Direct and Inverse Proportion
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Either paper 1 mark
\(a\) is inversely proportional to \(b\).
Circle the statement that is equivalent to this.
A) \(a\) is directly proportional to \(b\)
B) \(a\) is directly proportional to \(\dfrac{1}{ b }\)
C) \(a\) is directly proportional to \(b^2\)
D) \(a\) is directly proportional to \(\dfrac{1}{ b^2 }\)
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Either paper 4 marks
The diagram shows the position of a tap's dial. The tap is off at \(0^\circ\) and fully on at \(180^\circ\).
When fully on, water flows out of the tap at 24 litres per minute. The rate at which water flows out is in direct proportion to the angle of turn shown on the dial.
The water flows into a tank with a capacity of 115 litres.
Will it take less than 10 minutes to fill the tank? You must show your working.
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Either paper 3 marks
A chef has a tub of blueberries.
The chef uses all the blueberries and puts the same number on each dessert.
\(d\) is the number of desserts.
\(b\) is the number of blueberries on each dessert.
\(b\) is inversely proportional to \(d\), so
\(b = \dfrac{k}{ d }\)
where \(k\) is a constant.
(a)What does the constant \(k\) represent?
Tick the correct box.
☐ the number of blueberries in the tub
☐ the number of desserts
☐ the number of blueberries on each dessert
☐ none of the above
(b) Complete the table.b 3 d 80 48 8 -
Either paper 3 marksA train travels 42 miles. The table shows some values of the average speed of the train, in miles per hour, for different times taken.
Time (hours) Speed (mph) 1 42 2 21 3 4 5 6 7 (a)Three speeds are missing from the table. Work them out, and write the three in order: for 3 hours, for 4 hours and for 5 hours.
Give any decimal values correct to 1 decimal place.
(b)Draw a graph to show the average speed of the train for times taken between 1 hour and 6 hours. Plot the six points from your table on the grid, then join them with a smooth curve.
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Either paper 6 marks
\(T\) is directly proportional to \(M\).
When \(M = 5\), \(T = 25\).
(a)Find a formula for \(T\) in terms of \(M\).
(b)Work out the value of \(T\) when \(M = 2\).
(c)Work out the value of \(M\) when \(T = 85\).
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Either paper 4 marks
Here are two pairs of values of \(x\) and \(y\).
\(x = 8\), \(y = 32\)
\(x = 4\), \(y = 16\)
(a)Write down the value of \(y \div x\) for the first pair.
(b)Write down the value of \(y \div x\) for the second pair.
(c)Is \(y\) directly proportional to \(x\)? Show how you decide.
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Either paper 1 mark
\(a\) is directly proportional to \(\sqrt{b}\).
The value of \(b\) is multiplied by 64.
Marcus says: "\(a\) will be 32 times bigger."
Is he correct? Tick Yes or No. Give a reason for your answer.
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Either paper 3 marks
In this question, assume that each person works at the same rate.
14 people can unload a delivery lorry in 12 hours.
(a)If 8 people work on the same job, how many hours will it take?
(b)In fact, of the 8 people, 4 of them work at a slower rate. Everyone else works at the rate assumed in part (a).
What does this mean about the number of hours it will take? Tick one box.
☐ It is greater than the answer to (a)
☐ It is the same as the answer to (a)
☐ It is less than the answer to (a)
☐ It is not possible to say
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Either paper 7 marks
\(D\) is directly proportional to \(s^2\).
When \(s = 6\), \(D = 288\).
(a)Find a formula for \(D\) in terms of \(s\).
(b)Work out the value of \(D\) when \(s = 9\).
(c)Work out the positive value of \(s\) when \(D = 512\).
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Either paper 6 marks
\(y\) is inversely proportional to \(x\).
When \(x = 12\), \(y = 5\).
(a)Find a formula for \(y\) in terms of \(x\).
(b)Calculate the value of \(y\) when \(x = 5\).
(c)Calculate the value of \(x\) when \(y = 6\).
- B) \(a\) is directly proportional to \(\dfrac{1}{ b }\)
- No — the rate at \(75^\circ\) is \(10\) litres per minute, so \(10\) minutes delivers \(100\) litres, compared with the \(115\) litre tank.
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(a) the number of blueberries in the tub
(b) \(b = 5\) and \(b = 30\) -
(a) \(14\), \(10.5\) and \(8.4\) mph at 3, 4 and 5 hours
(b) The points \((1, 42)\), \((2, 21)\), \((3, 14)\), \((4, 10.5)\), \((5, 8.4)\) and \((6, 7)\), joined with one smooth curve that falls steeply and then levels off(b) -
(a) \(T = 5M\)
(b) \(10\)
(c) \(17\) -
(a) \(4\)
(b) \(4\)
(c) Yes — \(\dfrac{ 32 }{ 8 } = 4\) and \(\dfrac{ 16 }{ 4 } = 4\), which are equal, so \(y\) is directly proportional to \(x\). - No — the correct change is 8 times bigger, which does not match the claim of 32 times bigger.
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(a) \(21\ \text{hours}\)
(b) It is greater than the answer to (a) -
(a) \(D = 8s^2\)
(b) \(648\)
(c) \(8\) -
(a) \(y = \dfrac{ 60 }{ x }\)
(b) \(12\)
(c) \(10\)
Takes a couple of seconds the first time a sheet is made.