MetadataPastPaper.sampleTitle

To convert \(\frac{3}{8}\) to a decimal, divide 3 by 8. \(3 \div 8 = 0.375\). Alternatively, since \(\frac{1}{8} = 0.125\), then \(\frac{3}{8} = 3 \times 0.125 = 0.375\).

B1 for 0.375

Using the order of operations (BIDMAS): First evaluate the index/power: \(2^2 = 4\). Then perform the multiplication: \(3 \times 4 = 12\). Finally, perform the addition: \(5 + 12 = 17\).

B1 for 17

Collect the like terms: \(4x - x = 3x\) and \(3y + 2y = 5y\). Combining them gives \(3x + 5y\).

B1 for 3x + 5y

To simplify the ratio \(18 : 30\), find the highest common factor of 18 and 30, which is 6. Divide both sides of the ratio by 6: \(18 \div 6 = 3\) and \(30 \div 6 = 5\). This gives \(3 : 5\).

B1 for 3 : 5

The sum of the probabilities of an event happening and not happening is 1. Probability of not raining = \(1 - 0.15 = 0.85\).

B1 for 0.85

Add 4 to both sides of the equation: \(3x = 11 + 4\), so \(3x = 15\). Divide both sides by 3: \(x = \frac{15}{3} = 5\).

B1 for x = 5

A triangular prism has 2 triangular faces and 3 rectangular faces. Each of the 2 triangular faces has 3 vertices. Total number of vertices = \(2 \times 3 = 6\).

B1 for 6

There are 1000 grams in 1 kilogram. To convert 2.5 kilograms to grams, multiply by 1000: \(2.5 \times 1000 = 2500\text{g}\).

B1 for 2500g

Using BIDMAS, indices are calculated first: \(2^2 = 4\). Next, perform the multiplication: \(3 \times 4 = 12\). Finally, perform the addition: \(5 + 12 = 17\).

B1: for 17

First, convert 1.5 hours to minutes: \(1.5 \times 60 = 90\text{ minutes}\). The ratio is \(45 : 90\). Dividing both sides by 45 gives \(1 : 2\).

B1: for 1 : 2

\(0.375 = \frac{375}{1000}\). Dividing both the numerator and the denominator by 125 simplifies the fraction to \(\frac{3}{8}\).

B1: for \(\frac{3}{8}\)

The prime numbers on a six-sided dice are 2, 3, and 5. There are 3 prime numbers out of 6 possible outcomes. The probability is \(\frac{3}{6} = \frac{1}{2}\).

B1: for \(\frac{1}{2}\)

Collect like terms: \(3a + 2a = 5a\) and \(-4b - b = -5b\). Combining these gives \(5a - 5b\).

B1: for \(5a - 5b\)

Add 7 to both sides of the equation: \(4x = 20\). Divide both sides by 4: \(x = 5\).

B1: for \(x = 5\)

The first significant figure is 4 and the second is 6. The next digit is 8, which is 5 or more, so we round up the 6 to 7. This gives \(0.047\).

B1: for \(0.047\)

A triangular prism has 2 triangular bases (each with 3 edges) and 3 rectangular faces. The total number of edges is \(3 + 3 + 3 = 9\).

B1: for 9

First, find 10% of 240: \(240 \div 10 = 24\). Next, find 30% by multiplying by 3: \(24 \times 3 = 72\). Find 5% by halving 10%: \(24 \div 2 = 12\). Add 30% and 5% to find 35%: \(72 + 12 = 84\).

M1 for showing a method to find 10% and/or 5% of 240, or writing \(\frac{35}{100} \times 240\). A1 for 84.

Group the \(x\) terms together: \(6x - 2x = 4x\). Group the \(y\) terms together: \(8y - 3y = 5y\). Combine these to get \(4x + 5y\).

M1 for grouping like terms, showing either \(4x\) or \(+5y\) or writing \(6x - 2x\) and \(8y - 3y\) separately. A1 for \(4x + 5y\) (or equivalent such as \(5y + 4x\)).

Add 9 to both sides of the equation: \(4y = 15 + 9\) which simplifies to \(4y = 24\). Divide both sides by 4: \(y = 24 \div 4 = 6\).

M1 for adding 9 to both sides to get \(4y = 24\) (or showing \(4y = 15 + 9\)), or dividing by 4 as a correct first step. A1 for 6 or \(y = 6\).

First multiply 138 by 6: \(100 \times 6 = 600\), \(30 \times 6 = 180\), \(8 \times 6 = 48\). Add these together: \(600 + 180 + 48 = 828\). Since 13.8 has one decimal place, the final answer must have one decimal place: \(82.8\).

M1 for showing a complete multiplication method, such as \(138 \times 6 = 828\), or listing \(60 + 18 + 4.8\). A1 for 82.8.

The sum of all probabilities must equal 1. Sum of red and blue probabilities is \(0.45 + 0.3 = 0.75\). The probability of green is \(1 - 0.75 = 0.25\).

M1 for adding the two given probabilities: \(0.45 + 0.3\) or showing \(1 - (0.45 + 0.3)\). A1 for 0.25 (or equivalent fraction or percentage).

First find the total number of parts: \(3 + 5 = 8\). Find the value of one part: \(£72 \div 8 = £9\). Multiply the value of one part by the larger ratio share: \(5 \times £9 = £45\).

