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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":31,"initialQuestionWindow":[{"id":"1d2b9e81-6f4f-471e-a039-e2f9def4a426","revisionId":"93b61232-4bfc-4852-af33-c0e6e0e186d7","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"0239cc1e-ab7d-4e03-9fed-ad75e8433a92","label":"a","content":"5 screws and 3 nails cost £2.35.\n\n4 screws cost £1.40.\n\nWork out how much one nail costs.","markscheme":"- Finds cost of 1 screw: $1.40 \\div 4 = 0.35$ and then 5 screws: $0.35 \\times 5 = 1.75$ M1\n- Subtracts from total and divides by 3: $(2.35 - 1.75) \\div 3 = 0.20$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Calculates the total cost of the quantity of screws used in the first sentence (e.g., $1.40 \\div 4 \\times 5$)","code":"M"},{"id":"p2","marks":1,"description":"Calculates the cost of one nail by subtracting and dividing.","code":"A"}],"finalAnswers":["0.20","£0.20","20p"],"generatedAt":"2026-05-06T14:05:53.511Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"unitary_method","label":"Find one item first","steps":[{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Cost of 1 screw}=£1.40\\div 4$"},{"latex":"$$\\text{Cost of 1 screw}=£0.35$"}]},"order":0,"title":"Find one screw","description":"Divide the cost of 4 screws by 4 to find the cost of 1 screw. Keep this result for the next calculation; the method mark is awarded only after the cost of 5 screws is also found."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Cost of 5 screws}=£0.35\\times 5$"},{"latex":"$$\\text{Cost of 5 screws}=£1.75$"}]},"order":1,"title":"Find five screws","description":"Multiply the cost of 1 screw by 5. Showing both this calculation and the previous division earns M1."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Cost of 3 nails}=£2.35-£1.75$"},{"latex":"$$\\text{Cost of 3 nails}=£0.60$"}]},"order":2,"title":"Find three nails","description":"Subtract the cost of the 5 screws from the total cost. The amount left is the cost of all 3 nails."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{Cost of 1 nail}=\\pounds 0.60\\div 3$"},{"latex":"$$\\text{Cost of 1 nail}=\\pounds 0.20$"}]},"order":3,"title":"Find one nail","description":"Divide the remaining cost equally between the 3 nails. Completing the subtraction and this division to obtain £0.20 earns A1."}]}],"generatedAt":"2026-07-20T10:59:43.456Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"0239cc1e-ab7d-4e03-9fed-ad75e8433a92:arithmetic","label":"Arithmetic Method","steps":[{"type":"text","order":0,"title":"Find the cost of one screw","content":"We are given that 4 screws cost £1.40. To find the cost of a single screw, we divide this price by 4:\n\n$$\\text{Cost of 1 screw} = £1.40 \\div 4 = £0.35$$","highlights":[{"text":"4 screws cost £1.40","location":"specification"}]},{"type":"text","order":1,"title":"Find the cost of five screws","content":"Since we know one screw costs $£0.35$, we can multiply this price by 5 to find the total cost of 5 screws:\n\n$$\\text{Cost of 5 screws} = £0.35 \\times 5 = £1.75$$\n\n*(This earns the first method mark, **M1**)*"},{"type":"text","order":2,"title":"Find the cost of three nails","content":"We are told that 5 screws and 3 nails together cost £2.35.\n\nTo find the cost of just the 3 nails, we subtract the cost of the 5 screws ($£1.75$) from the overall total:\n\n$$\\text{Cost of 3 nails} = £2.35 - £1.75 = £0.60$$","highlights":[{"text":"5 screws and 3 nails cost £2.35","location":"specification"}]},{"type":"text","order":3,"title":"Find the cost of one nail","content":"Finally, we divide the cost of the 3 nails ($£0.60$) by 3 to find the cost of one nail:\n\n$$\\text{Cost of 1 nail} = £0.60 \\div 3 = £0.20$$\n\nSo, one nail costs **£0.20** (or **20p**).\n\n*(This earns the accuracy mark, **A1**)*"}]},{"id":"0239cc1e-ab7d-4e03-9fed-ad75e8433a92:algebraic","label":"Algebraic Method","steps":[{"type":"text","order":0,"title":"Define the variables","content":"To solve this problem algebraically, let's represent the unknown prices with variables:\n- Let $s$ be the cost of one screw (in £).\n- Let $n$ be the cost of one nail (in £)."