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one apple."},{"text":"The total amount or value for all the units of a quantity.","correct":false,"explanation":"This describes a total rather than a unit rate. The unit rate is the amount or value corresponding to one single unit of the quantity."},{"text":"The difference in value between two separate units of a quantity.","correct":false,"explanation":"A unit rate is not found by comparing two individual units. It gives the amount or value for one single unit of the quantity."},{"text":"The amount or value for one hundred units of a quantity.","correct":false,"explanation":"This confuses a unit rate with a rate per hundred, such as a percentage. A unit rate gives the amount or value for one single unit."}]},{"id":"0b134026-a393-ff33-4e22-02de5281c735","cardType":"cloze","points":null,"front":"In direct proportion, multiplying one quantity by a scale factor [ ] the other quantity by the same scale factor.","back":"In direct proportion, multiplying one quantity by a scale factor multiplies the other quantity by the same scale factor.","mcqOptions":[{"text":"multiplies","correct":true,"explanation":"Directly proportional quantities change by the same scale factor. If one quantity is multiplied by a number, the other quantity is multiplied by that number too."},{"text":"divides","correct":false,"explanation":"Dividing the other quantity by the scale factor describes inverse proportion. In direct proportion, the same scale factor multiplies both quantities."},{"text":"subtracts","correct":false,"explanation":"Subtracting a scale factor confuses multiplicative proportional change with additive change. In direct proportion, multiplying one quantity by a scale factor multiplies the other by the same factor."},{"text":"squares","correct":false,"explanation":"Squaring the other quantity would create a non-linear relationship. In direct proportion, both quantities are multiplied by exactly the same scale factor."}]},{"id":"8793024f-db5d-bc30-6127-c2b539fab47d","cardType":"basic","points":1,"front":"What is the general algebraic formula for direct proportion?","back":"$$y = kx$","mcqOptions":[{"text":"$$y = kx$","correct":true,"explanation":"In direct proportion, the ratio $y \\div x$ is constant. Calling this constant $k$ gives the general equation $y = kx$."},{"text":"$$y = \\frac{k}{x}$","correct":false,"explanation":"This is the general formula for inverse proportion, where $xy$ is constant. Direct proportion has a constant ratio $y \\div x$, so its formula is $y = kx$."},{"text":"$$y = x + k$","correct":false,"explanation":"Adding a constant gives a straight line that does not generally pass through the origin. Direct proportion requires $y \\div x$ to be constant, giving $y = kx$."},{"text":"$$y = kx^2$","correct":false,"explanation":"This represents proportionality to the square of $x$, not direct proportion to $x$. The direct proportion formula is $y = kx$."}]},{"id":"30d6fa93-5e9a-fff5-bfdb-c62bff3f00e2","cardType":"basic","points":1,"front":"A machine makes 800 cartons in 4 hours. How do you calculate the unit rate?","back":"$$800 \\div 4 = 200$ cartons per hour","mcqOptions":[{"text":"$$800 \\div 4 = 200$ cartons per hour","correct":true,"explanation":"The unit rate is the output in one hour, so divide 800 cartons by 4 hours. This gives 200 cartons per hour."},{"text":"$$800 \\times 4 = 3200$ cartons per hour","correct":false,"explanation":"Multiplying finds an output for four times as long instead of one hour. Divide 800 cartons by 4 hours to get 200 cartons per hour."},{"text":"$$800 \\div 2 = 400$ cartons per hour","correct":false,"explanation":"Dividing by 2 halves the total but does not account for all 4 hours. Divide 800 by 4 to obtain 200 cartons per hour."},{"text":"$$800 - 4 = 796$ cartons per hour","correct":false,"explanation":"Subtracting hours from cartons mixes unlike quantities and cannot produce a rate. Divide 800 cartons by 4 hours to get 200 cartons per hour."}]},{"id":"8287dbd2-9636-d068-97f2-3d137e605b7f","cardType":"basic","points":2,"front":"8 apples cost £1.20. Outline the steps to find the cost of 11 apples.","back":"

