct variation equation of the form:\n\n$$v = ku$$\n\nwhere $k$ is the constant of variation (or constant of proportionality) that we need to find."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\text{Using the pair } \bar{u} = \bar{4} \text{ and } \bar{v} = \bar{48}:"},{"latex":"$$$\n\\htmlData{anchor=v-sub}{\\color{@sky}{48}} = k \\times \\htmlData{anchor=u-sub}{\\color{@amber}{4}}\n$$"},{"latex":"$$$\nk = \\frac{\\color{@sky}{48}}{\\color{@amber}{4}} = \\htmlData{anchor=k-val}{\\color{@purple}{12}}\n$$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"u-sub"},{"color":"sky","style":"text-color","anchor":"v-sub"},{"color":"purple","style":"text-color","anchor":"k-val"}],"connections":[{"to":"k-val","from":"u-sub","color":"amber","label":null,"style":"solid"},{"to":"k-val","from":"v-sub","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Find the constant $k$","description":"We substitute a known pair of values from the table into our equation. Let's use $u = 4$ and $v = 48$ to solve for $k$."},{"type":"text","order":2,"title":"Verify the constant","content":"We can verify this value of $k$ using the second pair of values from the table, $u = 10$ and $v = 120$:\n\n$$v = 12 \\times 10 = 120$$\n\nThis confirms that our constant of variation is correct and earns us the first method mark (**M1**).","highlights":[{"text":"10","location":"specification"},{"text":"120","location":"specification"}]},{"type":"text","order":3,"title":"Write the final formula","content":"Now substitute $k = 12$ back into our initial equation $v = ku$ to get the final formula:\n\n$$v = 12u$$\n\nThis completes the question and earns the accuracy mark (**A1**)."}]},{"id":"7d9c33fa-e2f3-4cef-b1e8-ff28e168b822:linear_equation","label":"(ii) Linear Equation Approach","steps":[{"type":"text","order":0,"title":"Identify coordinate points","content":"To find a linear formula of the form $v = mu + c$, we can treat the pairs of values from the table as coordinates $(u, v)$. \n\nOur two coordinate points are:\n- $(4, 48)$\n- $(10, 120)$","highlights":[{"text":"u","location":"specification"},{"text":"4","location":"specification"},{"text":"10","location":"specification"},{"text":"v","location":"specification"},{"text":"48","location":"specification"},{"text":"120","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\text{Gradient } m = \\frac{v_2 - v_1}{u_2 - u_1}$"},{"latex":"$$m = \\frac{\\htmlData{anchor=v2}{\\color{@pink}{120}} - \\htmlData{anchor=v1}{\\color{@sky}{48}}}{\\htmlData{anchor=u2}{\\color{@amber}{10}} - \\htmlData{anchor=u1}{\\color{@teal}{4}}}$"},{"latex":"$$m = \\frac{72}{6} = 12$"}],"highlights":[{"color":"pink","style":"text-color","anchor":"v2"},{"color":"sky","style":"text-color","anchor":"v1"},{"color":"amber","style":"text-color","anchor":"u2"},{"color":"teal","style":"text-color","anchor":"u1"}],"connections":[{"to":"u2","from":"v2","color":"pink","label":null,"style":"dashed"}]},"order":1,"title":"Calculate the gradient","description":"We find the gradient $m$ of the line using the gradient formula, which represents the change in $v$ divided by the change in $u$."},{"type":"text","order":2,"title":"Find the constant term","content":"Now we substitute the gradient $m = 12$ and one of our points, say $(4, 48)$, into the equation $v = mu + c$ to find $c$:\n\n$$\\begin{aligned} v &= mu + c \\\\ 48 &= 12(4) + c \\\\ 48 &= 48 + c \\\\ c &= 0 \\end{aligned}$$\n\nThis shows that the line passes through the origin, meaning $v$ is directly proportional to $u$. This matches the markscheme requirement where we find $k = 12$ M1."},{"type":"text","order":3,"title":"Write the final formula","content":"Substituting $m = 12$ and $c = 0$ back into our linear equation gives us the final formula:\n\n$$v = 12u$$\n\nThis is our complete formula A1."