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quantities. Their values are connected by a constant multiplier called the constant of proportionality."},{"text":"Is equal to","correct":false,"explanation":"An equals sign, $=$, represents equality. The symbol $\\propto$ instead means that one quantity is proportional to another through a constant multiplier."},{"text":"Is approximately equal to","correct":false,"explanation":"Approximate equality is represented by $\\approx$, not $\\propto$. The symbol $\\propto$ states that two quantities are related by a constant proportion."},{"text":"Is not equal to","correct":false,"explanation":"Inequality is represented by symbols such as $\\ne$, $<$ or $>$. The symbol $\\propto$ means “is proportional to” and describes a scaling relationship."}]},{"id":"81308ac4-da19-b4a1-e29b-568dd8a67baf","cardType":"basic","points":1,"front":"What is the name of the constant $k$ in proportion equations?","back":"The constant of proportionality","mcqOptions":[{"text":"The constant of proportionality","correct":true,"explanation":"The value $k$ is called the constant of proportionality because it remains fixed and converts the proportional relationship into an equation."},{"text":"The coefficient of proportionality","correct":false,"explanation":"A coefficient is a broader term for a numerical multiplier, but in a proportion equation $k$ is specifically called the constant of proportionality."},{"text":"The gradient of proportionality","correct":false,"explanation":"The gradient describes the slope of a graph and only coincides with $k$ in some direct-proportion graphs. The general name for $k$ is the constant of proportionality."},{"text":"The variable of proportionality","correct":false,"explanation":"A variable can take different values, whereas $k$ remains fixed for a given proportional relationship. It is therefore the constant of proportionality."}]},{"id":"ac215568-c423-4b9e-e202-58c40ecb29aa","cardType":"basic","points":1,"front":"Write the formula for $A$ is directly proportional to the square of $B$.","back":"$$A = kB^2$","mcqOptions":[{"text":"$$A = kB^2$","correct":true,"explanation":"Direct proportionality to the square of $B$ means $A\\propto B^2$. Introducing the constant of proportionality $k$ gives $A=kB^2$."},{"text":"$$A = kB$","correct":false,"explanation":"This represents direct proportionality to $B$, not to its square. Squaring the stated variable gives the correct formula $A=kB^2$."},{"text":"$$A = \\frac{k}{B^2}$","correct":false,"explanation":"Placing $B^2$ in the denominator creates inverse proportionality to the square of $B$. Direct proportionality requires $A=kB^2$."},{"text":"$$A = kB^3$","correct":false,"explanation":"Cubing $B$ uses the wrong power and would make $A$ proportional to the cube of $B$. The stated square relationship is $A=kB^2$."}]},{"id":"6404186d-1aac-2cfa-7964-fa838ca105da","cardType":"basic","points":1,"front":"Write the formula for $P$ is inversely proportional to the cube of $Q$.","back":"$$P = \\frac{k}{Q^3}$","mcqOptions":[{"text":"$$P = \\frac{k}{Q^3}$","correct":true,"explanation":"Inverse proportionality to the cube of $Q$ means $P\\propto 1/Q^3$. Introducing the constant $k$ gives $P=\\frac{k}{Q^3}$."},{"text":"$$P = \\frac{k}{Q}$","correct":false,"explanation":"This omits the cube and represents inverse proportionality to $Q$ itself. Inverse proportionality to the cube requires $P=\\frac{k}{Q^3}$."},{"text":"$$P = kQ^3$","correct":false,"explanation":"Multiplying by $Q^3$ describes direct proportionality to the cube, reversing the stated relationship. For inverse proportionality, the cube belongs in the denominator."},{"text":"$$P = \\frac{k}{Q^2}$","correct":false,"explanation":"This uses a square rather than the stated cube, so the power is incorrect. The required inverse-cube formula is $P=\\frac{k}{Q^3}$."}]},{"id":"b05f83c4-85c9-ff41-7273-94ebc8fd6f85","cardType":"cloze","points":null,"front":"If $y$ is directly proportional to $x$, as $x$ increases, $y$ [ ] by the same scale factor.","back":"If $y$ is directly proportional to $x$, as $x$ increases, $y$ increases by the same scale factor.","mcqOptions":[{"text":"increases","correct":true,"explanation":"In direct proportion, $y=kx$, so multiplying $x$ by a scale factor multiplies $y$ by exactly the same factor. Therefore, as $x$ increases, $y$ increases."},{"text":"decreases","correct":false,"explanation":"Decreasing $y$ as $x$ increases describes an inverse relationship. In direct proportion, both quantities change in the same direction and by the same scale factor."},{"text":"stagnates","correct":false,"explanation":"A constant $y$ would mean it does not depend on $x$. In direct proportion, increasing $x$ increases $y$ by the same scale factor."},{"text":"fluctuates","correct":false,"explanation":"Direct proportion gives a consistent same-direction change, not repeated movement up and down. Since $y=kx$, increasing $x$ makes $y$ increase by the same scale factor."}]},{"id":"1bcc4889-53c3-4c24-f0e4-a485677027e1","cardType":"basic","points":3,"front":"What are the three main steps to solve most proportion problems?","back":"

