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Either paper 1 mark
Anya is asked to share 96 stickers between Owen and Priya in the ratio \(2 : 6\).
She writes:
Owen gets \(96 \times 2 = 192\) stickers
Priya gets \(96 \times 6 = 576\) stickers
Describe what Anya has done wrong.
Show answer
Anya multiplied the total by each ratio number, instead of first dividing by the total number of parts to find the value of one part. The correct shares are Owen 24 and Priya 72 stickers.Anya has multiplied the total by each ratio number, instead of first dividing by the total number of parts to find the value of one part.
The ratio has \(2 + 6 = 8\) parts, so one part is \(96 \div 8 = 12\) stickers.
Owen should get \(2 \times 12 = 24\) stickers and Priya should get \(6 \times 12 = 72\) stickers.
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Either paper 3 marks
Marcus makes green paint by mixing blue paint and yellow paint.
The graph shows how much of each colour of paint to use.
(a)Marcus uses 8 litres of yellow paint.
State how much blue paint Marcus uses.
(b)Marcus now uses 36 litres of yellow paint.
How much green paint does he make?
Show answer
(a) \(6\ \text{litres}\)
(b) \(63\ \text{litres}\)Read up from 8 litres of yellow paint to the line, then across to the vertical axis.
blue paint = \(6\) litres
The line passes through \((20, 15)\), so blue paint : yellow paint is \(3 : 4\) in its simplest form.
36 litres of yellow paint is off the graph, so scale up: \(36 \div 4 = 9\).
blue paint = \(3 \times 9 = 27\) litres
green paint = \(27 + 36 = 63\) litres
- (b) M1blue paint for 36 litres of yellow, \(27\) litres
- (b) A1total \(63\) litres
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Either paper 2 marks
In a bag, the number of green discs : the number of blue discs = \(13 : 8\)
Each statement below is about the discs in the bag.
(a)"In total there are 21 discs."
Tick one box: true, false or cannot tell.
(b)"There are more green discs than blue discs."
Tick one box: true, false or cannot tell.
Show answer
(a) Cannot tell. This total is in the list of possible totals, so it could be right, but the ratio does not make it certain.
(b) True. The colour named has the larger part of the ratio, so it has more discs in every possible bag.The ratio \(13 : 8\) only fixes how the numbers of discs compare. Every possible bag has \(13\) green discs and \(8\) blue discs, or the same multiple of each.
So the total number of discs could be \(21\), \(42\), \(63\), and so on, and the difference between the two colours could be \(5\), \(10\), \(15\), and so on.
This total is in the list of possible totals, so it could be right, but the ratio does not make it certain.
The answer is Cannot tell.
The ratio \(13 : 8\) fixes only the comparison, so the possible totals are \(21\), \(42\), \(63\), and so on, and the possible differences are \(5\), \(10\), \(15\), and so on.
The colour named has the larger part of the ratio, so it has more discs in every possible bag.
The answer is True.
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Either paper 2 marks
There are 140 counters in a bag. There are only red counters and blue counters in the bag.
number of red counters : number of blue counters = \(1 : 4\)
Noah has to work out how many red counters are in the bag.
Noah says, "There are 28 red counters in the bag, because 1 is \(\frac{1}{5}\) of 5 and 28 is \(\frac{1}{5}\) of 140."
Is Noah correct? You must give a reason for your answer.
Show answer
Yes. There are 28 red counters (140 ÷ 5 × 1). Noah compared the red share with all the parts in the bag, which is the whole.The ratio has \(1 + 4 = 5\) parts, so one part is \(140 \div 5 = 28\) counters.
The red counters are \(1\) of these parts, so there are \(1 \times 28 = 28\) red counters.
Noah says 28, and the true number is 28, so the answer to "Is Noah correct?" is: Yes.
Reason: Noah compared the red share with all the parts in the bag, which is the whole.
- M1number of red counters, \(140 \div 5 \times 1 = 28\)
- A1Yes, 28 compared with 28, with the reason that Noah compared the red share with all the parts in the bag, which is the whole
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Either paper 3 marks
The diagram shows two angles.
The angles are in the ratio \(1 : 9\).
Show that the smaller angle is \(36^\circ\). Give the angle fact that tells you the total of the two angles.
Show answer
Angles around a point add up to 360°, so the total is \(360^\circ\). One part is \(360 \div 10 = 36^\circ\), and \(1 \times 36 = 36^\circ\), which is the smaller angleAngles around a point add up to 360°, so the two angles share \(360^\circ\).
The ratio has \(1 + 9 = 10\) equal parts, so one part is
\(360 \div 10 = 36^\circ\)
The smaller angle is 1 of these parts:
\(1 \times 36 = 36^\circ\)
So the smaller angle is \(36^\circ\), as stated.
Check: the other angle is \(9 \times 36 = 324^\circ\), and \(36 + 324 = 360\).
- B1angle fact stated: Angles around a point add up to 360°, so the total is \(360^\circ\)
- M1one part \(360 \div 10 = 36^\circ\)
- A1\(1 \times 36 = 36^\circ\), meeting the stated smaller angle
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Either paper 5 marks
Amy and Ben share 70 counters in the ratio \(3 : 7\).