}]},{"id":"7d9c33fa-e2f3-4cef-b1e8-ff28e168b822:constant_ratio","label":"(ii) Finding a Constant Ratio","steps":[{"type":"text","order":0,"title":"Calculate the first ratio","content":"To see if there is a direct relationship between the variables, we can find the ratio of $v$ to $u$ for the first pair of values. \n\nFrom the table, when $u = 4$, we have $v = 48$. Let's calculate the ratio:\n\n$$\\frac{v}{u} = \\frac{48}{4}$$","highlights":[{"text":"4","location":"specification"},{"text":"48","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{\\htmlData{anchor=v1}{\\color{@sky}{48}}}{\\htmlData{anchor=u1}{\\color{@amber}{4}}} = \\htmlData{anchor=ratio1}{\\color{@purple}{12}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"v1"},{"color":"amber","style":"text-color","anchor":"u1"},{"color":"purple","style":"text-color","anchor":"ratio1"}],"connections":[]},"order":1,"title":"Simplify the first ratio","description":"Dividing $48$ by $4$ gives us a constant ratio of $12$."},{"type":"text","order":2,"title":"Calculate the second ratio","content":"Next, let's verify if this ratio remains constant for the second pair of values, where $u = 10$ and $v = 120$:\n\n$$\\frac{v}{u} = \\frac{120}{10} = 12$$","highlights":[{"text":"10","location":"specification"},{"text":"120","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\frac{v}{u} = \\htmlData{anchor=const}{\\color{@purple}{12}}$"},{"latex":"$$v = \\htmlData{anchor=final-formula}{\\color{@purple}{12}u}$"}],"highlights":[{"color":"purple","style":"text-color","anchor":"const"},{"color":"purple","style":"text-color","anchor":"final-formula"}],"connections":[{"to":"final-formula","from":"const","color":"purple","label":"multiply by u","style":"solid"}]},"order":3,"title":"Write the general formula","description":"Since the ratio $\\frac{v}{u}$ is consistently $12$, we can write this relationship as an equation. Multiplying both sides by $u$ gives us the final formula."},{"type":"text","order":4,"title":"Confirm markscheme criteria","content":"By identifying that the constant multiplier is $k = 12$, we earn the method mark M1.\n\nWriting the final formula earns the accuracy mark A1:\n\n$$v = 12u$$"}]}],"generatedAt":"2026-06-13T00:06:05.486Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"c70e9a7b-87d8-4c7c-ae0a-837f0b1c7820","revisionId":"b166665e-5d80-448e-a28c-6076d163478b","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"87c9d65a-3e48-4005-875a-01bb269e4c7e","label":"a","content":"$$r$ is inversely proportional to $s$\n\nWhen $r = 12$, $s = 3$\n\nFind the value of $r$ when $s = 9$","markscheme":"- Sets up equation $r = \\frac{k}{s}$ or $rs = k$ M1\n- Finds $k = 12 \\times 3 = 36$ M1\n- Substitutes $s = 9$ to find $r = \\frac{36}{9} = 4$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Sets up equation representing inverse proportionality (e.g., $r = k/s$ or $rs = k$).","code":"M"},{"id":"p2","marks":1,"description":"Calculates the constant of proportionality $k = 36$.","code":"M"},{"id":"p3","marks":1,"description":"Final value of $r$ when $s = 9$.","code":"A"}],"finalAnswers":["4"],"generatedAt":"2026-05-06T14:01:37.922Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"inverse_proportion","label":"Use the inverse proportion formula","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"For inverse proportion, the product of the two variables is constant. Calling this constant $k$, write\n\n$$rs=k$$\n\nEquivalently, $r=\\dfrac{k}{s}$. Setting up either form earns the first **M1** mark.","highlights":[{"text":"$$r$ is inversely proportional to $s$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=product_rule}{rs}=k$"},{"latex":"$$\\htmlData{anchor=known_product}{\\color{@sky}{12\\times 