Write the formula with $k$
Substitute values to find $k$
Substitute the new value and solve

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

Write the formula with $k$
Substitute values to find $k$
Substitute the new value and solve

","correct":true,"explanation":"A proportion problem is solved by forming the equation with $k$, using known paired values to calculate $k$, and then substituting the new value to find the unknown."},{"text":"

Write the formula without $k$
Substitute values to remove $k$
Substitute the new value and estimate

","correct":false,"explanation":"Eliminating $k$ before using known values loses the constant that defines the relationship. First write the formula, then find $k$, and finally substitute and solve."},{"text":"

Write the formula with $k$
Substitute the new value to find $k$
Substitute known values and solve

","correct":false,"explanation":"The new value cannot normally determine $k$ because its paired unknown is not yet known. Use the known pair to find $k$ before substituting the new value."},{"text":"

Write the formula with $k=1$
Substitute values to check $k$
Substitute the new value and solve

","correct":false,"explanation":"Assuming $k=1$ confuses proportionality with equality and ignores the known data. Calculate $k$ from the given pair, then use it with the new value."}]},{"id":"0e5fcd30-5663-09f7-ca35-aa71aadb960d","cardType":"basic","points":2,"front":"In the inverse proportion formula $y = \\frac{k}{x}$, what calculation involving $x$ and $y$ always equals $k$?","back":"The product $xy$ (or $x \\times y$)","mcqOptions":[{"text":"The product $xy$ (or $x \\times y$)","correct":true,"explanation":"Multiplying $y=\\frac{k}{x}$ by $x$ gives $xy=k$. Therefore, the product of corresponding $x$ and $y$ values remains constant."},{"text":"The sum $x+y$ (or $x + y$)","correct":false,"explanation":"The sum is not fixed in an inverse relationship and changes as $x$ and $y$ change. Rearranging $y=\\frac{k}{x}$ shows that the constant product is $xy=k$."},{"text":"The quotient $x/y$ (or $x \\div y$)","correct":false,"explanation":"The quotient is generally not constant because dividing $x$ by $y$ produces a changing value. Multiplying the equation by $x$ gives the constant product $xy=k$."},{"text":"The difference $x-y$ (or $x - y$)","correct":false,"explanation":"The difference between inversely proportional quantities does not stay fixed. From $y=\\frac{k}{x}$, multiplication by $x$ shows that their product $xy$ equals $k$."}]},{"id":"c5f2cc72-479a-28e6-f0c4-ebf76f419c80","cardType":"cloze","points":null,"front":"If $y$ is inversely proportional to $x$, as $x$ increases, $y$ [ ].","back":"If $y$ is inversely proportional to $x$, as $x$ increases, $y$ decreases.","mcqOptions":[{"text":"decreases","correct":true,"explanation":"For inverse proportion, $y=\\frac{k}{x}$. As a positive value of $x$ increases, the denominator grows, so the corresponding value of $y$ decreases."},{"text":"increases","correct":false,"explanation":"Increasing alongside $x$ describes direct rather than inverse proportion. In $y=\\frac{k}{x}$, a larger positive denominator produces a smaller value of $y$."},{"text":"stagnates","correct":false,"explanation":"A constant $y$ would not respond to changes in $x$. In inverse proportion, $xy=k$, so increasing positive $x$ forces $y$ to decrease."},{"text":"doubles","correct":false,"explanation":"Doubling occurs only under particular direct-proportion changes, not whenever $x$ increases. In inverse proportion, increasing positive $x$ makes $y$ decrease."}]},{"id":"393ad8b3-236c-f27a-6d03-8ed83c344756","cardType":"basic","points":2,"front":"Identify the two required features of a direct proportion graph ($y \\propto x$).","back":"