M1 for finding the value of one share: \(72 \div 8 = 9\), or showing either \(£27\) or \(£45\) as a part of the calculation. A1 for £45 (accept 45).

Use the formula for the area of a triangle: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\). Substituting the given values: \(\text{Area} = \frac{1}{2} \times 14 \times 9 = 7 \times 9 = 63\text{ cm}^2\).

M1 for substituting correctly into the triangle area formula: \(\frac{1}{2} \times 14 \times 9\) or \(14 \times 9 \div 2\) or showing \(126\). A1 for 63 (ignore units).

First, find \( 15\% \) of \( £80 \). Since \( 10\% = £8 \) and \( 5\% = £4 \), then \( 15\% = £8 + £4 = £12 \). Subtract this from the original price to find the sale price: \( £80 - £12 = £68 \). Finally, apply the loyalty card discount: \( £68 - £5 = £63 \).

M1: Calculates \( 15\% \) of \( £80 \) to get \( £12 \) (or shows \( 80 \times 0.15 \))

M1: Subtracts their \( £12 \) discount from \( £80 \) and then subtracts \( £5 \)

A1: 63 (or \( £63 \))

Find the unit cost of a pen: \( £4.80 \div 12 = £0.40 \). Find the cost of 3 pens: \( 3 \times £0.40 = £1.20 \). Find the unit cost of a pencil: \( £1.50 \div 5 = £0.30 \). Find the cost of 4 pencils: \( 4 \times £0.30 = £1.20 \). Add the two costs together: \( £1.20 + £1.20 = £2.40 \).

M1: Finds the unit cost of a pen (\( £0.40 \)) OR a pencil (\( £0.30 \))

M1: Finds the total cost of 3 pens (\( £1.20 \)) AND 4 pencils (\( £1.20 \)) and adds them

A1: 2.40 (accept \( £2.40 \) or 240p)

First, expand the brackets on the left side: \( 4x - 12 = 2x + 8 \). Next, subtract \( 2x \) from both sides: \( 2x - 12 = 8 \). Then, add 12 to both sides: \( 2x = 20 \). Finally, divide by 2: \( x = 10 \).

M1: Expands bracket correctly to get \( 4x - 12 \)

M1: Collects \( x \) terms on one side and constant terms on the other, e.g. \( 2x = 20 \) or \( 4x - 2x = 8 + 12 \)

A1: 10

The difference in ratio parts between girls and boys is \( 5 - 4 = 1 \) part. Since there are 120 more girls than boys, \( 1 \) part represents 120 students. The total number of parts is \( 4 + 5 = 9 \) parts. The total number of students is \( 9 \times 120 = 1080 \).

M1: Finds the difference in ratio parts is 1 part, or sets up an algebraic equation such as \( 5x - 4x = 120 \)

M1: Multiplies their value for 1 part (120) by 9, or calculates boys as 480 and girls as 600 and adds

A1: 1080

First, find the area of the rectangular garden: \( 12 \times 8 = 96 \, \text{m}^2 \). Next, find the area of the square patio: \( 4 \times 4 = 16 \, \text{m}^2 \). Subtract the area of the patio from the total area of the garden: \( 96 - 16 = 80 \, \text{m}^2 \).

M1: Method to find the total area of the garden: \( 12 \times 8 \) (or 96)

M1: Method to find the area of the square patio: \( 4 \times 4 \) (or 16) and subtracts from their garden area

A1: 80

The total probability of all counters is 1. The probability of choosing a green counter is \( 1 - (0.25 + 0.4) = 1 - 0.65 = 0.35 \). Since 14 counters represents \( 0.35 \) of the total, let \( T \) be the total number of counters: \( 0.35 \times T = 14 \). Therefore, \( T = 14 \div 0.35 = 40 \).

M1: Calculates the probability of a green counter: \( 1 - 0.25 - 0.4 = 0.35 \)

M1: Sets up the calculation \( 14 \div 0.35 \) or equivalent equation to find the total

A1: 40

If the mean of four numbers is 8, then their total sum must be \( 4 \times 8 = 32 \). The sum of the three given numbers is \( 5 + 9 + 11 = 25 \). The fourth number is the difference between the total sum and this sum: \( 32 - 25 = 7 \).

M1: Calculates the total sum of the four numbers: \( 4 \times 8 = 32 \)

M1: Sums the three given numbers (25) and subtracts from their total sum

A1: 7

To find the position \( n \) where both sequences have the same term, equate the two expressions: \( 3n + 2 = 50 - n \). Add \( n \) to both sides: \( 4n + 2 = 50 \). Subtract 2 from both sides: \( 4n = 48 \). Divide by 4: \( n = 12 \). Now substitute \( n = 12 \) back into either expression to find the term: \( 3(12) + 2 = 38 \) (or \( 50 - 12 = 38 \)).

M1: Equates the two algebraic expressions: \( 3n + 2 = 50 - n \)

M1: Solves to find \( n = 12 \) and attempts to substitute this into either expression

A1: 38 (must be the term value, not just \( n = 12 \))

First, find the number of coffees sold: \(\frac{3}{5}\) of \(80 = 80 \div 5 \times 3 = 16 \times 3 = 48\) coffees. Next, find the number of teas sold: \(80 - 48 = 32\) teas. Calculate the income from coffee: \(48 \times £2.50 = £120\). Calculate the income from tea: \(32 \times £1.80 = £57.60\). Finally, find the total income: \(£120 + £57.60 = £177.60\).