3}}=k$"},{"latex":"$$\\htmlData{anchor=constant}{\\color{@green}{36}}=k$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"known_product"},{"color":"green","style":"text-color","anchor":"constant"}],"connections":[{"to":"known_product","from":"product_rule","color":"sky","label":"substitute"},{"to":"constant","from":"known_product","color":"green","label":"multiply"}]},"order":1,"title":"Find the constant","description":"Substitute the known pair of values into rs = k. Correctly finding the constant earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$r=\\dfrac{36}{\\htmlData{anchor=variable_s}{s}}$"},{"latex":"$$r=\\dfrac{36}{\\htmlData{anchor=new_s}{\\color{@orange}{9}}}$"},{"latex":"$$r=\\htmlData{anchor=answer}{\\color{@green}{4}}$"}],"highlights":[{"color":"orange","style":"text-color","anchor":"new_s"},{"color":"green","style":"text-color-with-background","anchor":"answer"}],"connections":[{"to":"new_s","from":"variable_s","color":"orange","label":"substitute"}]},"order":2,"title":"Substitute the new value","description":"Now use the new value of s in r = k/s. Reaching the correct value earns A1."}]}],"generatedAt":"2026-07-19T18:57:44.237Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":null,"firstAttemptFullMarks":null,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/c70e9a7b-87d8-4c7c-ae0a-837f0b1c7820.mp4","durationSeconds":72,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/c70e9a7b-87d8-4c7c-ae0a-837f0b1c7820.jpg"}},{"id":"9dd9548c-a4cc-4572-ac05-70bfe6fe25e7","revisionId":"c5e09d28-f192-4da6-b51b-c9f994f7ea4c","questionNumber":"4","questionType":"LA","specification":"","difficulty":1,"difficultyBand":"easy","totalMarks":3,"sourceTitle":"Direct and Inverse Proportion","calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"bc2f5cea-8b7c-4d80-9d37-312b357b52f0","label":"","content":"$$e$ is directly proportional to $f$\n\nWhen $e = 3, f = 36$\n\nFind the value of $f$ when $e = 4$","markscheme":"- Sets up equation $e = kf$ or $\\frac{e}{f} = k$ M1\n- Finds $k = \\frac{3}{36} = \\frac{1}{12}$ or $f = 12e$ M1\n- Substitutes $e = 4$ to find $f = 48$ A1","marks":3,"order":0,"markingRubric":{"totalMarks":3,"points":[{"id":"p1","marks":1,"description":"Writes a correct proportionality statement such as $e = kf$.","code":"M"},{"id":"p2","marks":1,"description":"Calculates the constant of proportionality $k = \\frac{1}{12}$ or finds $f = 12e$.","code":"M"},{"id":"p3","marks":1,"description":"Final answer of 48.","code":"A"}],"finalAnswers":["48"],"generatedAt":"2026-05-05T15:47:10.128Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"constant_of_proportionality","label":"Use the proportionality constant","steps":[{"type":"text","order":0,"title":"Write the relationship","content":"“Directly proportional” means one variable is a constant multiple of the other. Introduce the constant of proportionality $k$ and write\n\n$$e=kf$$\n\nSetting up this equation earns the first **M1** mark.","highlights":[{"text":"$$e$ is directly proportional to $f$","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$e=kf$"},{"latex":"$$\\htmlData{anchor=substituted}{\\color{@sky}{3=36k}}$"},{"latex":"$$\\htmlData{anchor=divided}{\\color{@orange}{k=\\dfrac{3}{36}}}$"},{"latex":"$$k=\\dfrac{1}{12}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"substituted"},{"color":"orange","style":"text-color","anchor":"divided"}],"connections":[{"to":"divided","from":"substituted","color":"orange","label":"divide both sides by 36"}]},"order":1,"title":"Find the constant","description":"Use the given pair of values in the proportionality equation. This correct substitution and division earns the second M1 mark."