$1.20 \\div 8$ to find the cost of 1 apple
Multiply that result by 11

","mcqOptions":[{"text":"

$1.20 \\div 8$ to find the cost of 1 apple
Multiply that result by 11

","correct":true,"explanation":"Dividing £1.20 by 8 finds the cost of one apple. Multiplying that unit cost by 11 then gives the cost of 11 apples."},{"text":"

$1.20 \\div 11$ to find the cost of 1 apple
Multiply that result by 8

","correct":false,"explanation":"This reverses the known and required numbers. Divide the known cost by 8 apples to find one apple, then multiply that cost by 11."},{"text":"

$1.20 \\times 8$ to find the cost of 1 apple
Divide that result by 11

","correct":false,"explanation":"Multiplying by 8 does not find a unit cost. Divide £1.20 by 8 to find one apple’s cost, then multiply by 11."},{"text":"

$1.20 \\div 8$ to find the cost of 1 apple
Divide that result by 11

","correct":false,"explanation":"The first step correctly finds the cost of one apple, but dividing again makes the value smaller. Multiply the unit cost by 11 to price 11 apples."}]},{"id":"1b00802c-314b-c6ad-697b-ad55822aea0e","cardType":"basic","points":2,"front":"The mass of a wire $m$ is $m=25l$. Find length $l$ when $m=175$.","back":"$$l = 7$ ($175 \\div 25$)","mcqOptions":[{"text":"$$l = 7$ ($175 \\div 25$)","correct":true,"explanation":"Substituting $m=175$ gives $175=25l$. Dividing both sides by 25 isolates $l$, so the wire’s length is 7."},{"text":"$$l = 4375$ ($175 \\times 25$)","correct":false,"explanation":"Multiplying by 25 applies the coefficient twice rather than undoing it. Since $175=25l$, divide 175 by 25 to obtain $l=7$."},{"text":"$$l = 150$ ($175 - 25$)","correct":false,"explanation":"Subtracting 25 treats the coefficient as an added amount. In $175=25l$, 25 multiplies $l$, so divide 175 by 25 to get $l=7$."},{"text":"$$l = 8$ ($175 \\div 25$)","correct":false,"explanation":"This is an arithmetic slip because $25 \\times 7=175$, whereas $25 \\times 8=200$. Dividing 175 by 25 therefore gives $l=7$."}]},{"id":"98f76e6d-fd8c-d15b-9985-25c3f766a835","cardType":"basic","points":3,"front":"4 packets of rice and 3 tins of soup cost £5.10. 5 packets of rice cost £2.25. How do you find the cost of 1 tin of soup?","back":"

Find cost of 1 rice ($2.25 \\div 5$)
Find cost of 4 rice
Subtract 4 rice cost from £5.10
Divide remainder by 3

","mcqOptions":[{"text":"

Find cost of 1 rice ($2.25 \\div 5$)
Find cost of 4 rice
Subtract 4 rice cost from £5.10
Divide remainder by 3

","correct":true,"explanation":"The price of one rice packet lets you calculate four packets. Removing that cost from £5.10 leaves the cost of three soups, which is divided by 3."},{"text":"

Find cost of 1 rice ($2.25 \\div 5$)
Find cost of 4 rice
Add 4 rice cost to £5.10
Divide result by 3

","correct":false,"explanation":"Adding the rice cost counts it twice, although it is already included in £5.10. Subtract the four-rice cost, then divide the remainder by 3."},{"text":"

Find cost of 1 rice ($2.25 \\div 5$)
Find cost of 3 rice
Subtract 3 rice cost from £5.10
Divide remainder by 4

","correct":false,"explanation":"This swaps the numbers of rice packets and soup tins. Remove the cost of 4 rice packets from £5.10, then divide the remainder by 3 soups."},{"text":"

Find cost of 5 rice (£2.25)
Find cost of 4 rice
Subtract 4 rice cost from £5.10
Multiply remainder by 3

