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bg-background","children":["$","$L39",null,{"backHref":"/gcse/maths/aqa/questions","basePath":"/gcse/maths/aqa/grade-5-5-2-direct-and-inverse-proportion/questions","formulaUrl":"https://www.aqa.org.uk/files/resources.mathematics.AQA-8300H-FS-INS-2025_PDF/c87a0d4512612ebfb053370b84e9339e546e9e27.pdf","initialIndex":35,"initialQuestionWindow":[{"id":"8fd42485-cefe-4784-8b11-fb1854268377","revisionId":"0137abb8-0c77-4a02-a1ce-32382d049400","questionNumber":null,"questionType":"LA","specification":"Four mathematical sketches are shown below, labeled **M**, **N**, **O**, and **P**.\n\n![Graph M](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/8fd42485-cefe-4784-8b11-fb1854268377/image-1-c8b940ec4a456230800abaed.svg)\n**M**\n\n![Graph N](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/8fd42485-cefe-4784-8b11-fb1854268377/image-2-5d7bbfa4090c6e6c8b13842a.svg)\n**N**\n\n![Graph O](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/8fd42485-cefe-4784-8b11-fb1854268377/image-3-f2088633a9352f3eb487547e.svg)\n**O**\n\n![Graph P](https://assets.mathsgenie.co.uk/question-image-redraws/sol-max-v1/8fd42485-cefe-4784-8b11-fb1854268377/image-4-bad78130de40a55efb90e66d.svg)\n**P**","difficulty":3,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":false,"parts":[{"id":"8e938423-6f32-4516-a07e-f551b5227b02","label":"a","content":"Match each statement with its correct graph letter.\n\n| Proportionality type | Graph letter |\n|---|---|\n| Direct proportion ($y = kx$) | |\n| Inverse proportion ($y = \\frac{k}{x}$) | |\n| Direct square proportion ($y = kx^2$) | |\n| Inverse square proportion ($y = \\frac{k}{x^2}$) | |","markscheme":"- Direct proportion ($y = kx$): **O**\n- Inverse proportion ($y = \\frac{k}{x}$): **M**\n- Direct square proportion ($y = kx^2$): **P**\n- Inverse square proportion ($y = \\frac{k}{x^2}$): **N**\n\nAward 2 marks for all 4 correct, 1 mark for 2 or 3 correct. B2","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Any 2 or 3 relationships correctly identified.","code":"B"},{"id":"p2","marks":1,"description":"All 4 relationships correctly identified.","code":"B"}],"finalAnswers":["Direct: O, Inverse: M, Direct Square: P, Inverse Square: N"],"generatedAt":"2026-05-06T14:05:59.259Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"recognise_graph_shapes","label":"Recognise the graph shapes","steps":[{"type":"graph","order":1,"title":"Match the direct graphs","viewport":{"top":1.5,"left":-2,"right":2,"bottom":-1.5},"axisLabels":{"x":"x","y":"y"},"description":"A direct-proportion graph is a straight line through the origin, so it is O. A direct-square graph is a parabola with its turning point at the origin; P opens upwards, so its constant is positive. The examples are scaled so their only visible meeting point is the origin.","expressions":[{"id":"expr-0","color":"sky","latex":"y=0.5x\\left\\{-2\\le x\\le2\\right\\}"},{"id":"expr-1","color":"purple","latex":"y=0.2x^2\\left\\{-2\\le x\\le2\\right\\}"},{"id":"expr-2","color":"sky","label":"O: straight line","latex":"(-1.5,-0.75)","pointStyle":"point","labelOrientation":"below"},{"id":"expr-3","color":"purple","label":"P: parabola","latex":"(1.4,0.392)","pointStyle":"point","labelOrientation":"right"},{"id":"expr-4","color":"green","label":"origin","latex":"(0,0)","pointStyle":"point","labelOrientation":"left"}],"qualitative":true},{"type":"graph","order":2,"title":"Distinguish inverse graphs","viewport":{"top":6,"left":-5,"right":5,"bottom":-5},"axisLabels":{"x":"x","y":"y"},"description":"The sketches use a positive constant $k$. In $y=\\frac{k}{x}$, changing the sign of $x$ changes the sign of $y$, so its branches lie in quadrants I and III: this is M. In $y=\\frac{k}{x^2}$, $x^2$ is positive on either side of zero, so $y$ stays above the $x$-axis and the graph is symmetric