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$e=\\dfrac{1}{12}f$"},{"latex":"$$\\htmlData{anchor=before_multiply}{\\color{@sky}{4=\\dfrac{1}{12}f}}$"},{"latex":"$$\\htmlData{anchor=after_multiply}{\\color{@green}{f=48}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"before_multiply"},{"color":"green","style":"text-color-with-background","anchor":"after_multiply"}],"connections":[{"to":"after_multiply","from":"before_multiply","color":"green","label":"multiply both sides by 12"}]},"order":2,"title":"Substitute the new value","description":"Now use the new value of e and solve for f. Reaching the correct value earns the A1 mark."}]}],"generatedAt":"2026-07-19T07:02:03.887Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"bc2f5cea-8b7c-4d80-9d37-312b357b52f0:algebraic-constant","label":"Constant of Proportionality Method","steps":[{"type":"text","order":0,"title":"Set up direct proportion","content":"Since $e$ and $f$ are directly proportional, we can write this relationship as:\n\n$$f = ke$$\n\nwhere $k$ is the constant of proportionality. This equation tells us that $f$ is always a constant multiple of $e$.","highlights":[{"text":"e is directly proportional to f","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\text{We are given } \framebox{\\htmlData{anchor=e-val}{\\color{@sky}{e = 3}}} \text{ when } \framebox{\\htmlData{anchor=f-val}{\\color{@amber}{f = 36}}}"},{"latex":"\text{Substitute these into } \\htmlData{anchor=f-eq}{\\color{@amber}{f}} = k\\htmlData{anchor=e-eq}{\\color{@sky}{e}}:"},{"latex":"\\color{@amber}{36} = k \\times \\color{@sky}{3}"},{"latex":"k = \\frac{36}{3} = 12"}],"highlights":[{"color":"sky","style":"box","anchor":"e-val"},{"color":"amber","style":"box","anchor":"f-val"}],"connections":[{"to":"e-eq","from":"e-val","color":"sky","label":null,"style":"solid"},{"to":"f-eq","from":"f-val","color":"amber","label":null,"style":"solid"}]},"order":1,"title":"Find the constant","description":"Substitute the known values of $e$ and $f$ to solve for our constant $k$."},{"type":"text","order":2,"title":"Write the full formula","content":"Now that we know the constant of proportionality is $k = 12$, we can write down the complete formula linking $f$ and $e$:\n\n$$f = 12e$$"},{"type":"text","order":3,"title":"Calculate the final value","content":"To find the value of $f$ when $e = 4$, substitute $e = 4$ into our formula:\n\n$$\\begin{aligned} f &= 12 \\times 4 \\\\ f &= 48 \\end{aligned}$$\n\nSo, when $e = 4$, $f = 48$.","highlights":[{"text":"Find the value of f when e = 4","location":"specification"}]}]},{"id":"bc2f5cea-8b7c-4d80-9d37-312b357b52f0:fractional-proportion","label":"Direct Ratio Method","steps":[{"type":"text","order":0,"title":"Identify the relationship","content":"Since $e$ is directly proportional to $f$, the ratio between them remains constant. This means we can set up a direct proportion:\n\n$$\\frac{e_1}{f_1} = \\frac{e_2}{f_2}$$","highlights":[{"text":"directly proportional","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Given values: } \\htmlData{anchor=e1}{\\color{@sky}{e_1 = 3}}, \\; \\htmlData{anchor=f1}{\\color{@amber}{f_1 = 36}}$"},{"latex":"$$\\text{New values: } \\htmlData{anchor=e2}{\\color{@purple}{e_2 = 4}}, \\; \\htmlData{anchor=f2}{\\color{@teal}{f_2 = ?}}$"},{"latex":"$$$\\frac{\\htmlData{anchor=e1-sub}{\\color{@sky}{3}}}{\\htmlData{anchor=f1-sub}{\\color{@amber}{36}}} = \\frac{\\htmlData{anchor=e2-sub}{\\color{@purple}{4}}}{\\htmlData{anchor=f2-sub}{\\color{@teal}{f}}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"e1"},{"color":"amber","style":"text-color","anchor":"f1"},{"color":"purple","style":"text-color","anchor":"e2"}],"connections":[{"to":"e1-sub","from":"e1","color":"sky","label":null,"style":"solid"},{"to":"f1-sub","from":"f1","color":"amber","label":null,"style":"solid"},{"to":"e2-sub","from":"e2","color":"purple","label":null,"style":"solid"}]},"order":1,"title":"Set up the proportion","description":"Substitute the known values from the question into our proportion ratio."