","correct":false,"explanation":"The remainder is the combined cost of three soup tins, so multiplying by 3 increases it incorrectly. Divide that remainder by 3 to find one tin’s cost."}]},{"id":"13492127-2b75-8d7d-93e0-61d7b910b414","cardType":"cloze","points":null,"front":"In inverse proportion, multiplying one quantity by a scale factor [ ] the other quantity by that same scale factor.","back":"In inverse proportion, multiplying one quantity by a scale factor divides the other quantity by that same scale factor.","mcqOptions":[{"text":"divides","correct":true,"explanation":"In inverse proportion, the product of the two quantities remains constant. Multiplying one quantity by a scale factor therefore divides the other by that factor."},{"text":"multiplies","correct":false,"explanation":"Multiplying both quantities by the same scale factor describes direct proportion. In inverse proportion, multiplying one quantity by a factor divides the other by that factor."},{"text":"subtracts","correct":false,"explanation":"Subtracting the scale factor confuses multiplicative proportional change with additive change. In inverse proportion, the other quantity is divided by the same scale factor."},{"text":"squares","correct":false,"explanation":"Squaring the other quantity does not preserve the constant product required by inverse proportion. The other quantity must be divided by the same scale factor."}]},{"id":"b06f631c-091d-8d50-f231-b8ea7c80a145","cardType":"basic","points":1,"front":"What is the general algebraic formula for inverse proportion?","back":"$$y = \\frac{k}{x}$","mcqOptions":[{"text":"$$y = \\frac{k}{x}$","correct":true,"explanation":"In inverse proportion, the product $xy$ remains constant. Writing this constant as $k$ gives $xy=k$, which rearranges to $y = \\frac{k}{x}$."},{"text":"$$y = kx$","correct":false,"explanation":"This is the formula for direct proportion, where the ratio $\\frac{y}{x}$ remains constant. In inverse proportion, $xy$ is constant, so $y = \\frac{k}{x}$."},{"text":"$$y = \\frac{x}{k}$","correct":false,"explanation":"This makes $y$ increase as $x$ increases, so it describes a direct relationship with constant $\\frac{1}{k}$. Inverse proportion requires $y = \\frac{k}{x}$."},{"text":"$$y = k + x$","correct":false,"explanation":"Adding a constant produces a linear relationship with a fixed difference, not inverse proportion. For inverse proportion, the product $xy$ is constant, giving $y = \\frac{k}{x}$."}]},{"id":"0937ce86-0c73-ee5f-3638-43a765fe63f2","cardType":"basic","points":1,"front":"3 painters take 8 days to paint a house. How do you find the total amount of painter-days required for the job?","back":"$$3 \\times 8 = 24$ painter-days","mcqOptions":[{"text":"$$3 \\times 8 = 24$ painter-days","correct":true,"explanation":"The total work is found by multiplying the number of painters by the number of days. Therefore, the job requires $3 \\times 8 = 24$ painter-days."},{"text":"$$3 + 8 = 11$ painter-days","correct":false,"explanation":"Adding painters and days mixes two different quantities and does not measure total work. Painter-days are calculated by multiplication, so the job requires $3 \\times 8 = 24$ painter-days."},{"text":"$$8 \\div 3 = 2.67$ painter-days","correct":false,"explanation":"Dividing the days by the painters gives days per painter rather than the total work completed. Total painter-days equal painters multiplied by days, giving $3 \\times 8 = 24$ painter-days."},{"text":"$$8 - 3 = 5$ painter-days","correct":false,"explanation":"Subtracting the number of painters from the days does not represent total work. Painter-days combine the quantities by multiplication, so the required amount is $3 \\times 8 = 24$ painter-days."