about the $y$-axis: this is N.","expressions":[{"id":"expr-0","color":"orange","latex":"y=\\frac{1}{x}\\left\\{-5\\le x\\le-0.18\\right\\}"},{"id":"expr-1","color":"orange","latex":"y=\\frac{1}{x}\\left\\{0.18\\le x\\le5\\right\\}"},{"id":"expr-2","color":"red","latex":"y=\\frac{10}{x^2}\\left\\{-5\\le x\\le-0.4\\right\\}"},{"id":"expr-3","color":"red","latex":"y=\\frac{10}{x^2}\\left\\{0.4\\le x\\le5\\right\\}"},{"id":"expr-4","color":"orange","label":"M: y = k/x","latex":"(-2,-0.5)","pointStyle":"point","labelOrientation":"below"},{"id":"expr-5","color":"red","label":"N: y = k/x²","latex":"(-3,1.1111111111)","pointStyle":"point","labelOrientation":"above"}],"qualitative":true},{"type":"text","order":3,"title":"Record every match","content":"Enter the letters in the table:\n\n| Proportionality type | Graph letter |\n|---|---|\n| $y=kx$ | $O$ |\n| $y=\\frac{k}{x}$ | $M$ |\n| $y=kx^2$ | $P$ |\n| $y=\\frac{k}{x^2}$ | $N$ |\n\nAll four correct matches earn both marks. Two or three correct matches earn one mark, so fill every row even if you are unsure about one pair."}]}],"generatedAt":"2026-07-19T16:33:24.954Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"8e938423-6f32-4516-a07e-f551b5227b02:shape_recognition","label":"Graph Shape Recognition","steps":[{"type":"text","order":0,"title":"Analyze the direct relationships","content":"To match these equations to their corresponding graphs, we can recognize the standard geometric shapes associated with each type of algebraic relationship.\n\nLet's first look at the two **direct** proportionalities:\n1. **Direct proportion** ($y = kx$): This represents a linear relationship. The graph of any equation in this form is a **straight line** that passes through the origin $(0,0)$.\n2. **Direct square proportion** ($y = kx^2$): This represents a quadratic relationship. Since $y$ depends on $x^2$, the graph is a **parabolic curve** that starts at the origin $(0,0)$ and curves upwards with an increasing gradient."},{"type":"text","order":1,"title":"Match the direct graphs","content":"Using our shape recognition:\n- **Graph O** is a straight line passing through the origin. Therefore, **Graph O** represents **direct proportion** ($y = kx$).\n- **Graph P** starts at the origin and curves upwards increasingly steeply. Therefore, **Graph P** represents **direct square proportion** ($y = kx^2$).","highlights":[{"text":"Direct proportion ($y = kx$)","location":"specification"},{"text":"Direct square proportion ($y = kx^2$)","location":"specification"}]},{"type":"text","order":2,"title":"Analyze the inverse relationships","content":"Now let's look at the remaining two graphs, **M** and **N**, which both represent **inverse** relationships. \n\nIn both cases, as $x$ increases, $y$ decreases, creating a curve that is asymptotic to both axes (it gets closer and closer to the axes but never touches them).\n\nTo distinguish between them, we can compare how fast they decrease:\n- **Inverse proportion** ($y = \\frac{k}{x}$): The curve decreases at a steady, standard rate.\n- **Inverse square proportion** ($y = \\frac{k}{x^2}$): Because the denominator is squared, as $x$ increases, the denominator grows much faster. This causes $y$ to drop much more rapidly. The curve is **steeper** and hugs the axes more closely."},{"type":"text","order":3,"title":"Match the inverse graphs","content":"Comparing our two curves:\n- **Graph N** is visibly steeper and drops much faster than Graph **M**.\n- Therefore, **Graph M** represents **inverse proportion** ($y = \\frac{k}{x}$).