Straight line
Passes through the origin (0,0)

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

Straight line
Passes through the origin (0,0)

","correct":true,"explanation":"Direct proportion has the form $y = kx$, which produces a straight line. Substituting $x = 0$ gives $y = 0$, so the line passes through the origin."},{"text":"

Straight line
Crosses the y-axis above (0,0)

","correct":false,"explanation":"A non-zero y-intercept indicates a relationship of the form $y = kx + c$, not direct proportion. A direct proportion graph must be straight and pass through the origin."},{"text":"

Curved line
Passes through the origin (0,0)

","correct":false,"explanation":"Passing through the origin alone is insufficient; the curved shape is the misconception here. Since direct proportion has equation $y = kx$, its graph must also be a straight line."},{"text":"

Straight line
Passes through the point (0,1)

","correct":false,"explanation":"The point (0,1) gives a y-intercept of 1, so the variables are not directly proportional. The required straight line must instead pass through the origin (0,0)."}]},{"id":"e1dd5d50-6d71-5964-64c3-8d43bf18b974","cardType":"basic","points":2,"front":"Describe the graph of $y \\propto x^2$.","back":"A U-shaped curve (parabola) with its vertex at the origin.","mcqOptions":[{"text":"A U-shaped curve (parabola) with its vertex at the origin.","correct":true,"explanation":"The equation has the form $y = kx^2$. For positive $k$, this is a parabola whose minimum point is the origin and whose two sides rise symmetrically."},{"text":"A straight line (linear graph) passing through the origin.","correct":false,"explanation":"A straight line through the origin represents $y \\propto x$, not $y \\propto x^2$. Squaring $x$ produces a U-shaped parabola with its vertex at the origin."},{"text":"A reciprocal curve with branches approaching both axes.","correct":false,"explanation":"Branches that approach the axes are characteristic of an inverse relationship such as $y \\propto \\frac{1}{x}$. A square relationship produces a U-shaped parabola at the origin."},{"text":"A V-shaped graph with its turning point at the origin.","correct":false,"explanation":"A V-shaped graph represents an absolute-value relationship such as $y = k|x|$. Squaring produces a smooth U-shaped parabola rather than two straight sections meeting at the origin."}]},{"id":"edf8e2ff-9477-8bf9-130e-c17683cacd6b","cardType":"basic","points":2,"front":"What does the graph of $y \\propto \\frac{1}{x}$ look like?","back":"Two curved branches (typically in the top-right and bottom-left quadrants).","mcqOptions":[{"text":"Two curved branches (typically in the top-right and bottom-left quadrants).","correct":true,"explanation":"For $y = \\frac{k}{x}$ with positive $k$, $x$ and $y$ have the same sign. This creates two curved branches that approach but never reach either axis."},{"text":"One U-shaped curve with its vertex at the origin.","correct":false,"explanation":"A U-shaped curve is associated with a square relationship such as $y \\propto x^2$. A reciprocal relationship instead has two separate curved branches approaching the coordinate axes."},{"text":"One straight diagonal line passing through the origin.","correct":false,"explanation":"A straight line through the origin represents direct proportion, $y \\propto x$. For $y \\propto \\frac{1}{x}$, the changing reciprocal rate creates two separate curved branches."},{"text":"Two curved branches that cross both coordinate axes.","correct":false,"explanation":"The branches approach the axes asymptotically but never cross them, because $x = 0$ is undefined and $y$ cannot equal zero. They typically occupy the top-right and bottom-left quadrants."}]},{"id":"ff886f59-ef41-b1e0-8590-a406355f1d4d","cardType":"basic","points":2,"front":"How does the graph of $y \\propto \\frac{1}{x^2}$ differ from $y \\propto \\frac{1}{x}$ regarding the x-axis?","back":"Both branches are above the x-axis (symmetric about the y-axis).","mcqOptions":[{"text":"Both branches are above the x-axis (symmetric about the y-axis).","correct":true,"explanation":"Because $x^2$ is positive for every non-zero $x$, $\\frac{k}{x^2}$ is positive when $k$ is positive. The equal outputs for $x$ and $-x$ give y-axis symmetry."},{"text":"Both branches are below the x-axis (symmetric about the y-axis).","correct":false,"explanation":"This incorrectly treats the square as making the reciprocal negative. For positive $k$, $x^2$ is always positive, so both branches lie above the x-axis and are symmetric about the y-axis."