}]},{"id":"c16f1ff0-1cf4-043a-a310-6b6ced7b522c","cardType":"basic","points":2,"front":"3 painters take 8 days to paint a house. How long would 4 painters take?","back":"6 days ($24 \\div 4$)","mcqOptions":[{"text":"6 days ($24 \\div 4$)","correct":true,"explanation":"The job requires 24 painter-days in total. Sharing this work equally among 4 painters gives $24 \\div 4 = 6$, so they take 6 days."},{"text":"8 days ($24 \\div 3$)","correct":false,"explanation":"This repeats the calculation for the original 3 painters rather than using the new workforce. With 4 painters, the fixed 24 painter-days give $24 \\div 4 = 6$ days."},{"text":"12 days ($24 \\div 2$)","correct":false,"explanation":"This effectively treats the increase from 3 to 4 painters as though only 2 painters were working. The correct calculation is $24 \\div 4 = 6$ days."},{"text":"10 days ($24 - 4 - 10$)","correct":false,"explanation":"Subtracting the number of painters does not model inverse proportion or distribute the workload. The fixed total of 24 painter-days must be divided by 4, giving 6 days."}]},{"id":"b80d91f2-2e0d-3ae2-b52f-93c8159daea9","cardType":"basic","points":1,"front":"What is the formula for $y$ being directly proportional to $x$ squared?","back":"$$y = kx^2$","mcqOptions":[{"text":"$$y = kx^2$","correct":true,"explanation":"Direct proportion to $x$ squared means the ratio $\\frac{y}{x^2}$ is constant. Calling that constant $k$ gives the formula $y = kx^2$."},{"text":"$$y = kx$","correct":false,"explanation":"This represents $y$ being directly proportional to $x$, not to its square. Direct proportion to $x^2$ requires the squared term, giving $y = kx^2$."},{"text":"$$y = \\frac{k}{x^2}$","correct":false,"explanation":"Placing $x^2$ in the denominator represents inverse proportion to $x^2$. Direct proportion means multiplying the constant by $x^2$, so the correct formula is $y = kx^2$."},{"text":"$$y = kx^3$","correct":false,"explanation":"This describes direct proportion to the cube of $x$, which changes more rapidly than a squared relationship. Direct proportion to $x^2$ is written $y = kx^2$."}]},{"id":"b9d3c64b-47fb-462b-82dd-ec1037d03d01","cardType":"cloze","points":null,"front":"If $y$ is inversely proportional to $x^2$, the formula is $\\text{[...]}$.","back":"If $y$ is inversely proportional to $x^2$, the formula is $\\boxed{y = \\frac{k}{x^2}}$.","mcqOptions":[{"text":"$$y = \\frac{k}{x^2}$","correct":true,"explanation":"Inverse proportion places the relevant power of $x$ in the denominator. Since the relationship is inverse to $x^2$, the formula is $y = \\frac{k}{x^2}$."},{"text":"$$y = kx^2$","correct":false,"explanation":"Multiplying by $x^2$ describes direct proportion to $x^2$, so $y$ increases when the magnitude of $x$ increases. Inverse proportion requires $y = \\frac{k}{x^2}$."},{"text":"$$y = \\frac{k}{x}$","correct":false,"explanation":"This describes inverse proportion to $x$, but it omits the required square. Being inversely proportional to $x^2$ means the formula is $y = \\frac{k}{x^2}$."},{"text":"$$y = \\frac{k}{x^3}$","correct":false,"explanation":"This uses a cube and therefore represents inverse proportion to $x^3$. The stated relationship is inverse proportion to the square, so the formula is $y = \\frac{k}{x^2}$."}]},{"id":"36f0543d-710d-e278-a3d7-35ad31326696","cardType":"basic","points":1,"front":"In the formula $F = \\frac{48}{d^2}$, what must you do to $d=4$ before dividing?","back":"Square it (to get 16).","mcqOptions":[{"text":"Square it (to get 16).","correct":true,"explanation":"The denominator is $d^2$, so substituting $d=4$ requires calculating $4^2=16$ before dividing. This makes the expression $F=\\frac{48}{16}$."},{"text":"Double it (to get 8).","correct":false,"explanation":"Doubling confuses the exponent 2 with multiplication by 2. The exponent means multiply 4 by itself, so $4^2=16$ before calculating $48 \\div 16$."