M1: For finding the number of coffees sold (48) or teas sold (32). M1: For a method to calculate the income of either drink (e.g. \(48 \times 2.5\) or \(32 \times 1.8\)). M1: For finding both coffee income (£120) and tea income (£57.60). A1: For the correct total of £177.60 (accept 177.6 or 177.60).

First, calculate the total area of the garden: \(12 \times 8 = 96\text{ m}^2\). Next, calculate the area of the patio: \(5 \times 4 = 20\text{ m}^2\). Work out the area covered with grass: \(96 - 20 = 76\text{ m}^2\). Determine how many boxes of seed are needed: \(76 \div 15 = 5\) remainder \(1\). Since you cannot buy a fraction of a box, you must buy 6 boxes of grass seed. Finally, calculate the total cost: \(6 \times £6.50 = £39\).

M1: For finding the area of the grass, \(96 - 20 = 76\text{ m}^2\) (or showing \(12 \times 8\) and \(5 \times 4\)). M1: For dividing their grass area by 15 (e.g. \(76 \div 15\)). A1: For correctly identifying that 6 boxes are needed (rounding up 5.06... to 6). A1: For the correct total cost of £39 (accept £39.00 or 39).

To make 40 flapjacks instead of 8, the ingredients need to be scaled up.

**Step 1: Find the scale factor**
\(40 \div 8 = 5\)

**Step 2: Calculate the required amount of each ingredient**
* Oats: \(120\text{ g} \times 5 = 600\text{ g}\)
* Butter: \(60\text{ g} \times 5 = 300\text{ g}\)
* Golden syrup: \(50\text{ g} \times 5 = 250\text{ g}\)

**Step 3: Determine the number of packs needed and their cost**
* Oats: Needs \(600\text{ g}\). Sold in \(500\text{ g}\) bags.
Sam must buy \(2\) bags.
Cost: \(2 \times \pounds 1.20 = \pounds 2.40\)

* Butter: Needs \(300\text{ g}\). Sold in \(250\text{ g}\) blocks.
Sam must buy \(2\) blocks.
Cost: \(2 \times \pounds 1.50 = \pounds 3.00\)

* Golden syrup: Needs \(250\text{ g}\). Sold in \(200\text{ g}\) tins.
Sam must buy \(2\) tins.
Cost: \(2 \times \pounds 1.10 = \pounds 2.20\)

**Step 4: Calculate total cost**
\(\text{Total Cost} = \pounds 2.40 + \pounds 3.00 + \pounds 2.20 = \pounds 7.60\)

**Step 5: Calculate total revenue**
Sam sells 40 flapjacks at \(30\text{ p}\) each.
\(\text{Total Revenue} = 40 \times 30\text{ p} = 1200\text{ p} = \pounds 12.00\)

**Step 6: Calculate profit**
\(\text{Profit} = \text{Total Revenue} - \text{Total Cost}\)
\(\text{Profit} = \pounds 12.00 - \pounds 7.60 = \pounds 4.40\)

* **M1**: Correct method to scale recipe by factor of 5 (e.g. shows at least one of \(600\text{ g}\) of oats, \(300\text{ g}\) of butter, or \(250\text{ g}\) of syrup).
* **M1**: Correctly identifies the number of packs needed for all 3 ingredients (2 bags of oats, 2 blocks of butter, and 2 tins of syrup).
* **M1**: Correctly calculates the total cost of ingredients: \(2 \times 1.20 + 2 \times 1.50 + 2 \times 1.10 = \pounds 7.60\) (or equivalent in pence).
* **M1**: Correctly calculates total revenue: \(40 \times 0.30 = \pounds 12.00\) (or equivalent in pence).
* **A1**: Correct final profit of \(\pounds 4.40\) (or \(440\text{p}\) with correct units shown).

To make 40 flapjacks instead of 8, the ingredients need to be scaled up.

**Step 1: Find the scale factor**
\(40 \div 8 = 5\)

**Step 2: Calculate the required amount of each ingredient**
* Oats: \(120\text{ g} \times 5 = 600\text{ g}\)
* Butter: \(60\text{ g} \times 5 = 300\text{ g}\)
* Golden syrup: \(50\text{ g} \times 5 = 250\text{ g}\)

**Step 3: Determine the number of packs needed and their cost**
* Oats: Needs \(600\text{ g}\). Sold in \(500\text{ g}\) bags.
Sam must buy \(2\) bags.
Cost: \(2 \times \pounds 1.20 = \pounds 2.40\)

* Butter: Needs \(300\text{ g}\). Sold in \(250\text{ g}\) blocks.
Sam must buy \(2\) blocks.
Cost: \(2 \times \pounds 1.50 = \pounds 3.00\)

* Golden syrup: Needs \(250\text{ g}\). Sold in \(200\text{ g}\) tins.
Sam must buy \(2\) tins.
Cost: \(2 \times \pounds 1.10 = \pounds 2.20\)

**Step 4: Calculate total cost**
\(\text{Total Cost} = \pounds 2.40 + \pounds 3.00 + \pounds 2.20 = \pounds 7.60\)

**Step 5: Calculate total revenue**
Sam sells 40 flapjacks at \(30\text{ p}\) each.
\(\text{Total Revenue} = 40 \times 30\text{ p} = 1200\text{ p} = \pounds 12.00\)