},{"text":"The branches are on opposite sides of the x-axis (symmetric about the origin).","correct":false,"explanation":"Origin symmetry applies to $y \\propto \\frac{1}{x}$ because changing the sign of $x$ changes the sign of $y$. Squaring removes that sign change, placing both branches above the x-axis."},{"text":"Both branches cross the x-axis (symmetric about the y-axis).","correct":false,"explanation":"A reciprocal-square graph approaches $y = 0$ but never reaches it, so neither branch crosses the x-axis. Both branches remain above it and are symmetric about the y-axis."}]},{"id":"af3315fb-cdf7-de44-8d7d-ceb00542f0f3","cardType":"cloze","points":null,"front":"A straight line graph that [ ] does not represent direct proportion.","back":"A straight line graph that misses the origin does not represent direct proportion.","mcqOptions":[{"text":"misses the origin","correct":true,"explanation":"A direct proportion has equation $y = kx$, so its graph must pass through (0,0). Any straight line that misses the origin has a non-zero intercept and is not direct proportion."},{"text":"passes through the origin","correct":false,"explanation":"Passing through the origin is essential rather than disqualifying. Together with being a straight line, this shows a relationship of the form $y = kx$, which is direct proportion."},{"text":"has a positive gradient","correct":false,"explanation":"A positive gradient does not prevent direct proportion; for example, $y = 2x$ is directly proportional. The feature that rules it out is missing the origin, indicating a non-zero intercept."},{"text":"slopes downwards","correct":false,"explanation":"A downward-sloping line can still represent direct proportion when $k$ is negative, provided it passes through the origin. A straight line fails the test only when it misses the origin."}]},{"id":"3602a700-1c73-85c6-6da1-f7e25c2db7ab","cardType":"basic","points":1,"front":"Write the formula for $y$ being directly proportional to the square root of $x$.","back":"$$y = k\\sqrt{x}$","mcqOptions":[{"text":"$$y = k\\sqrt{x}$","correct":true,"explanation":"Direct proportionality to the square root means $y \\propto \\sqrt{x}$. Replacing the proportionality sign with an equals sign requires the constant of proportionality $k$, giving $y = k\\sqrt{x}$."},{"text":"$$y = \\frac{k}{\\sqrt{x}}$","correct":false,"explanation":"Placing $\\sqrt{x}$ in the denominator changes the relationship to inverse proportion with the square root. Direct proportion requires multiplication by $\\sqrt{x}$, so the formula is $y = k\\sqrt{x}$."},{"text":"$$y = kx^2$","correct":false,"explanation":"This formula makes $y$ directly proportional to the square of $x$, not its square root. The square-root relationship is written $y = k\\sqrt{x}$."},{"text":"$$y = \\sqrt{kx}$","correct":false,"explanation":"Putting $k$ inside the root makes the effective multiplier $\\sqrt{k}$ rather than the stated constant $k$. The standard proportionality equation is $y = k\\sqrt{x}$."}]},{"id":"16802f8c-c99b-23a1-8499-75ff2058e264","cardType":"basic","points":2,"front":"How do you use a table of values to determine if $y \\propto x^2$ or $y \\propto x^3$?","back":"Test if $\\frac{y}{x^2}$ or $\\frac{y}{x^3}$ gives the same constant $k$ for every pair.","mcqOptions":[{"text":"Test if $\\frac{y}{x^2}$ or $\\frac{y}{x^3}$ gives the same constant $k$ for every pair.","correct":true,"explanation":"Rearranging $y = kx^2$ gives $k = \\frac{y}{x^2}$, while rearranging $y = kx^3$ gives $k = \\frac{y}{x^3}$. The correct model has a constant ratio for every pair."},{"text":"Test if $\\frac{x}{y^2}$ or $\\frac{x}{y^3}$ gives the same constant $k$ for every pair.","correct":false,"explanation":"This reverses the variables and squares or cubes the wrong one. Rearranging the proposed relationships requires testing $\\frac{y}{x^2}$ and $\\frac{y}{x^3}$ for a constant value."},{"text":"Test if $\\frac{y}{x}$ or $\\frac{x}{y}$ gives the same constant $k$ for every pair.","correct":false,"explanation":"These ratios test simple direct proportion between $x$ and $y$, not square or cube proportion. The relevant tests are $\\frac{y}{x^2}$ and $\\frac{y}{x^3}$."},{"text":"Test if $y-x^2$ or $y-x^3$ gives the same constant $k$ for every pair.","correct":false,"explanation":"Subtracting tests an additive relationship such as $y = x^2 + k$, rather than proportionality. For $y = kx^2$ or $y = kx^3$, divide by the relevant power of $x$."