},{"text":"Cube it (to get 64).","correct":false,"explanation":"Cubing uses the power 3 rather than the power 2 shown in the formula. The denominator requires $4^2=16$, not $4^3=64$."},{"text":"Leave it (as 4).","correct":false,"explanation":"Using 4 unchanged ignores the exponent in $d^2$. You must square the substituted value, giving $4^2=16$, before dividing 48 by the denominator."}]},{"id":"01382179-2981-813f-757b-6186de9c3ef3","cardType":"basic","points":2,"front":"Calculate $F$ in $F = \\frac{48}{d^2}$ when $d = 4$.","back":"$$F = 3$ ($48 \\div 16$)","mcqOptions":[{"text":"$$F = 3$ ($48 \\div 16$)","correct":true,"explanation":"Substituting $d=4$ gives $d^2=4^2=16$. Therefore, $F=\\frac{48}{16}=3$."},{"text":"$$F = 12$ ($48 \\div 4$)","correct":false,"explanation":"This divides by $d$ and ignores the square in the denominator. Since $d^2=4^2=16$, the correct calculation is $48 \\div 16=3$."},{"text":"$$F = 6$ ($48 \\div 8$)","correct":false,"explanation":"This doubles 4 instead of squaring it, confusing an exponent of 2 with multiplication by 2. The denominator is $4^2=16$, so $F=3$."},{"text":"$$F = 0.75$ ($48 \\div 64$)","correct":false,"explanation":"This cubes 4, producing 64, even though the formula contains $d^2$. Squaring gives $4^2=16$, so $F=48 \\div 16=3$."}]},{"id":"873b7be9-7e49-e4d3-a7b5-1d75e37ac8da","cardType":"basic","points":1,"front":"What does the symbol $\\propto$ mean?","back":"is proportional to","mcqOptions":[{"text":"is proportional to","correct":true,"explanation":"The symbol $\\propto$ states that one variable is proportional to another. The precise direct or inverse relationship is then indicated by the accompanying algebraic expression."},{"text":"is equal to","correct":false,"explanation":"Equality is represented by $=$, whereas $\\propto$ indicates a proportional relationship involving an unspecified constant of proportionality. Therefore, the symbol means “is proportional to”."},{"text":"is inversely proportional to","correct":false,"explanation":"The symbol alone does not specify inverse proportion. An inverse relationship must be shown by a reciprocal expression, such as $y \\propto \\frac{1}{x}$; $\\propto$ means “is proportional to”."},{"text":"is approximately equal to","correct":false,"explanation":"Approximate equality is represented by $\\approx$, not $\\propto$. The proportionality symbol indicates that two quantities are related by a constant multiplier and means “is proportional to”."}]},{"id":"661dfe40-b5bd-2eae-29bb-7c7f4195192c","cardType":"basic","points":2,"front":"Describe the graph shape for $y \\propto x$.","back":"A straight line passing through the origin.","mcqOptions":[{"text":"A straight line passing through the origin.","correct":true,"explanation":"Direct proportion has the form $y = kx$, which is linear and gives $y = 0$ when $x = 0$. Its graph is therefore a straight line through the origin."},{"text":"A straight line not passing through the origin.","correct":false,"explanation":"A straight line missing the origin has a non-zero intercept, so its equation is $y = kx + c$, not direct proportion. For $y \\propto x$, the line passes through the origin."},{"text":"A curved line passing through the origin.","correct":false,"explanation":"Confusing direct proportion with a power relationship produces a curved graph. Since $y = kx$ has a constant gradient, its graph is a straight line through the origin."},{"text":"A straight line passing through the y-intercept.","correct":false,"explanation":"Treating the graph as an ordinary linear equation can suggest a separate y-intercept. In direct proportion, the y-intercept is zero, so the straight line passes through the origin."