},{"type":"text","order":1,"title":"Formulate the equations","content":"From the given information, we can write down a system of two linear equations:\n\n1. From \"5 screws and 3 nails cost £2.35\":\n$$5s + 3n = 2.35$$\n\n2. From \"4 screws cost £1.40\":\n$$4s = 1.40$$","highlights":[{"text":"5 screws and 3 nails cost £2.35","location":"specification"},{"text":"4 screws cost £1.40","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\text{Using the second equation: } 4\text{\\htmlData{anchor=eq2-s}{s}} = 1.40$"},{"latex":"$$s = \\frac{1.40}{4}$"},{"latex":"$$s = \\htmlData{anchor=s-val}{\\color{@sky}{0.35}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"eq2-s"},{"color":"sky","style":"text-color","anchor":"s-val"}],"connections":[{"to":"s-val","from":"eq2-s","color":"sky","label":"solve","style":"solid"}]},"order":2,"title":"Solve for s","description":"We find the cost of a single screw by dividing the cost of 4 screws by 4."},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\text{Substitute } s = \\htmlData{anchor=s-sub-val}{\\color{@sky}{0.35}} \text{ into } 5\\htmlData{anchor=s-var}{\\color{@sky}{s}} + 3n = 2.35$"},{"latex":"$$5(\\htmlData{anchor=s-sub-num}{\\color{@sky}{0.35}}) + 3n = 2.35$"},{"latex":"$$1.75 + 3n = 2.35$"},{"latex":"$$3n = 2.35 - 1.75$"},{"latex":"$$3n = 0.60$"},{"latex":"$$n = \\htmlData{anchor=n-val}{\\color{@green}{0.20}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"s-sub-val"},{"color":"sky","style":"text-color","anchor":"s-sub-num"},{"color":"green","style":"text-color","anchor":"n-val"}],"connections":[{"to":"s-sub-num","from":"s-sub-val","color":"sky","label":"substitute","style":"solid"}]},"order":3,"title":"Substitute and solve","description":"We substitute the value of a screw into the first equation to solve for the cost of one nail."},{"type":"text","order":4,"title":"State the final cost","content":"One nail costs £0.20 (or 20p)."}]}],"generatedAt":"2026-06-12T11:22:03.026Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":80},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/1d2b9e81-6f4f-471e-a039-e2f9def4a426.mp4","durationSeconds":90,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/1d2b9e81-6f4f-471e-a039-e2f9def4a426.jpg"}},{"id":"698d3433-1ad5-4ca5-a6cc-b797ee69db57","revisionId":"d28bb936-0353-40ff-a1ad-dd171d29b065","questionNumber":null,"questionType":"LA","specification":"","difficulty":2,"difficultyBand":"hard","totalMarks":2,"sourceTitle":null,"calculatorUse":"non_calc","isCalculation":true,"parts":[{"id":"3f2bdd6f-b852-4f1c-8559-44dcfa25bc58","label":"a","content":"It takes 10 builders 18 days to lay 5000 bricks.\n\nWork out how long it would take 15 builders to lay 5000 bricks.","markscheme":"- Calculates total builder days: $10 \\times 18 = 180$ M1\n- Divides by 15: $180 \\div 15 = 12$ A1","marks":2,"order":0,"markingRubric":{"totalMarks":2,"points":[{"id":"p1","marks":1,"description":"Evidence of calculating total man-days (e.g., $10 \\times 18$)","code":"M"},{"id":"p2","marks":1,"description":"Correct final answer of 12","code":"A"}],"finalAnswers":["12","12 days"],"generatedAt":"2026-05-06T14:05:50.866Z","modelVersion":"gemini-3-flash-preview"},"tableImages":[],"explanation":{"methods":[{"id":"builder_days","label":"Use builder-days","steps":[{"type":"text","order":0,"title":"Spot inverse proportion","content":"The number of bricks stays fixed. More builders means fewer days, so builders and days are **inversely proportional**: their product stays constant.","highlights":[{"text":"15 builders to lay 5000 bricks","location":"specification"}]},{"type":"text","order":1,"title":"Find total builder-days","content":"Multiply the original number of builders by the number of days:\n\n$$10\\text{ builders}\\times18\\text{ days}=180\\text{ builder-days}$$\n\nThis calculates the total amount of work and earns **M1**."},{"type":"text","order":2,"title":"Divide between builders","content":"Divide the total builder-days by $15$:\n\n$$180\\text{ builder-days}\\div15\\text{ builders}=12\\text{ days}$$\n\nThis calculation earns **A1**, so the final answer is $12\\text{ days}$."}]}],"generatedAt":"2026-07-20T11:06:16.453Z","modelVersion":"gpt-5.6-sol","explanationVersion":"v3"},"options":[]}],"options":[],"explanation":{"methods":[{"id":"3f2bdd6f-b852-4f1c-8559-44dcfa25bc58:builder_days","label":"Builder-Days Method","steps":[{"type":"text","order":0,"title":"Analyze the problem","content":"We need to find out how many days it will take for $15$ builders to lay $5,000$ bricks, given that $10$ builders take $18$ days to lay the same number of bricks.