Direct and inverse proportion
Ratio, proportion and rates of change
- Higher
- Non-calculator
- 8 questions
- 29 marks
These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.
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$y$ is directly proportional to $x$. When $x = 4$, $y = 12$. Find $y$ when $x = 7$.
2 marks
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$y$ is directly proportional to $x$. When $x = 5$, $y = 20$. Find $x$ when $y = 8$.
2 marks
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$y$ is inversely proportional to $x$. When $x = 4$, $y = 6$. Find $y$ when $x = 8$.
3 marks
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$y$ is proportional to $x^{2}$. When $x = 3$, $y = 18$. Find $y$ when $x = 5$.
3 marks
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The cooking time $T$ hours is directly proportional to the square root of the mass $m$ kg. When $m = 9$, $T = 4.5$. Find $T$ when $m = 16$, and find $m$ when $T = 7.5$.
4 marks
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$y$ is inversely proportional to $x^{2}$. When $x = 2$, $y = 12$. Find $y$ when $x = 4$, and find $x$ when $y = 3$, taking $x$ positive.
4 marks
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The pressure $P$ of a gas is inversely proportional to its volume $V$. When $V = 4$, $P = 6$. Find a formula for $P$, the pressure when $V = 8$, and the volume when $P = 2.4$.
5 marks
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The cost $C$ pence of a circular badge is proportional to the square of its radius $r$ cm. A badge of radius $2$ cm costs $80$p. Find a formula for $C$, the cost of a badge of radius $5$ cm, the radius of a badge costing £$3.20$, and how many times as expensive a $6$ cm badge is as a $2$ cm badge.
6 marks
Mark scheme
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Answer. $21$
- $y = 3x$
- When $x = 7$, $y = 21$.
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Answer. $2$
- $y = 4x$
- $x = 2$
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Answer. $3$
- $xy = 24$
- $y = \dfrac{24}{8} = 3$
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Answer. $50$
- $y = 2x^{2}$
- $y = 2 \times 25 = 50$
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Answer. $6$ hours, and $25$ kg
- $T = 1.5\sqrt{m}$
- When $m = 16$, $T = 6$.
- $1.5\sqrt{m} = 7.5$ gives $m = 25$.
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Answer. $y = 3$ and $x = 4$
- $yx^{2} = 48$
- When $x = 4$, $y = 3$.
- When $y = 3$, $x^{2} = 16$, so $x = 4$.
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Answer. $P = \dfrac{24}{V}$, pressure $3$, volume $10$
- $P = \dfrac{24}{V}$
- When $V = 8$, $P = 3$.
- When $P = 2.4$, $V = 10$.
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Answer. $C = 20r^{2}$. A $5$ cm badge costs £$5$. The radius is $4$ cm. It is $9$ times as expensive.
- $20 \times 2^{2} = 80$, so $C = 20r^{2}$.
- Radius $5$: $20 \times 25 = 500$p $= £5$.
- £$3.20 = 320$p, so $20r^{2} = 320$ and $r = 4$.
- The scale factor of the radii is $3$, so the costs are in the ratio $9:1$.