}]},{"id":"df76bacc-dd34-56e3-fd75-a4e8c94e4f8e","cardType":"basic","points":1,"front":"Describe the graph shape for $y \\propto x^2$.","back":"A U-shaped parabola through the origin.","mcqOptions":[{"text":"A U-shaped parabola through the origin.","correct":true,"explanation":"The relationship has the form $y = kx^2$. For positive $k$, squaring positive and negative x-values gives the U-shaped symmetry of a parabola with its vertex at the origin."},{"text":"An inverted parabola through the origin.","correct":false,"explanation":"An inverted parabola would require a negative proportionality constant. The standard graph intended for $y \\propto x^2$ uses positive $k$, giving a U-shaped parabola through the origin."},{"text":"A V-shaped graph through the origin.","correct":false,"explanation":"A V-shaped graph is associated with $y = k|x|$, where each side is straight. Squaring x instead produces a smooth U-shaped parabola through the origin."},{"text":"A U-shaped parabola above the origin.","correct":false,"explanation":"Shifting the vertex above the origin would add a constant, giving $y = kx^2 + c$. Direct square proportion has no added constant, so the parabola passes through the origin."}]},{"id":"3c363c46-7dc8-d7c0-5525-0390e376c65f","cardType":"cloze","points":null,"front":"The graph for inverse proportion ($y \\propto \\frac{1}{x}$) has two curved branches in [ ].","back":"The graph for inverse proportion ($y \\propto \\frac{1}{x}$) has two curved branches in opposite quadrants.","mcqOptions":[{"text":"opposite quadrants","correct":true,"explanation":"For $y = \\frac{k}{x}$, x and y have matching signs when $k>0$ and opposite signs when $k<0$. In either case, the branches occupy diagonally opposite quadrants."},{"text":"adjacent quadrants","correct":false,"explanation":"Adjacent quadrants would place both branches on the same side of one axis. For $y = \\frac{k}{x}$, changing the sign of x also changes the sign of y, producing opposite quadrants."},{"text":"the upper quadrants","correct":false,"explanation":"Placing both branches above the x-axis wrongly assumes y is always positive. In inverse proportion, y changes sign when x changes sign, so the branches lie in opposite quadrants."},{"text":"the lower quadrants","correct":false,"explanation":"Placing both branches below the x-axis wrongly assumes y is always negative. Since $y = \\frac{k}{x}$ changes sign across $x=0$, its branches occupy opposite quadrants."}]},{"id":"a478fd0e-58fd-6df4-8f17-920380ca5b17","cardType":"basic","points":1,"front":"Why is the graph for $y \\propto \\frac{1}{x^2}$ entirely above the x-axis (for $k > 0$)?","back":"Because $x^2$ is always positive.","mcqOptions":[{"text":"Because $x^2$ is always positive.","correct":true,"explanation":"For every allowed value $x \\ne 0$, squaring x gives a positive denominator. Since $k>0$, $y = \\frac{k}{x^2}$ is positive and lies above the x-axis."},{"text":"Because $x$ is always positive.","correct":false,"explanation":"The domain includes both positive and negative non-zero x-values, so x itself is not always positive. Squaring makes $x^2$ positive, ensuring $\\frac{k}{x^2}>0$ when $k>0$."},{"text":"Because $x^2$ is always negative.","correct":false,"explanation":"This reverses the sign rule for squaring: the square of any non-zero real number is positive, not negative. Therefore $\\frac{k}{x^2}$ is positive when $k>0$."},{"text":"Because $k$ is always negative.","correct":false,"explanation":"The question explicitly specifies $k>0$, so treating k as negative reverses the stated condition. The graph is above the x-axis because both k and $x^2$ are positive."}]}]}],"examCountdown":{"examDate":"2027-05-14","paperLabel":"Paper 1"},"heading":"Direct and Inverse Proportion","title":"Direct and Inverse Proportion Flashcards","topbar":{"backHref":"/gcse/maths/ocr/flashcards"},"topicSidebar":"$L37"}]}],"topicId":"115f7de9-632c-463e-b062-c76452f70147","walled":"$L38"}],"$L39","$L3a"]