(a)Calculate how many counters Ben gets.
(b)Ben then gives 11 of his counters to Amy.
Work out the new ratio of Amy's counters to Ben's counters. Give your answer in its simplest form.
Show answer
(a) \(49\)
(b) \(16 : 19\)The ratio has \(3 + 7 = 10\) parts altogether.
One part is \(70 \div 10 = 7\) counters.
Ben gets 7 parts: \(7 \times 7 = 49\)
Amy started with 3 parts, which is \(3 \times 7 = 21\) counters, so Amy now has \(21 + 11 = 32\) counters.
Ben now has \(49 - 11 = 38\) counters.
The new ratio is \(32 : 38\). Dividing both parts by 2 gives
\(32 : 38 = 16 : 19\)
- (a) M1\(10\) parts altogether
- (a) M1one part \(7\)
- (a) A1\(49\) counters
- (b) M1new amounts \(32\) and \(38\)
- (b) A1simplest form \(16 : 19\)
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Either paper 4 marks
Callum and Aisha share some money in the ratio \(5 : 7\).
Callum gets £60.
(a)Work out how much money Aisha gets.
(b)Work out how much money they shared altogether.
Show answer
(a) \(\text{£}84\)
(b) \(\text{£}144\)Callum's £60 is 5 parts of the ratio, so one part is
\(£60 \div 5 = £12\)
Aisha gets 7 parts: \(7 \times £12 = £84\)
Altogether the ratio has \(5 + 7 = 12\) parts, and each part is worth £12:
\(12 \times £12 = £144\)
Check: \(£60 + £84 = £144\)
- (a) M1one part \(£60 \div 5 = £12\)
- (a) A1£84
- (b) M1\(5 + 7 = 12\) parts altogether
- (b) A1£144
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Either paper 5 marks
Grace, Harry and Isla share £340 in the ratio \(4 : 6 : 7\).
(a)Calculate how much money Isla gets.
(b)Calculate how much more money Isla gets than Grace.
Show answer
(a) \(\text{£}140\)
(b) \(\text{£}60\)The ratio has \(4 + 6 + 7 = 17\) parts altogether.
One part is \(£340 \div 17 = £20\)
Isla gets 7 parts: \(7 \times £20 = £140\)
Grace gets 4 parts: \(4 \times £20 = £80\)
\(£140 - £80 = £60\)
This is the same as the \(7 - 4 = 3\) extra parts Isla has: \(3 \times £20 = £60\)
- (a) M1\(17\) parts altogether
- (a) M1one part £\(20\)
- (a) A1£\(140\)
- (b) M1Grace gets £\(80\)
- (b) A1difference £\(60\)
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Either paper 4 marks
The three angles of a triangle are in the ratio \(9 : 17 : 19\).
(a)Work out the size of each of the three angles.
(b)Write down whether the triangle is acute-angled, right-angled or obtuse-angled.
Show answer
(a) \(36^\circ\), \(68^\circ\) and \(76^\circ\)
(b) acute-angledThe angles of a triangle add up to \(180^\circ\), and the ratio splits that into \(9 + 17 + 19 = 45\) parts.
One part is \(180 \div 45 = 4\) degrees.
\(9 \times 4 = 36\), \(17 \times 4 = 68\), \(19 \times 4 = 76\)
Check: \(36 + 68 + 76 = 180\)
A triangle is named by its largest angle: below \(90^\circ\) it is acute-angled, exactly \(90^\circ\) it is right-angled, and above \(90^\circ\) it is obtuse-angled.
The largest angle is \(76^\circ\), which is less than \(90^\circ\), so the triangle is acute-angled.
- (a) M1total parts \(9 + 17 + 19 = 45\)
- (a) M1one part \(180 \div 45 = 4\) degrees
- (a) A1angles \(36^\circ\), \(68^\circ\) and \(76^\circ\)
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Either paper 2 marks
Sam and Priya throw darts at a target.
Sam's ratio of hits to misses is \(5 : 2\)
Priya's ratio of hits to misses is \(6 : 1\)
Priya says that because 6 is bigger than 5, she must have more hits than Sam.
Give an example to show that this might not be true.
Show answer
e.g. Priya: 6 hits, 1 miss (\(6 : 1\)). Sam: 30 hits, 12 misses (\(5 : 2\) scaled up). Sam has more hits than Priya, even though 5 is smaller than 6.A ratio only fixes the split between hits and misses, not the actual totals — any multiple of \(6 : 1\) is a valid result for Priya, and any multiple of \(5 : 2\) is valid for Sam.
Give Priya the smallest possible totals: \(6\) hits, \(1\) miss.
Scale Sam's ratio up instead: \(5 \times 6 : 2 \times 6 = 30 : 12\)
Sam has \(30\) hits, which is more than Priya's \(6\) hits — so the claim can be false.
- M1a valid pair of totals for Sam in the ratio \(5 : 2\), scaled up to \(30 : 12\)
- A1comparing 30 hits for Sam with 6 hits for Priya, so the claim can be false