\n\nSince the number of bricks ($5,000$) is constant, we can focus entirely on the relationship between the number of builders and the number of days.","highlights":[{"text":"10 builders","location":"specification"},{"text":"18 days","location":"specification"},{"text":"15 builders","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"text":"First, find the total amount of work required for the job. We measure this in \"builder-days\" by multiplying the number of builders by the number of days:"},{"latex":"$$$\text{Total builder-days} = \text{\\htmlData{anchor=builders-1}{10\\ builders}} \\times \\text{\\htmlData{anchor=days-1}{18\\ days}}$$"},{"latex":"$$$\text{Total builder-days} = \\htmlData{anchor=total-days}{\\color{@sky}{180\\ builder-days}}$$"}],"highlights":[{"color":"sky","style":"box","anchor":"total-days"}],"connections":[{"to":"total-days","from":"builders-1","color":"sky","label":"multiply","style":"solid"},{"to":"total-days","from":"days-1","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Calculate total builder-days","description":"Calculating the total effort in terms of builder-days helps us keep track of the constant amount of work needed to complete the job."},{"type":"hyper_latex","block":{"nodes":[{"text":"Now, we divide this total workload by the new number of builders ($15$) to find the number of days they will need:"},{"latex":"$$$\\text{Days} = \\frac{\\text{\\htmlData{anchor=total-days-2}{\\color{@sky}{180\\ builder-days}}}}{\\text{\\htmlData{anchor=new-builders}{15\\ builders}}}$$"},{"latex":"$$$\\text{Days} = \\htmlData{anchor=final-ans}{\\color{@green}{12\\ days}}$$"}],"highlights":[{"color":"green","style":"box","anchor":"final-ans"}],"connections":[{"to":"final-ans","from":"total-days-2","color":"green","label":"divide by 15","style":"solid"}]},"order":2,"title":"Calculate the new duration","description":"Dividing the total work required by the number of available workers gives the total number of days needed to complete the job."},{"type":"text","order":3,"title":"State the final answer","content":"It would take **$12$ days** for $15$ builders to lay $5,000$ bricks.\n\n### Markscheme breakdown:\n* **$1$ mark** (Method mark) for calculating the total builder-days: $10 \\times 18 = 180$\n* **$1$ mark** (Accuracy mark) for dividing by $15$ to get $12$"}]},{"id":"3f2bdd6f-b852-4f1c-8559-44dcfa25bc58:inverse_proportion","label":"Inverse Proportion Scaling","steps":[{"type":"text","order":0,"title":"Analyze the given relationship","content":"The question asks us to find how long it takes $15$ builders to lay $5000$ bricks, given that $10$ builders take $18$ days to lay the same number of bricks.\n\nSince the number of bricks remains constant ($5000$ bricks), we only need to look at the relationship between the number of **builders** and the number of **days**.\n\nAs the number of builders increases, the work gets done faster, so the number of days decreases. This is an example of **inverse proportion**.","highlights":[{"text":"10 builders","location":"specification"},{"text":"18 days to lay 5000 bricks","location":"specification"},{"text":"15 builders to lay 5000 bricks","location":"specification"}]},{"type":"hyper_latex","block":{"nodes":[{"latex":"$$\\text{New Days} = \\text{Original Days} \\times \\frac{\\text{Initial Builders}}{\\text{Final Builders}}$"},{"latex":"$$D = \\htmlData{anchor=days}{\\color{@sky}{18}} \\times \\frac{\\htmlData{anchor=init}{\\color{@amber}{10}}}{\\htmlData{anchor=final}{\\color{@purple}{15}}}$"}],"highlights":[{"color":"sky","style":"text-color","anchor":"days"},{"color":"amber","style":"text-color","anchor":"init"},{"color":"purple","style":"text-color","anchor":"final"}],"connections":[{"to":"init","from":"days","color":"sky","label":null,"style":"solid"}]},"order":1,"title":"Set up inverse scaling","description":"Because the relationship is inversely proportional, we multiply the original number of days by the ratio of the initial number of builders to the final number of builders."},{"type":"text","order":2,"title":"Calculate the final answer","content":"Let's calculate the expression:\n\n$$\\begin{aligned} D &= 18 \\times \\frac{10}{15} \\\\ D &= 18 \\times \\frac{2}{3} \\\\ D &= 6 \\times 2 \\\\ D &= 12 \\text{ days} \\end{aligned}$$\n\nSo, it will take **$12$ days** for $15$ builders to lay $5000$ bricks."}]}],"generatedAt":"2026-06-12T18:57:31.374Z","modelVersion":"gemini-3.5-flash","explanationVersion":"v2"},"firstAttemptFullMarks":{"percent":95},"walkthroughVideo":{"url":"https://assets.mathsgenie.co.uk/walkthroughs/videos/698d3433-1ad5-4ca5-a6cc-b797ee69db57.mp4","durationSeconds":71,"posterUrl":"https://assets.mathsgenie.co.uk/walkthroughs/videos/698d3433-1ad5-4ca5-a6cc-b797ee69db57.jpg"}},{"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** 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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 10

Direct and Inverse Proportion Questions

Direct and Inverse Proportion

Question 10

Direct and Inverse Proportion Questions