**Step 6: Calculate profit**
\(\text{Profit} = \text{Total Revenue} - \text{Total Cost}\)
\(\text{Profit} = \pounds 12.00 - \pounds 7.60 = \pounds 4.40\)

* **M1**: Correct method to scale recipe by factor of 5 (e.g. shows at least one of \(600\text{ g}\) of oats, \(300\text{ g}\) of butter, or \(250\text{ g}\) of syrup).
* **M1**: Correctly identifies the number of packs needed for all 3 ingredients (2 bags of oats, 2 blocks of butter, and 2 tins of syrup).
* **M1**: Correctly calculates the total cost of ingredients: \(2 \times 1.20 + 2 \times 1.50 + 2 \times 1.10 = \pounds 7.60\) (or equivalent in pence).
* **M1**: Correctly calculates total revenue: \(40 \times 0.30 = \pounds 12.00\) (or equivalent in pence).
* **A1**: Correct final profit of \(\pounds 4.40\) (or \(440\text{p}\) with correct units shown).

To find a multiple of 14, we can multiply 14 by integers: \(14 \times 1 = 14\), \(14 \times 2 = 28\), \(14 \times 3 = 42\). Therefore, 42 is a multiple of 14.

B1 for 42 (C)

Collect like terms: \(3x - x = 2x\) and \(4y + 2y = 6y\). Combining these gives \(2x + 6y\).

B1 for \(2x + 6y\) (A)

There are 1000 metres in 1 kilometre. Multiply 4.5 by 1000: \(4.5 \times 1000 = 4500\text{ m}\).

B1 for 4,500 m (C)

To convert a fraction to a percentage, multiply by 100: \(\frac{7}{20} \times 100 = 7 \times 5 = 35\%\).

B1 for 35% (B)

The numbers greater than 4 on a standard dice are 5 and 6. This gives 2 favourable outcomes out of 6 possible outcomes. The probability is \(\frac{2}{6} = \frac{1}{3}\).

B1 for \(\frac{1}{3}\) (B)

Divide both parts of the ratio by their highest common factor, which is 6: \(12 \div 6 = 2\) and \(30 \div 6 = 5\). The simplest form is \(2 : 5\).

B1 for \(2 : 5\) (C)

A triangular prism has 2 triangular faces connected by 3 rectangular faces. Each triangular face has 3 vertices, giving a total of \(2 \times 3 = 6\) vertices.

B1 for 6 (B)

Add 3 to both sides to get \(4x = 20\). Divide by 4 to get \(x = 5\).

B1 for \(x = 5\) (B)

To simplify the expression, group the like terms together: \(5a - 2a = 3a\) and \(-3b + 7b = 4b\). Combining these terms gives \(3a + 4b\).

B1 for correct answer of \(3a + 4b\) (Option a)

The probability of an event not happening is calculated by subtracting the probability of the event happening from 1. Thus, \(1 - 0.14 = 0.86\).

B1 for correct answer of \(0.86\) (Option c)

The probability of the die not landing on a 6 is \(1 - 0.35 = 0.65\).

To find the expected number of times, multiply this probability by the total number of rolls:
\(120 \times 0.65 = 78\).

Alternatively, find the expected number of times it lands on a 6:
\(120 \times 0.35 = 42\).

Subtract this from the total rolls:
\(120 - 42 = 78\).

M1 for \(120 \times (1 - 0.35)\) or \(120 \times 0.65\) or \(120 - (120 \times 0.35)\) or 42 seen
A1 for 78

First, calculate the multiplication inside the square root:
\(4.2 \times 3.5 = 14.7\)

Next, add this to 15.6:
\(15.6 + 14.7 = 30.3\)

Then, find the square root:
\(\sqrt{30.3} = 5.504543...\)

Rounding to 1 decimal place gives 5.5.

M1 for finding 30.3 or showing \(5.504...\)
A1 for 5.5

Multiply both sides of the equation by 2:
\(3x + 5 = 11.5 \times 2\)
\(3x + 5 = 23\)

Subtract 5 from both sides:
\(3x = 18\)

Divide both sides by 3:
\(x = 6\)

M1 for a correct first step to clear the fraction (e.g. \(3x + 5 = 23\)) or for dividing terms by 2 (e.g. \(1.5x + 2.5 = 11.5\))
A1 for 6

First, calculate 15% of £84:
\(0.15 \times 84 = 12.60\)

Subtract the reduction from the original price:
\(84 - 12.60 = 71.40\)

Alternatively, calculate 85% of £84:
\(84 \times 0.85 = 71.40\)

M1 for a correct method to find 15% of 84 (e.g. \(12.6\)) or to find 85% of 84 (e.g. \(84 \times 0.85\))
A1 for 71.40 (accept 71.4)

First, find the total number of parts:
\(3 + 5 = 8\) parts.

Next, find the value of one part:
\(144 \div 8 = 18\).

The difference in parts between the larger and smaller shares is:
\(5 - 3 = 2\) parts.

Now, multiply the difference in parts by the value of one part:
\(2 \times 18 = 36\).

Alternatively:
Smaller share: \(3 \times 18 = 54\)
Larger share: \(5 \times 18 = 90\)
Difference: \(90 - 54 = 36\).

