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7 Here is a rule to work out the cost of buying a number of floorboards.
Cost = £55 \(\times\) number of floorboards
Toby buys a number of floorboards.
The cost is £330
Work out the number of floorboards Toby buys. (2)
| Answer | Mark | Mark scheme |
|---|---|---|
| 6 | M1 | for \(330 \div 55\) |
| A1 | cao |
Build up methods are allowed
Condone one error in build up method
\(55 \times 6 = 330\), 330 or 55 on answer line scores M1A0
ignore use of £ eg £6
16
Assume the job needs completing in 20 days.
How many workers are needed? [3 marks]
What does this mean about the number of workers that are needed?
Tick one box. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(20 = \dfrac{800}{w}\) or \((w =)\ \dfrac{800}{d}\) | M1 | oe |
| \(800 \div 20\) | M1dep | oe |
| 40 | A1 |
| Answer | Mark | Comments |
|---|---|---|
| It is greater than the answer to part (a) | B1 |
10 Work out the value of \(\quad 2(a^2 + 3a) \quad\) when \(\quad a = 4\) [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: substitutes first | ||
| \(4^2\) or 16 or 32 or \(3 \times 4\) or 12 or 24 | M1 | oe implied by 28 |
| \(2 \times (4 \times 4 + 3 \times 4)\) | M1dep | oe |
| 56 | A1 | |
| Alternative method 2: multiplies out first | ||
| \(2 \times a^2\) or \(2 \times 3a\) | M1 | oe |
| \(2 \times 4 \times 4 + 2 \times 3 \times 4\) | M1dep | oe |
| 56 | A1 | |
| Alternative method 3: factorises first | ||
| \(2a(a + 3)\) or \(2 \times a\) or \(a + 3\) | M1 | oe |
| \(2 \times 4 \times (4 + 3)\) | M1dep | oe |
| 56 | A1 | |
| Inclusion of \(a\) can score a maximum of one mark eg1 \(4^2 = 16 \quad 16 \times 2 = 32a\) eg2 \(56a\) | M1 only M1 only |
| \(2 \times 4^2 + 2 \times 3a\) is not enough for the second mark of Alt 2 | M1M0A0 |
23 The total cost of a taxi ride is calculated by adding
a fixed charge of £4
and
a charge of £2 per mile.
Write a formula to work out the total cost, £\(C\), of a journey of \(m\) miles. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(2m + 4\) or \(2 \times m + 4\) | B2 | oe B1 \(2m\) or \(m2\) or \(200m + 400\) or \(km + 4\) with k an integer, \(k \ne 0\) or 2 |
| Ignore any attempt to substitute a value for \(m\) after \(2m\) seen for B1 | |
| Condone \(m2 + 4\) (not testing simplification) | B2 |
| Accept use of another letter for \(m\) other than \(c\) eg1 \(2x + 4\) eg2 \(2c + 4\) | B2 B0 |
| Condone inclusion of £ sign for B1 eg1 £\(2m + 4\) eg2 £\(2m\) | B1 B1 |
12
| Answer | Mark | Comments |
|---|---|---|
| \((-4)^2 + 7 \times -4\) or \(-4(-4 + 7)\) or 16 or \(-28\) | M1 | oe eg \((-4)^2 + 7(-4)\) |
| \(-12\) | A1 | SC1 \(-44\) |
| SC is for \(-4^2 + 7 \times -4 = -16 - 28 = -44\) | |
| Embedded 16 or \(-28\) seen eg \(16 + 7x\) without correct answer | M1A0 |
| Values may be implied eg1 \((-4)^2 + 7 \times 4 = 44\) 16 is implied eg2 only answer 44 | M1A0 M0A0 |
| Further correct work eg \(16 - 28 = -12\), Answer \(3x\) | M1A1 |
| Further incorrect work eg \(16 - 28 = -12\), Answer \(-12x\) | M1A0 |
| \(+-28\) is the same as \(-28\) | M1 |
| Only \(-4^2 + 7 \times -4\) | M0A0 |
| \(-4^2 + 7 \times -4 = -16 + 28 = 12\) | M0A0 |
| \(16x\) does not imply 16 | M0A0 |
| Answer | Mark | Comments |
|---|---|---|
| \(y + 1\) or \(1 + y\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(4a + 8\) or \(8 + 4a\) or \(5a\) | M1 | |
| \(5a + 8\) or \(8 + 5a\) | A1 |
| Further incorrect work or simplification eg \(5a + 8\), Answer \(13a\) | M1A0 |