\n- **Graph N** represents **inverse square proportion** ($y = \\frac{k}{x^2}$).","highlights":[{"text":"Inverse proportion ($y = \\frac{k}{x}$)","location":"specification"},{"text":"Inverse square proportion ($y = \\frac{k}{x^2}$)","location":"specification"}]},{"type":"text","order":4,"title":"Summary of answers","content":"Here is our completed table matching each proportion type to its graph letter:\n\n| Proportionality type | Graph letter |\n| :--- | :---: |\n| Direct proportion ($y = kx$) | **O** |\n| Inverse proportion ($y = \\frac{k}{x}$) | **M** |\n| Direct square proportion ($y = kx^2$) | **P** |\n| Inverse square proportion ($y = \\frac{k}{x^2}$) | **N** |"}]},{"id":"8e938423-6f32-4516-a07e-f551b5227b02:elimination_and_behavior","label":"Process of Elimination and Rates of Change","steps":[{"type":"text","order":0,"title":"Identify Direct Relationships","content":"First, let's look at the two **direct proportion** relationships where $y$ increases as $x$ increases:\n\n* **Direct proportion** ($y = kx$): This is a linear relationship, which is represented by a straight line starting at the origin $(0,0)$. Looking at our graphs, **Graph O** is a straight line.\n* **Direct square proportion** ($y = kx^2$): This is a non-linear (quadratic) relationship, which is represented by a curve that starts at the origin $(0,0)$ and curves upwards as $x$ increases. Looking at our graphs, **Graph P** shows this upward curve.\n\nNow we can match these two:\n* $y = kx \\rightarrow$ **O**\n* $y = kx^2 \\rightarrow$ **P**","highlights":[{"text":"Direct proportion ($y = kx$)","location":"specification"},{"text":"Direct square proportion ($y = kx^2$)","location":"specification"}]},{"type":"text","order":1,"title":"Identify Inverse Relationships","content":"Next, let's look at the remaining two options, which are **inverse relationships**. In both of these, $y$ decreases as $x$ increases, meaning the curve starts high on the left and slopes downwards towards the $x$-axis on the right:\n\n* **Inverse proportion** ($y = \\frac{k}{x}$)\n* **Inverse square proportion** ($y = \\frac{k}{x^2}$)\n\nThis leaves us with **Graph M** and **Graph N**. Both curves decrease as $x$ increases, so we need to compare their rates of change to differentiate them.","highlights":[{"text":"Inverse proportion ($y = \\frac{k}{x}$)","location":"specification"},{"text":"Inverse square proportion ($y = \\frac{k}{x^2}$)","location":"specification"}]},{"type":"text","order":2,"title":"Compare Rates of Change","content":"To see which graph decreases faster, let's compare how quickly the values drop for each equation as $x$ increases (assuming a constant $k = 1$):\n\n$$\\begin{array}{c|c|c} x & y = \\frac{1}{x} & y = \\frac{1}{x^2} \\\\ \\hline 1 & 1 & 1 \\\\ 2 & 0.5 & 0.25 \\\\ 3 & 0.33 & 0.11 \\\\ 4 & 0.25 & 0.0625 \\end{array}$$\n\nNotice that the inverse square relationship $y = \\frac{k}{x^2}$ drops **much more sharply and rapidly** toward the axes compared to $y = \\frac{k}{x}$. \n\nComparing our remaining graphs:\n* **Graph N** drops much faster and hugs the axes more closely than Graph M, so it must represent the inverse square proportion ($y = \\frac{k}{x^2}$).\n* **Graph M** decreases more gradually, so it represents the standard inverse proportion ($y = \\frac{k}{x}$)."},{"type":"text","order":3,"title":"State the Final Answer","content":"By pairing our analysis together, we get the final completed table:\n\n| Proportionality type | Graph letter |\n|---|---|\n| Direct proportion ($y = kx$) | **O** |\n| Inverse proportion ($y = \\frac{k}{x}$) | **M** |\n| Direct square proportion ($y = kx^2$) | **P** |\n| Inverse square proportion ($y = \\frac{k}{x^2}$) | **N** |"}]}],"generatedAt":"2026-06-12T22:52:34.184Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":null,"walkthroughVideo":null},{"id":"ac7a77ed-4975-405a-a4f9-3ec8a4872dba","revisionId":"1447eecc-fad0-47be-8ba2-a30b98d25793","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"afdb4bd9-0024-49ba-9993-7da617995a7e","label":"a","content":"It takes 12 workers 5 days to build a wall.