M1 for a correct method to find the value of one share, e.g. \(144 \div 8\) (or 18), or for finding both shares (54 and 90)
A1 for 36

First, find the radius of the semi-circle:
\(r = 12 \div 2 = 6\text{ cm}\)

The area of a full circle is \(\pi r^2\):
\(\text{Area} = \pi \times 6^2 = 36\pi \approx 113.097\text{ cm}^2\)

The area of the semi-circle is half of the full circle's area:
\(\text{Area of semi-circle} = 36\pi \div 2 = 18\pi \approx 56.548...\text{ cm}^2\)

Rounding to 1 decimal place gives 56.5.

M1 for \(\pi \times 6^2 \div 2\) or \(18\pi\) or showing the full area of the circle as \(\approx 113.1\) (or using \(r = 6\))
A1 for 56.5

First, find the total sum of the four numbers:
\(4 \times 8.5 = 34\)

Next, find the sum of the three known numbers:
\(6 + 11 + 7 = 24\)

Subtract the sum of the known numbers from the total sum to find the fourth number:
\(34 - 24 = 10\).

M1 for \(4 \times 8.5\) (or 34) or for \(6 + 11 + 7 + x = 4 \times 8.5\)
A1 for 10

Find the number of packs sold:
\(30 \div 5 = 6\) packs.

Calculate the total selling price:
\(6 \times £2.80 = £16.80\).

Calculate the profit made:
\(£16.80 - £12.00 = £4.80\).

Calculate the percentage profit:
\(\frac{4.80}{12.00} \times 100 = 40\%\).

M1: For finding the total selling price, \(6 \times 2.80\) or \(16.80\)
M1: For a complete method to find the percentage profit, e.g., \(\frac{16.80 - 12}{12} \times 100\) or \(\frac{4.80}{12} \times 100\)
A1: \(40\%\) (accept \(40\))

Calculate the area of the rectangle:
\(\text{Area} = 6 \times 1.5 = 9\text{ m}^2\).

Calculate the area of the triangle:
\(\text{Area} = \frac{1}{2} \times 1.5 \times 2 = 1.5\text{ m}^2\).

Calculate the total area:
\(9 + 1.5 = 10.5\text{ m}^2\).

Calculate the area of one tile in square metres:
\(50\text{ cm} = 0.5\text{ m}\)
\(\text{Tile Area} = 0.5 \times 0.5 = 0.25\text{ m}^2\).

Calculate the total number of tiles needed:
\(10.5 \div 0.25 = 42\).

M1: For finding the total area of the patio, \(10.5\text{ m}^2\)
M1: For dividing their total area by the area of one tile (e.g. \(10.5 \div 0.25\) or \(105,000 \div 2500\))
A1: \(42\)

Find the difference in ratio parts between Chris and Alan:
\(8 - 3 = 5\) parts.

Set up the equation to find the value of 1 part:
\(5\text{ parts} = £120\)
\(1\text{ part} = 120 \div 5 = £24\).

Calculate the total number of parts:
\(3 + 5 + 8 = 16\) parts.

Calculate the total money shared:
\(16 \times £24 = £384\).

M1: For finding that \(5\) parts is equal to \(£120\) or writing \(8x - 3x = 120\)
M1: For finding the value of one part as \(£24\) or for multiplying their part value by \(16\)
A1: \(£384\) (accept \(384\))

Set up an expression for the perimeter of the rectangle:
\(\text{Perimeter} = 2 \times (\text{width} + \text{length})\)
\(\text{Perimeter} = 2 \times (x + 2x + 5) = 2(3x + 5) = 6x + 10\).

Set the expression equal to the given perimeter and solve for \(x\):
\(6x + 10 = 46\)
\(6x = 36\)
\(x = 6\).

M1: For setting up a correct algebraic equation for the perimeter, e.g., \(2(x + 2x + 5) = 46\) or \(6x + 10 = 46\)
M1: For correctly simplifying to find \(6x = 36\) or showing a correct method to solve their linear equation
A1: \(6\)

First, find the sum of the remaining probabilities for C and D:
\(1 - (0.2 + 0.35) = 1 - 0.55 = 0.45\).

Since the probability of landing on C is twice that of D, let the probability of D be \(p\) and C be \(2p\):
\(2p + p = 0.45\)
\(3p = 0.45 \implies p = 0.15\).
So, \(P(C) = 2 \times 0.15 = 0.30\).

Now, calculate the expected frequency of C in 250 spins:
\(\text{Expected frequency} = 250 \times 0.30 = 75\).

M1: For subtracting the given probabilities from 1, i.e., \(1 - 0.55 = 0.45\)
M1: For finding the probability of C is \(0.30\) (or \(30\%\))
A1: \(75\)

First, calculate the total number of goals scored across all matches:
\((0 \times 4) + (1 \times 6) + (2 \times 7) + (3 \times 3)\)
\(= 0 + 6 + 14 + 9 = 29\text{ goals}\).

The total number of matches is given as 20 (or calculated as \(4 + 6 + 7 + 3 = 20\)).

Calculate the mean:
\(\text{Mean} = \frac{29}{20} = 1.45\).

M1: For attempting to find the total number of goals (at least 3 correct products summed)
M1: For dividing their total goals by the sum of the frequencies (which should be 20)
A1: \(1.45\)

Substitute the given values into the formula:
\(v^2 = 5^2 + 2 \times 4.5 \times 16\).

Evaluate each part:
\(v^2 = 25 + 144\)
\(v^2 = 169\).

Find the square root to determine the positive value of \(v\):
\(v = \sqrt{169} = 13\).