},{"type":"text","order":2,"title":"Solve by cross-multiplying","content":"To solve for $f$, we cross-multiply the terms of our proportion:\n\n$$\\begin{aligned} 3 \\times f &= 36 \\times 4 \\\\ 3f &= 144 \\end{aligned}$$\n\nNow, divide both sides by $3$ to isolate $f$:\n\n$$f = \\frac{144}{3} = 48$$"},{"type":"text","order":3,"title":"Final Answer","content":"The value of $f$ when $e = 4$ is **$48$**.\n\n*Note on exam marking:* Setting up a correct direct ratio equation such as $\\frac{3}{36} = \\frac{4}{f}$ is equivalent to setting up the formula $e = kf$ and will secure you the method marks (M1 M1), leading to the final accuracy mark (A1) for the correct answer of $48$."}]},{"id":"bc2f5cea-8b7c-4d80-9d37-312b357b52f0:unitary-method","label":"Unitary Method","steps":[{"type":"text","order":0,"title":"Understand the relationship","content":"We are given that $e$ is directly proportional to $f$. This means that as $e$ increases or decreases, $f$ changes by the same ratio. \n\nWe know that when $e = 3$, $f = 36$. We can use the **unitary method** to find the value of $f$ for a single unit of $e$.","highlights":[{"text":"e is directly proportional to f","location":"specification"},{"text":"When e = 3, f = 36","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"To find the value of $f$ when $e = 1$, we divide the given value of $f$ by the corresponding value of $e$:"},{"latex":"$$f \text{ when } e = 1 \text{ is } \frac{\text{\\htmlData{anchor=f-initial}{36}}}{\text{\\htmlData{anchor=e-initial}{3}}}$"},{"latex":"$$\frac{36}{3} = \text{\\htmlData{anchor=unit-val}{12}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"f-initial"},{"color":"amber","style":"text-color","anchor":"e-initial"},{"color":"green","style":"box","anchor":"unit-val"}],"connections":[{"to":"unit-val","from":"f-initial","color":"sky","label":"divide","style":"solid"}]},"order":1,"title":"Find value for one unit","description":"Dividing the total value of $f$ by $e$ gives us the value of $f$ per unit of $e$."},{"type":"hyper_latex","block":{"nodes":[{"text":"Now, to find $f$ when $e = 4$, we multiply the unit value of $f$ by $4$:"},{"latex":"$$f = \text{\\htmlData{anchor=unit-val-2}{\\color{@green}{12}}} \\times \\text{\\htmlData{anchor=e-target}{\\color{@purple}{4}}}$"},{"latex":"$$f = \text{\\htmlData{anchor=final-ans}{\\color{@orange}{48}}}$"}],"highlights":[{"color":"green","style":"text-color","anchor":"unit-val-2"},{"color":"purple","style":"text-color","anchor":"e-target"},{"color":"orange","style":"box","anchor":"final-ans"}],"connections":[{"to":"final-ans","from":"unit-val-2","color":"orange","label":"multiply","style":"solid"}]},"order":2,"title":"Calculate the final value","description":"Multiplying the single-unit value of $12$ by the target value $4$ gives the final value of $f$."},{"type":"text","order":3,"title":"State the final answer","content":"So, when $e = 4$, the value of $f$ is **$48$**.\n\nIn an exam, setting up the proportional relationship either via the unitary method or using the constant of proportionality $k$ would earn you full marks."}]}],"generatedAt":"2026-06-13T00:06:29.850Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/9dd9548c-a4cc-4572-ac05-70bfe6fe25e7.mp4","durationSeconds":86,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/9dd9548c-a4cc-4572-ac05-70bfe6fe25e7.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","ca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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 10

Direct and Inverse Proportion Questions

Direct and Inverse Proportion

Question 10

Direct and Inverse Proportion Questions