\n\nWork out how many days it would take 15 workers to build the same wall.","markscheme":"- Calculates total man-days: $12 \\times 5 = 60$ M1\n- Divides by number of workers: $60 \\div 15 = 4$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Evidence of calculating total work (e.g., $12 \\times 5$)","code":"M"},{"id":"p2","marks":1,"description":"Correct final answer of 4 days","code":"A"}],"finalAnswers":["4","4 days"],"generatedAt":"2026-05-06T14:05:50.856Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"worker_days_solution","label":"Use worker-days","steps":[{"type":"text","order":0,"title":"Keep the work fixed","content":"The wall is unchanged, so the total amount of work stays fixed. Measure this work in **worker-days**: one worker-day means the work done by one worker in one day.","highlights":[{"text":"the same wall","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=original}{\\color{@sky}{12\\times5}}"},{"latex":"\\htmlData{anchor=work}{\\color{@green}{60\\text{ worker-days}}}"}],"highlights":[{"color":"sky","anchor":"original"},{"color":"green","style":"text-color-with-background","anchor":"work"}],"connections":[{"to":"work","from":"original","color":"green","label":"calculate total work"}]},"order":1,"title":"Find total worker-days","description":"Multiply the original number of workers by the original number of days. This earns the method mark, M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"\\htmlData{anchor=division}{\\color{@amber}{60\\div15}}"},{"latex":"\\htmlData{anchor=days}{\\color{@green}{4\\text{ days}}}"}],"highlights":[{"color":"amber","anchor":"division"},{"color":"green","style":"text-color-with-background","anchor":"days"}],"connections":[{"to":"days","from":"division","color":"green","label":"divide by workers"}]},"order":2,"title":"Share between workers","description":"With the larger workforce, divide the fixed amount of work by the number of workers. The correct calculation and result earn A1."},{"type":"text","order":3,"title":"Check the direction","content":"There are more workers, so the job should take fewer days. The answer is therefore sensible: it would take $4$ days.","highlights":[{"text":"15 workers","location":"specification"}]}]}],"generatedAt":"2026-07-19T07:33:47.457Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"afdb4bd9-0024-49ba-9993-7da617995a7e:total_work","label":"Total Work Approach","steps":[{"type":"text","order":0,"title":"Understand the total work","content":"To solve this problem using the **Total Work Approach**, we first find the total size of the job. We measure this in **worker-days**, which represents the total effort needed if one worker did the entire job alone.","highlights":[{"text":"12 workers 5 days","location":"specification"}]},{"type":"text","order":1,"title":"Calculate total worker-days","content":"We calculate the total work by multiplying the number of workers by the number of days:\n\n$$\\text{Total Work} = 12 \\text{ workers} \\times 5 \\text{ days} = 60 \\text{ worker-days}$$\n\nCalculating this correctly earns you the first mark (**M1**)."},{"type":"text","order":2,"title":"Divide by new workers","content":"We now share this total workload of $60$ worker-days among the $15$ workers. To find the number of days it will take them, we divide the total work by the new number of workers:\n\n$$\\text{Days} = \\frac{60 \\text{ worker-days}}{15 \\text{ workers}} = 4 \\text{ days}$$\n\nThis gives us our final answer of **4 days**, earning you the second mark (**A1**).","highlights":[{"text":"15 workers","location":"specification"}]}]},{"id":"afdb4bd9-0024-49ba-9993-7da617995a7e:inverse_proportion_formula","label":"Inverse Proportion Equation","steps":[{"type":"text","order":0,"title":"Understand the inverse relationship","content":"This is an **inverse proportion** problem because as the number of workers increases, the time taken to build the wall decreases.