AQA · Mathematics · 8300
GCSE Maths: Proportion: inverse
Ratio & Proportion · AQA GCSE Maths (8300). Cards are generated from the Corbettmaths textbook exercise, practice questions and answers for Proportion: inverse (Corbettmaths Video 255), and every card links back to the exact page it came from. Practice tests generate fresh variations of these questions, so the numbers change every attempt. Source material © Corbettmaths — used with attribution.
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What does it mean when variable A is directly proportional to variable B?
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When A increases, B increases at a constant rate. The relationship can be written as A = kB, where k is a constant.
What does it mean when variable T is inversely proportional to variable N?
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When T increases, N decreases (and vice versa). The relationship can be written as T = k/N, where k is a constant.
If y is directly proportional to x², what is the general formula?
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y = kx², where k is a constant of proportionality.
In a spring, the tension T is directly proportional to the extension y. This relationship is an example of ?.
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Hooke's law
If A is directly proportional to B³, and B is doubled, by what factor does A increase?
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A increases by a factor of 8 (since 2³ = 8).
If y is inversely proportional to x², and x is doubled, by what factor does y change?
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y is divided by 4 (or multiplied by 1/4), since doubling x makes x² four times larger.
What real-world relationship is modeled by force being inversely proportional to the square of distance?
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The force exerted by a magnet on a metal object (or gravitational/electromagnetic forces following inverse square laws).
If the falling distance d is directly proportional to t², what physical law does this represent?
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The motion of a falling object under constant acceleration (gravity), where d = ½gt².
When finding a constant of proportionality, what information do you need?
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You need one pair of corresponding values for the two variables in the relationship.
If A is directly proportional to B, and you double B, what happens to A?
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A also doubles, since the relationship is linear (A = kB).
If H varies directly with the cube of c, what is the general form of the equation?
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H = kc³, where k is the constant of proportionality.
If force F is inversely proportional to the square of distance d, what is the general form of the equation?
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F = k/d², where k is the constant of proportionality.
When distance d is halved in the equation F = k/d², what happens to the force F?
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F becomes 4 times larger (quadruples). Since d is halved, d² becomes one-quarter, so F = k/(d²/4) = 4k/d².
If C is directly proportional to the square root of y, what is the general form of the equation?
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C = k√y, where k is the constant of proportionality.
If time t is inversely proportional to the square of the number of staff s, what is the general form of the equation?
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t = k/s², where k is the constant of proportionality.
If w is increased by 10% and w is directly proportional to x³, what is the percentage increase in x?
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Approximately 3.23%. Since w ∝ x³, if w increases by 10%, then 1.1 = (new x/old x)³, so new x/old x = ∛1.1 ≈ 1.0323.
If x is increased by 10% and w is directly proportional to x³, what is the percentage increase in w?
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33.1%. If x increases by 10%, then new w = k(1.1x)³ = 1.331kx³ = 1.331w, which is a 33.1% increase.
If A is directly proportional to the cube root of B, what is the general form of the equation?
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A = k∛B or A = kB^(1/3), where k is the constant of proportionality.
If W is directly proportional to the square of x, what is the relationship expressed algebraically?
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W = kx² (where k is a constant of proportionality)
If W = kx² and increasing x by 2 makes W three times larger, what equation relates the original and new values?
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k(x+2)² = 3kx²
When W = kx² and k(x+2)² = 3kx², what simplified equation results after canceling k and expanding?
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x² + 4x + 4 = 3x², which simplifies to 2x² - 4x - 4 = 0 or x² - 2x - 2 = 0
What is the exact positive solution to x² - 2x - 2 = 0?
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x = 1 + √3 (using the quadratic formula, and taking the positive root since x must be positive)
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