M1: For correct substitution of the numbers into the formula, e.g., \(5^2 + 2 \times 4.5 \times 16\)
M1: For simplifying to get \(v^2 = 169\)
A1: \(13\)

Calculate the time taken for the first journey:
\(\text{Time}_1 = \frac{\text{Distance}}{\text{Speed}} = \frac{45}{50} = 0.9\text{ hours}\).

Calculate the time taken for the second journey:
\(\text{Time}_2 = \frac{36}{40} = 0.9\text{ hours}\).

Calculate the total time:
\(0.9 + 0.9 = 1.8\text{ hours}\).

Convert the decimal hours to hours and minutes:
\(1\text{ hour}\) and \(0.8 \times 60 = 48\text{ minutes}\).
Total time is \(1\text{ hour and } 48\text{ minutes}\).

M1: For a correct calculation to find the time of one of the journeys, e.g. \(45 \div 50\) or \(36 \div 40\)
M1: For adding both times to get \(1.8\text{ hours}\)
A1: \(1\text{ hour } 48\text{ minutes}\) (accept \(1\text{ hr } 48\text{ mins}\))

First, find the number of boys and girls by sharing 450 in the ratio \(4 : 5\). Total parts = \(4 + 5 = 9\). Value of one part = \(450 \div 9 = 50\). Number of boys = \(4 \times 50 = 200\). Number of girls = \(5 \times 50 = 250\). Next, find the number of boys who walk to school: \(40\%\) of \(200 = 80\). Next, find the number of girls who walk to school: \(30\%\) of \(250 = 75\). Finally, add these to find the total: \(80 + 75 = 155\).

**M1** Method to find the number of boys and girls, e.g., \(450 \div 9 \times 4\) or \(200\) or \(250\) seen. **M1** Method to find \(40\%\) of their boys and \(30\%\) of their girls and add them, e.g., \(0.40 \times 200 + 0.30 \times 250\). **A1** \(155\)

To scale the recipe from 12 pancakes to 30 pancakes, calculate the scale factor:
\(30 \div 12 = 2.5\)

Calculate the total ingredients required for 30 pancakes:
- Eggs: \(4 \times 2.5 = 10\) eggs
- Flour: \(300\text{g} \times 2.5 = 750\text{g}\)
- Milk: \(500\text{ml} \times 2.5 = 1250\text{ml}\)

Calculate how much more she needs of each ingredient:
- Eggs: She has 3, needs 10, so she needs to buy \(10 - 3 = 7\) more eggs. Since they are sold in boxes of 6, she must buy 2 boxes (12 eggs). Cost = \(2 \times £1.50 = £3.00\).
- Flour: She has 400g, needs 750g, so she needs to buy \(750\text{g} - 400\text{g} = 350\text{g}\) more flour. She must buy 1 bag of 1 kg. Cost = \(1 \times £1.20 = £1.20\).
- Milk: She has 300ml, needs 1250ml, so she needs to buy \(1250\text{ml} - 300\text{ml} = 950\text{ml}\) more milk. Since milk is sold in 500ml bottles, she must buy 2 bottles (1000ml). Cost = \(2 \times £0.60 = £1.20\).

Total cost = \(£3.00 + £1.20 + £1.20 = £5.40\).

M1: For scaling the ingredient requirements for 30 pancakes (10 eggs, 750g flour, 1250ml milk) OR for calculating the extra amounts of ingredients required (7 eggs, 350g flour, 950ml milk).
M1: For determining the correct number of packs/boxes to buy: 2 boxes of eggs, 1 bag of flour, and 2 bottles of milk.
M1: For calculating the total cost of any two of the ingredients correctly (e.g., eggs = £3.00, flour = £1.20, milk = £1.20).
A1: For £5.40 (accept 5.40, or 540p if units are written clearly).

First, split the L-shaped lawn into two rectangles to find the total area:
- Vertical split: Left rectangle has width \(12 - 7 = 5\) m and height \(4\) m. Area = \(5 \times 4 = 20\text{ m}^2\).
Right rectangle has width \(7\) m and height \(8\) m. Area = \(7 \times 8 = 56\text{ m}^2\).
Total Area = \(20 + 56 = 76\text{ m}^2\).

Second, find the number of boxes of lawn seed required:
\(76 \div 15 = 5.066...\) boxes.
As only whole boxes can be bought, Sarah needs 6 boxes of seed.

Third, apply the special offer 'Buy 2 get 1 free':
Every group of 3 boxes contains 1 free box, meaning you only pay for 2 boxes.
For 6 boxes, there are exactly 2 groups of 3.
Therefore, Sarah only pays for \(2 \times 2 = 4\) boxes.

Calculate the total cost:
\(4 \times £8.50 = £34.00\).

M1: For a complete method to find the area of the lawn, e.g. \((12 \times 4) + (7 \times 4)\) or \((5 \times 4) + (7 \times 8)\).
A1: For area = 76
M1: For \(76 \div 15\) showing that 6 boxes are needed, and applying the 'Buy 2 get 1 free' offer to show that only 4 boxes need to be paid for.
A1: For £34.00 (accept 34 or £34).