}]},{"id":"fa28886e-0550-263e-20e7-48bb60710d23","cardType":"cloze","points":null,"front":"When solving $\\sqrt{s} = 3$, the value of $s$ is [ ].","back":"When solving $\\sqrt{s} = 3$, the value of $s$ is 9.","mcqOptions":[{"text":"9","correct":true,"explanation":"Squaring both sides of $\\sqrt{s} = 3$ gives $s = 3^2 = 9$. Checking confirms that the principal square root of 9 is 3."},{"text":"6","correct":false,"explanation":"This comes from doubling 3 instead of squaring it. To undo a square root, square both sides: $s = 3^2 = 9$."},{"text":"3","correct":false,"explanation":"This leaves the square root operation unresolved. Squaring both sides of $\\sqrt{s} = 3$ gives $s = 9$, and indeed $\\sqrt{9} = 3$."},{"text":"$$\\sqrt{3}$","correct":false,"explanation":"Taking another square root applies the operation again rather than reversing it. The inverse operation is squaring, so $s = 3^2 = 9$."}]},{"id":"204133b6-1706-b431-81ff-3d228300f891","cardType":"basic","points":2,"front":"If $r = k\\sqrt{s}$ and $r=10$ when $s=25$, calculate $k$.","back":"$$k = 2$ ($10 = k \\times 5$)","mcqOptions":[{"text":"$$k = 2$ ($10 = k \\times 5$)","correct":true,"explanation":"Since $\\sqrt{25}=5$, substitution gives $10=5k$. Dividing both sides by $5$ gives the constant of proportionality $k=2$."},{"text":"$$k = 50$ ($10 \\times 5 = k$)","correct":false,"explanation":"This multiplies by $5$ instead of dividing by it. Since $10=5k$, dividing both sides by $5$ gives the correct value $k=2$."},{"text":"$$k = 0.4$ ($10 = 25k$)","correct":false,"explanation":"This uses $s=25$ directly and ignores the square root. The equation requires $\\sqrt{25}=5$, so $10=5k$ and therefore $k=2$."},{"text":"$$k = 5$ ($10 = 2k$)","correct":false,"explanation":"This reverses the roles of the known output and the constant when rearranging. Substitution actually gives $10=5k$, which leads to $k=2$."}]},{"id":"08e77e67-7657-8f40-aecb-1683ff08b55b","cardType":"basic","points":1,"front":"What is the common mistake when translating \"$p$ is directly proportional to $q$\" into an equation?","back":"Swapping variables; the first variable mentioned should be on the left ($p = kq$).","mcqOptions":[{"text":"Swapping variables; the first variable mentioned should be on the left ($p = kq$).","correct":true,"explanation":"The statement identifies $p$ as the dependent variable, so it belongs on the left. Direct proportion means multiplying $q$ by a constant: $p=kq$."},{"text":"Keeping variables in order; the second variable mentioned should be on the left ($q = kp$).","correct":false,"explanation":"Putting the second variable on the left reverses the stated dependency. As $p$ is directly proportional to $q$, the correct translation is $p=kq$."},{"text":"Using a constant of proportionality; directly proportional variables should be written without $k$ ($p = q$).","correct":false,"explanation":"Directly proportional variables are not necessarily equal, so omitting $k$ incorrectly assumes a constant of $1$. The general equation is $p=kq$."},{"text":"Multiplying by the second variable; direct proportion requires division by $q$ ($p = \\frac{k}{q}$).","correct":false,"explanation":"Dividing a constant by $q$ describes inverse proportion, not direct proportion. The first variable remains on the left and the correct equation is $p=kq$."}]},{"id":"5c23b4ff-9d93-879c-901a-139d7ced879d","cardType":"cloze","points":null,"front":"In inverse proportion, if the input gets larger, the output should usually get [ ].","back":"In inverse proportion, if the input gets larger, the output should usually get smaller.","mcqOptions":[{"text":"smaller","correct":true,"explanation":"In inverse proportion, increasing the input reduces the output so that their product remains constant. Therefore, the output usually gets smaller."},{"text":"larger","correct":false,"explanation":"An output getting larger with the input describes direct rather than inverse proportion. In inverse proportion, increasing the input makes the output smaller."},{"text":"unchanged","correct":false,"explanation":"An unchanged output would indicate no dependence on the input. In inverse proportion, the output must get smaller as the input gets larger."},{"text":"equal","correct":false,"explanation":"Inverse proportion does not mean that the input and output become equal. As the input gets larger, the output usually gets smaller to maintain a constant product."}]},{"id":"7637a0fb-604c-829d-817c-b04ea56fcb07","cardType":"basic","points":3,"front":"Compare the relationship of variables in $y = kx$ versus $y = \\frac{k}{x}$.","back":"