\n\nWhen two quantities are inversely proportional, their product is constant. We can represent this relationship using the inverse proportion equation:\n\n$$W_1 \\times D_1 = W_2 \\times D_2$$\n\nWhere:\n* $W_1$ and $D_1$ are the initial number of workers and days.\n* $W_2$ and $D_2$ are the new number of workers and days.","highlights":[{"text":"It takes 12 workers 5 days to build a wall.","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\htmlData{anchor=W1}{\\color{@sky}{12}} \\text{ workers} \\times \\htmlData{anchor=D1}{\\color{@amber}{5}} \\text{ days} = \\htmlData{anchor=W2}{\\color{@purple}{15}} \\text{ workers} \\times \\htmlData{anchor=D2}{\\color{@teal}{x}} \\text{ days}$"},{"latex":"$$$12 \\times 5 = 15 \\times x$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"W1"},{"color":"amber","style":"text-color","anchor":"D1"},{"color":"purple","style":"text-color","anchor":"W2"},{"color":"teal","style":"text-color","anchor":"D2"}],"connections":[{"to":"W2","from":"W1","color":"sky","label":"Workers increase","style":"dashed"}]},"order":1,"title":"Set up the equation","description":"We substitute the known values into our equation, letting $x$ be the unknown number of days."},{"type":"text","order":2,"title":"Simplify and solve for x","content":"First, calculate the left side of the equation (the total work in 'man-days'):\n\n$$12 \\times 5 = 60$$\n\nNow, substitute this back to solve for $x$:\n\n$$\\begin{aligned} 60 &= 15 \\times x \\\\ x &= \\frac{60}{15} \\\\ x &= 4 \\end{aligned}$$\n\nThis means it will take **4 days** for 15 workers to build the wall.","highlights":[{"text":"Work out how many days it would take 15 workers to build the same wall.","location":"specification"}]}]},{"id":"afdb4bd9-0024-49ba-9993-7da617995a7e:scaling_factor","label":"Scaling Factor Method","steps":[{"type":"text","order":0,"title":"Identify the relationship","content":"The number of workers and the time taken to build the wall are in **inverse proportion**.\n\nThis means that if we increase the number of workers, the time taken to build the wall will decrease by the same proportion (or scale factor).","highlights":[{"text":"12 workers 5 days","location":"specification"},{"text":"15 workers","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\text{Original workers} = \\htmlData{anchor=orig-workers}{\\color{@sky}{12}}$$"},{"latex":"$$$\\text{New workers} = \\htmlData{anchor=new-workers}{\\color{@pink}{15}}$$"},{"latex":"$$$\\text{Scale Factor} = \\frac{\\htmlData{anchor=new-sub}{\\color{@pink}{15}}}{\\htmlData{anchor=orig-sub}{\\color{@sky}{12}}} = \\htmlData{anchor=sf-val}{\\color{@purple}{1.25}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"orig-workers"},{"color":"pink","style":"text-color","anchor":"new-workers"},{"color":"purple","style":"text-color","anchor":"sf-val"}],"connections":[{"to":"orig-sub","from":"orig-workers","color":"sky","label":null,"style":"solid"},{"to":"new-sub","from":"new-workers","color":"pink","label":null,"style":"solid"}]},"order":1,"title":"Calculate the scale factor","description":"First, find the scale factor by which the number of workers increases. We divide the new number of workers by the original number of workers."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$$\\text{Original days} = \\htmlData{anchor=orig-days}{\\color{@amber}{5}}$$"},{"latex":"$$$\\text{Scale Factor} = \\htmlData{anchor=sf-ref}{\\color{@purple}{1.25}}$$"},{"latex":"$$$\\text{New days} = \\frac{\\htmlData{anchor=days-sub}{\\color{@amber}{5}}}{\\htmlData{anchor=sf-sub}{\\color{@purple}{1.25}}} = \\htmlData{anchor=ans-val}{\\color{@green}{4}}$$"}],"highlights":[{"color":"amber","style":"text-color","anchor":"orig-days"},{"color":"purple","style":"text-color","anchor":"sf-ref"},{"color":"green","style":"text-color","anchor":"ans-val"}],"connections":[{"to":"days-sub","from":"orig-days","color":"amber","label":null,"style":"solid"},{"to":"sf-sub","from":"sf-ref","color":"purple","label":null,"style":"solid"}]},"order":2,"title":"Adjust the time","description":"Since the relationship is inversely proportional, multiplying the workers by $1.25$ means we must divide the number of days by $1.25$."