1. Find the volume of the pool: \(\text{Volume} = 15 \times 8 \times 1.5 = 180\text{ m}^3\). 2. Convert the volume to litres: \(180 \times 1000 = 180,000\text{ litres}\). 3. Calculate the target volume (\(80\%\) of the capacity): \(180,000 \times 0.80 = 144,000\text{ litres}\). 4. Work out the combined rate of both hoses: \(\text{Combined rate} = 12 + 18 = 30\text{ litres per minute}\). 5. Calculate the total time in minutes to fill the target volume: \(\text{Time} = 144,000 \div 30 = 4800\text{ minutes}\). 6. Convert the time from minutes to hours: \(\text{Time in hours} = 4800 \div 60 = 80\text{ hours}\).

M1: For calculating the volume of the pool: \(15 \times 8 \times 1.5\) or \(180\text{ m}^3\). M1: For converting their volume to litres (multiplied by 1000) OR for finding \(80\%\) of their volume. M1: For finding the target volume of \(144,000\text{ litres}\) (or \(144\text{ m}^3\)). M1: For dividing their target volume in litres by the combined rate of \(30\text{ litres per minute}\). A1: For 80.

1. Calculate the total weekly wages of the original 8 assistants: \(\text{Total wage} = 8 \times 320 = \text{\pounds}2560\). 2. Calculate the new mean wage for all 10 assistants (\(2.5\%\) higher than \(\text{\pounds}320\)): \(\text{New mean} = 320 \times 1.025 = \text{\pounds}328\). 3. Calculate the new total weekly wages for all 10 assistants: \(\text{New total wage} = 10 \times 328 = \text{\pounds}3280\). 4. Find the combined weekly wages of the two new assistants: \(\text{Combined wage} = 3280 - 2560 = \text{\pounds}720\). 5. Find the wage of the second new assistant (\(x\)): \(x = 720 - 345 = 375\).

M1: For finding the original total weekly wage: \(8 \times 320 = 2560\). M1: For finding the new mean wage: \(320 \times 1.025\) or \(328\). M1: For finding the new total wage of all 10 assistants: \(\text{their } 328 \times 10\) or \(3280\). M1: For a complete method to find \(x\): \(\text{their } 3280 - \text{their } 2560 - 345\). A1: For 375.

To convert the decimal 0.175 to a fraction, we can write it as \(\frac{175}{1000}\). Simplifying this by dividing both the numerator and the denominator by their greatest common divisor, 25, we get: \(\frac{175 \div 25}{1000 \div 25} = \frac{7}{40}\).

B1 for \(\frac{7}{40}\)

The outcomes on a fair six-sided dice are 1, 2, 3, 4, 5, and 6. The prime numbers in this set are 2, 3, and 5 (three outcomes). The probability of rolling a prime number is \(\frac{3}{6} = \frac{1}{2}\).

B1 for \(\frac{1}{2}\)

Add 7 to both sides of the equation: \(3x = 8 + 7\), which simplifies to \(3x = 15\). Divide both sides by 3 to find \(x\): \(x = \frac{15}{3} = 5\).

B1 for \(x = 5\)

A polygon with 9 sides is called a nonagon. (A hexagon has 6 sides, an octagon has 8 sides, and a decagon has 10 sides.)

B1 for Nonagon

The actual distance in centimetres is \(6 \times 25000 = 150000\text{ cm}\). Convert centimetres to metres: \(150000 \div 100 = 1500\text{ m}\). Convert metres to kilometres: \(1500 \div 1000 = 1.5\text{ km}\).

B1 for \(1.5\text{ km}\)

To find the mean, add the numbers together and divide by the total count: \(\text{Sum} = 4 + 7 + 8 + 11 + 15 = 45\). There are 5 numbers, so \(\text{Mean} = 45 \div 5 = 9\).

B1 for 9

Group the like terms: \(4a + 2a = 6a\) and \(-3b + 5b = 2b\). Combining these terms gives the simplified expression \(6a + 2b\).

B1 for \(6a + 2b\)

The volume of a cuboid is found by multiplying its length, width, and height: \(\text{Volume} = 5\text{ cm} \times 3\text{ cm} \times 4\text{ cm} = 60\text{ cm}^3\).

B1 for \(60\text{ cm}^3\)

An increase of \(3.5\%\) is equivalent to finding \(100\% + 3.5\% = 103.5\%\) of the original amount. To write \(103.5\%\) as a multiplier, we divide by \(100\) to get \(1.035\). Therefore, the new value is found by multiplying the original investment by \(1.035\), which is \(240 \times 1.035\).

B1 for choosing the correct option C (\(240 \times 1.035\)).

First, expand both brackets: \(3(2x - 5) = 6x - 15\) and \(-2(x - 1) = -2x + 2\). Next, collect the like terms: \(6x - 2x - 15 + 2 = 4x - 13\). Therefore, the correct simplified expression is \(4x - 13\).

B1 for choosing the correct option B (\(4x - 13\)).

First, convert #250 to Euros: \(250 \times 1.16 = 290\) Euros. Next, subtract the spent Euros: \(290 - 180 = 110\) Euros. Finally, convert 110 Euros back to Pounds: \(110 \div 1.16 \approx 94.8275...\) Pounds. To the nearest penny, this is #94.83.

M1 for \(250 \times 1.16 - 180\) or 110 seen OR for dividing their remaining Euros by 1.16. A1 for 94.83 (accept #94.83).

A 15% discount means the sale price is 85% of the original price. Let the original price be \(x\). \(0.85x = 57.80\). Therefore, \(x = 57.80 \div 0.85 = 68\). The original price was #68.

M1 for \(57.80 \div 0.85\) or equivalent. A1 for 68 (accept 68.00 or #68).