$y=kx$: variables scale in the same direction
$y=\\frac{k}{x}$: variables scale in opposite directions

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

$y=kx$: variables scale in the same direction
$y=\\frac{k}{x}$: variables scale in opposite directions

","correct":true,"explanation":"For $y=kx$, multiplying $x$ by a factor multiplies $y$ by the same factor. For $y=\\frac{k}{x}$, increasing $x$ decreases $y$ proportionally."},{"text":"

$y=kx$: variables scale in the opposite direction
$y=\\frac{k}{x}$: variables scale in the same direction

","correct":false,"explanation":"This reverses the two relationships. Direct proportion makes the variables scale together, while inverse proportion makes one decrease by the corresponding factor when the other increases."},{"text":"

$y=kx$: variables scale in the same direction
$y=\\frac{k}{x}$: variables scale in the same direction

","correct":false,"explanation":"The direct-proportion statement is right, but the inverse-proportion statement is not. In $y=\\frac{k}{x}$, increasing $x$ causes $y$ to decrease, so they scale oppositely."},{"text":"

$y=kx$: variables scale in the opposite direction
$y=\\frac{k}{x}$: variables scale in opposite directions

","correct":false,"explanation":"The inverse-proportion statement is right, but the direct-proportion statement is not. In $y=kx$, scaling $x$ by a factor scales $y$ by that same factor."}]}]}],"examCountdown":null,"heading":"Direct and Inverse Proportion","title":"Direct and Inverse Proportion Flashcards","topbar":{"backHref":"/gcse/maths/ccea/flashcards"},"topicSidebar":"$L3a"}]}],"topicId":"3af7bdb8-3293-6856-1bdb-f2f5a9e7ba14","walled":"$L3b"}],"$L3c","$L3d"]