},{"type":"text","order":3,"title":"State final answer","content":"It will take 15 workers **4 days** to build the same wall."}]}],"generatedAt":"2026-06-13T01:39:34.033Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":88},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/ac7a77ed-4975-405a-a4f9-3ec8a4872dba.mp4","durationSeconds":77,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/ac7a77ed-4975-405a-a4f9-3ec8a4872dba.jpg"}},{"id":"ae40d109-7527-4556-be8d-bfeb1bcecdf0","revisionId":"31f08a64-1c31-4b03-9d4d-384b113e30f8","questionNumber":null,"questionType":"LA","specification":"","difficulty":1,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"f8fb8e53-5e0a-4be8-af83-0090b08b1d3e","label":"a","content":"A typist types 1200 words in 15 minutes.\n\nWork out how many minutes it would take the typist to type 2000 words.","markscheme":"- Finds the typing speed per minute: $1200 \\div 15 = 80$ words per minute M1\n- Calculates time for 2000 words: $2000 \\div 80 = 25$ minutes A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Evidence of finding the words per minute rate (e.g., $1200 \\div 15$ or $2000/80$).","code":"M"},{"id":"p2","marks":1,"description":"Correct final answer of 25 minutes.","code":"A","acceptable":["25","25 minutes"]}],"finalAnswers":["25"],"generatedAt":"2026-05-06T14:05:50.629Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"unit_rate","label":"Find the unit rate","steps":[{"type":"text","order":0,"title":"Find one minute","content":"First find the **unit rate**, meaning the number of words typed in one minute. Divide the number of words by the number of minutes:\n\n$$1200 \\div 15 = 80$$\n\nThe typist types $80$ words per minute. This earns **M1** for using the correct method."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$2000 \\div 80 = \\htmlData{anchor=time}{t}$"},{"latex":"$$\\htmlData{anchor=result}{t=25\\text{ minutes}}$"}],"connections":[{"to":"result","from":"time","color":"green","label":"calculate"}]},"order":1,"title":"Find the time","description":"Now divide the required number of words by the typing speed. This earns A1 for calculating 2000 ÷ 80 = 25 minutes."}]}],"generatedAt":"2026-07-20T10:49:04.226Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"f8fb8e53-5e0a-4be8-af83-0090b08b1d3e:unitary_words_per_minute","label":"Unitary Method (Rate per Minute)","steps":[{"type":"text","order":0,"title":"Calculate the typing speed","content":"To find out how many words the typist can type in one minute, we divide the total number of words by the time taken.\n\n$$\\text{Typing speed} = \\frac{1200 \\text{ words}}{15 \\text{ minutes}}$$\n\n$$\\text{Typing speed} = 80 \\text{ words per minute}$$","highlights":[{"text":"types 1200 words in 15 minutes","location":"specification"}]},{"type":"text","order":1,"title":"Find the total time","content":"Now, we divide the target word count of $2000$ words by the typing speed of $80$ words per minute to calculate the total time required.\n\n$$\\text{Time taken} = \\frac{2000 \\text{ words}}{80 \\text{ words per minute}}$$\n\n$$\\text{Time taken} = 25 \\text{ minutes}$$","highlights":[{"text":"type 2000 words","location":"specification"}]}]},{"id":"f8fb8e53-5e0a-4be8-af83-0090b08b1d3e:scaling_ratio","label":"Scaling Factor Approach","steps":[{"type":"text","order":0,"title":"Identify the key information","content":"We are given the following information about the typist:\n\n* **Original number of words:** $1200\\text{ words}$\n* **Original time taken:** $15\\text{ minutes}$\n* **Target number of words:** $2000\\text{ words}$\n\nOur goal is to find how many minutes it will take to type the target of $2000$ words using the **scaling factor approach**.","highlights":[{"text":"A typist types 1200 words in 15 minutes.","location":"specification"},{"text":"Work out how many minutes it would take the typist to type 2000 words.","location":"specification"}]},{"type":"text","order":1,"title":"Calculate the scale factor","content":"First, we determine how many times larger the target word count is compared to the original word count. This is called the **scale factor** ($k$).