11 A window cleaner uses this formula.
| \(C = 2W + 5\) |
\(C\) = cost, in £, for the customer
\(W\) = number of windows to be cleaned
Show why this cost must be incorrect. [1 mark]
| Answer | Mark | Comments |
|---|---|---|
| \(2 \times 6\) or 12 or 1700 | M1 | oe |
| 17(.00) | A1 | accept 1700p |
| Answer | Mark | Comments |
|---|---|---|
| 9.5 or \(9\frac{1}{2}\) or valid reason | B1 | eg shows that 9 windows cost £23 and 10 windows cost £25 or you can’t have half a window |
| Calculated values must be correct | |
| (19, 21,) 23, 25 (, 27…) | B1 |
| Even number \(+\) 5 can’t be even / can’t end in a 4 | B1 |
| There’s no whole number you can times by 2 to get to 19 There’s no number you can times by 2 to get to 19 | B1 B0 |
| Even \(+\) 5 \(=\) odd, 24 is even Even \(+\) 5 \(=\) odd | B1 B0 |
| Costs are all always odd numbers, 24 is even Costs are all always odd numbers | B1 B0 |
| 19 is a prime number and so doesn’t divide exactly by 2 19 is a prime number | B1 B0 |
5
Work out the value of \(d\) when \(\quad g = 15 \quad\) and \(\quad h = 63\) [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(15^2 - 2 \times 63\) or 225 or 126 | M1 | \(-126\) implies M1 |
| 99 | A1 |
| Answer | Mark | Comments |
|---|---|---|
| \(m - k\) or \(-k + m\) | B1 |
19 The value of \(x\) can be 2 or 5
The value of \(y\) can be 3 or 12
You must show your working. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| 6, 15, 24, 60 in any order | B2 | B1 for 6, 15, 24, 60 with no more than one additional value or three correct values with no more than one incorrect value |
| Ignore repeated values for B2 and B1 | |
| 6, 10, 15, 24, 60 | B1 |
| 6, 10, 15, 24 | B1 |
| 6, 10, 15, 24, 36 | B0 |
| \(2 \times 3,\ 5 \times 3,\ 2 \times 12,\ 5 \times 12\) | B0 |
| \(6xy,\ 15xy,\ 24xy,\ 60xy\) | B0 |
| Answer | Mark | Comments |
|---|---|---|
| \(\dfrac{2 - 12}{2}\) or one correctly evaluated trial with correct substitutions for \(x = 2\) or 5 and \(y = 3\) or 12 or two correct values from \(-\dfrac{10}{2},\ -\dfrac{1}{2},\ -\dfrac{7}{5},\ \dfrac{2}{5}\) oe or two correct values from \(-5,\ -0.5,\ -1.4,\ 0.4\) oe | M1 | \(\dfrac{2 - 3}{2} = -\dfrac{1}{2}\) oe or \(\dfrac{5 - 12}{5} = -\dfrac{7}{5}\) oe or \(\dfrac{5 - 3}{5} = \dfrac{2}{5}\) oe |
| \(-\dfrac{10}{2}\) or \(-5\) | A1 |
| Two separate correct values can be in either fraction or decimal form | |
| \(2 - 12 \div 2 = -5\) (recovered) | M1A1 |
| \(2 - 12 \div 2\) | M0A0 |
| An example of an incorrect substitution with different values of \(x\) eg \(\dfrac{5 - 12}{2} = -\dfrac{7}{2}\) |
12 A football team has \(P\) points.
\[P = 3W + D\]\(W\) is the number of wins
\(D\) is the number of draws
How many points does the team have? [1 mark]
11 games were draws.
How many games has this team lost? [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| 20 | B1 | allow \(P = 20\) |
| Answer | Mark | Comments |
|---|---|---|
| \(53 - 11\) or 42 or \(33 \times 3\) or 99 or \(11 \times 2\) or \(33 - 11\) or 22 | M1 | |
| their 42 \(\div\) 3 or 14 or their 99 \(-\) 53 \(-\) their 22 or (their 22 \(\times\) 3) \(-\) their 42 or 24 | M1dep | oe eg build up - allow one error |
| \(33 - 11 -\) their 14 or their 24 \(\div\) 3 | M1dep | dep on M1M1 |
| 8 | A1 |
| \(3 \times 14 + 11 = 53\) | M2 |
7 Here is a number machine.
| Answer | Mark | Comments |
|---|---|---|
| 10 | B1 |
| Answer | Mark | Comments |
|---|---|---|
| –14 | B1 |
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