\n\n$$\\text{Scale Factor } (k) = \\frac{\\text{Target words}}{\\text{Original words}}$$\n\nSubstitute the values into the formula:\n\n$$k = \\frac{2000}{1200}$$\n\nSimplify the fraction by dividing the numerator and denominator by $400$:\n\n$$k = \\frac{5}{3}$$"},{"type":"text","order":2,"title":"Apply the scale factor","content":"Since the number of words increases by a scale factor of $\\frac{5}{3}$, the time required will also increase by the same scale factor:\n\n$$\\text{New Time} = \\text{Original Time} \\times k$$\n\nMultiply the original time ($15\\text{ minutes}$) by our scale factor:\n\n$$\\text{New Time} = 15 \\times \\frac{5}{3}$$\n\nTo calculate this easily, divide $15$ by $3$ first, and then multiply by $5$:\n\n$$\\text{New Time} = 5 \\times 5 = 25\\text{ minutes}$$"},{"type":"text","order":3,"title":"Final answer","content":"It would take the typist **25 minutes** to type 2000 words."}]},{"id":"f8fb8e53-5e0a-4be8-af83-0090b08b1d3e:cross_multiplication","label":"Direct Proportion Equation","steps":[{"type":"text","order":0,"title":"Define the variables","content":"Let's define our unknown value. Let $x$ be the number of minutes it takes to type 2000 words.\n\nSince the typing rate is constant, the number of words typed is directly proportional to the time spent typing. This means the ratio of words to minutes remains constant:\n\n$$\\frac{\\text{Words}_1}{\\text{Minutes}_1} = \\frac{\\text{Words}_2}{\\text{Minutes}_2}$$"},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Case 1: } \\htmlData{anchor=w1}{\\color{@sky}{1200}} \\text{ words in } \\htmlData{anchor=t1}{\\color{@amber}{15}} \\text{ minutes}$"},{"latex":"$$\\text{Case 2: } \\htmlData{anchor=w2}{\\color{@purple}{2000}} \\text{ words in } \\htmlData{anchor=t2}{\\color{@teal}{x}} \\text{ minutes}$"},{"latex":"$$$\\frac{\\htmlData{anchor=ratio1-top}{\\color{@sky}{1200}}}{\\htmlData{anchor=ratio1-bot}{\\color{@amber}{15}}} = \\frac{\\htmlData{anchor=ratio2-top}{\\color{@purple}{2000}}}{\\htmlData{anchor=ratio2-bot}{\\color{@teal}{x}}}$$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"w1"},{"color":"amber","style":"text-color","anchor":"t1"},{"color":"purple","style":"text-color","anchor":"w2"},{"color":"teal","style":"text-color","anchor":"t2"}],"connections":[{"to":"ratio1-top","from":"w1","color":"sky","label":"words","style":"solid"},{"to":"ratio1-bot","from":"t1","color":"amber","label":"minutes","style":"solid"},{"to":"ratio2-top","from":"w2","color":"purple","label":"words","style":"solid"},{"to":"ratio2-bot","from":"t2","color":"teal","label":"minutes","style":"solid"}]},"order":1,"title":"Set up the proportion","description":"We substitute the given values from the question into our proportion equation."},{"type":"text","order":2,"title":"Cross-multiply and solve","content":"To solve for $x$, we cross-multiply to eliminate the fractions:\n\n$$\\begin{aligned} 1200 \\times x &= 2000 \\times 15 \\\\[10pt] 1200x &= 30000 \\end{aligned}$$\n\nNow, divide both sides by $1200$ to isolate $x$:\n\n$$\\begin{aligned} x &= \\frac{30000}{1200} \\\\[10pt] x &= 25 \\end{aligned}$$\n\nIt takes **$25$ minutes** to type 2000 words.","calculator":[{"gdc":{"expression":"30000 / 1200","instructions":"Type 30000 / 1200 and press ENTER."},"ti84":{"expression":"30000 / 1200","instructions":"Type 30000 / 1200 and press ENTER."},"desmos":{"expression":"30000 / 1200","instructions":"Type 30000 / 1200 in the input box."},"description":"Use a calculator to easily verify the division."}]}]}],"generatedAt":"2026-06-13T01:33:36.515Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":82},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/ae40d109-7527-4556-be8d-bfeb1bcecdf0.mp4","durationSeconds":66,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/ae40d109-7527-4556-be8d-bfeb1bcecdf0.jpg"}}],"subjectCode":"gcse_maths__aqa","summaries":[{"id":"8f0008a5-3ac6-4daa-ab98-050efff8c8c8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"2efa34ad-c4e1-4fb2-99c7-e40e22697f46","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"6334644e-dba1-4be7-b3e0-97b7a6fc7377","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"71c255df-ec36-40dc-a20c-787ef9075fa1","difficultyBand":"easy","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"a4bda0dc-d8a6-4b15-bfe5-c086660f9be8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"b08c8c9b-351b-4fbe-a354-99f17519cfb7","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"b2a500a8-943b-4475-bdba-acc34e9cbfec","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"b4bd99ed-85f3-499c-806e-e8605d933975","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"bb00807a-a770-4cd5-a9a3-4f7f1fae43d5","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"c8d3cd6a-a050-4279-b1c8-092a2e0831ba","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"cc641143-4555-4183-9360-09a0b92261e8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"d2aac61c-aaaa-4b70-b87a-2b4ead388c73","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"dcd403ca-e74e-4712-be67-2420d24b535b","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"62593c20-a6cd-4a9d-b44e-91ff8dcfe767","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"182f75f6-f7cf-4350-b93a-923288d6b828","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"2118d5a1-2797-4c94-94ed-de149bd9311d","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"316cae9e-40e9-4ab8-9f61-be628b25354f","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":false},{"id":"3a4a646e-1d18-4987-8c4a-2532b3136883","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"3dea0308-e49d-4726-9111-5e36ecdf40ed","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"55f524a9-359f-4c43-987b-043a18fb86b5","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"673dfdcd-9896-4a4e-a3c5-915dc304449b","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"704f2fe1-bf7d-4d5c-a1f4-751ef6502ba3","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"b52d2ac2-2a2c-4a79-96a8-3b8d64b91913","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"bdeb869d-75a4-49dd-8e82-7368d5a1e68a","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"c3ce6dda-d4a3-4979-8dfe-53b8434404d8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"f0e45ce5-0e33-4809-9d77-5b9f3307f761","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"ff691b7d-38fb-402c-84df-553e039916b8","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"95284614-40ff-4ac9-97f3-a71a07fedfb2","difficultyBand":"hard","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"15dd20f3-0925-40c1-b6bc-5580f693d254","difficultyBand":"medium","calculatorUse":"non_calc","totalMarks":2,"questionType":"LA","isCalculation":true},{"id":"1d2b9e81-6f4f-471e-a039-e2f9def4a426","difficultyBand":"hard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and Inverse Proportion","topicId":"10d067f8-e6d9-45de-8116-a7f0fc44a036","topicSidebar":"$L3a","topicSlug":"grade-5-5-2-direct-and-inverse-proportion","topicTitle":"Direct and Inverse Proportion"}]}],"$L3b","$L3c"]}]

Direct and Inverse Proportion

Question 12

Direct and Inverse Proportion Questions

Direct and Inverse Proportion

Question 12

Direct and Inverse Proportion Questions