The formula for the area of a trapezium is \(\text{Area} = \frac{1}{2}(a+b)h\). Substituting the given values: \(\text{Area} = \frac{1}{2}(8.4 + 12.6) \times 5.5 = \frac{1}{2}(21) \times 5.5 = 10.5 \times 5.5 = 57.75\text{ cm}^2\).

M1 for \(0.5 \times (8.4 + 12.6) \times 5.5\) or \(10.5 \times 5.5\). A1 for 57.75.

Expand the brackets: \(8x - 12 = 14\). Add 12 to both sides: \(8x = 26\). Divide by 8: \(x = \frac{26}{8} = 3.25\).

M1 for \(8x - 12 = 14\) or \(2x - 3 = 3.5\). A1 for 3.25 (accept \(3 \frac{1}{4}\) or \(\frac{13}{4}\)).

The expected number of heads is the total number of spins multiplied by the probability of landing on heads: \(240 \times 0.35 = 84\).

M1 for \(240 \times 0.35\). A1 for 84.

The sum of all five numbers is \(12 \times 5 = 60\). The sum of the four given numbers is \(8 + 15 + 11 + 14 = 48\). The fifth number is \(60 - 48 = 12\).

M1 for \(12 \times 5\) or 60 seen OR \(8 + 15 + 11 + 14 = 48\) seen. A1 for 12.

Let the number of small boxes bought be 5x and the number of large boxes bought be 2x. The number of candles in the small boxes is 8 * 5x = 40x. The number of candles in the large boxes is 15 * 2x = 30x. The total number of candles is 40x + 30x = 70x. We are given that 70x = 210, which gives x = 3. Therefore, the number of small boxes is 5 * 3 = 15, and the number of large boxes is 2 * 3 = 6. The total number of boxes bought is 15 + 6 = 21.

M1 for expressing the total number of candles in terms of a variable, e.g. 8 * 5x + 15 * 2x = 210, or for finding the ratio of candles in small boxes to large boxes as 40 : 30. M1 for finding the value of the multiplier (x = 3) or the number of one type of box (e.g. 15 small boxes or 6 large boxes). A1 for 21.

The area of the rectangle is 12 m * 6 m = 72 m^2. The semicircle has a diameter equal to the width of the rectangle (6 m), so its radius is 3 m. The area of the semicircle is 0.5 * pi * r^2 = 0.5 * pi * 3^2 = 4.5 * pi ≈ 14.137 m^2. The total area of the garden is 72 + 14.137 = 86.137 m^2. Rounding to 1 decimal place gives 86.1 m^2.

M1 for finding the area of the rectangle: 12 * 6 = 72. M1 for a complete method to find the area of the semicircle: 0.5 * pi * 3^2 (approx 14.1). A1 for 86.1 (accept 86.14 or answer in the range [86.1, 86.2]).

The total number of possible outcomes when spinning the spinner and rolling the die is 3 * 6 = 18. Let's find the winning outcomes: spinning an A gives 6 outcomes (A1, A2, A3, A4, A5, A6); rolling a 6 with any other spin gives 2 outcomes (B6, C6). The total number of winning outcomes is 6 + 2 = 8. Therefore, the probability that Chloe wins is 8/18, which simplifies to 4/9. Alternatively, the probability of losing is P(not A and not 6) = 2/3 * 5/6 = 10/18 = 5/9, so the probability of winning is 1 - 5/9 = 4/9.

M1 for identifying the total number of outcomes (18) or identifying individual probabilities P(A) = 1/3 and P(6) = 1/6. M1 for a complete method to find the number of winning outcomes (8) or calculating 1 - (2/3 * 5/6). A1 for 4/9 or any equivalent fraction (e.g. 8/18), decimal (0.44 or 0.444...), or percentage (44.4%).

We first find the total number of books read by multiplying each number of books by its frequency: 0 * 5 = 0, 1 * 12 = 12, 2 * 11 = 22, 3 * 8 = 24, 4 * 4 = 16. The total number of books read is 0 + 12 + 22 + 24 + 16 = 74. The total number of students is 40. The mean is 74 / 40 = 1.85.

M1 for finding at least three correct products of the number of books and their frequencies (e.g. 12, 22, 24, 16). M1 for dividing the total sum of books by 40. A1 for 1.85.

First, expand the bracket on the left side: 5x - 15 < 2x + 6. Next, subtract 2x from both sides of the inequality: 3x - 15 < 6. Then, add 15 to both sides: 3x < 21. Finally, divide both sides by 3 to find x < 7.

M1 for correct expansion of the bracket: 5x - 15 < 2x + 6. M1 for isolating the x terms on one side, e.g. 3x < 21 or -21 < -3x. A1 for x < 7 (accept 7 > x).

The revenue from selling the first 45 hoodies is 45 * £24 = £1080. The price of a discounted hoodie is £24 * (1 - 0.30) = £16.80. The revenue from the remaining 15 hoodies is 15 * £16.80 = £252. The total revenue is £1080 + £252 = £1332. The profit is £1332 - £900 = £432. The percentage profit is (432 / 900) * 100 = 48%.

M1 for finding the revenue from the first 45 hoodies (£1080) or finding the discounted price of a hoodie (£16.80). M1 for a complete method to find the total revenue (£1332) or total profit (£432). A1 for 48.

First, find the total number of parts in the ratio: \(3 + 2 + 1 = 6\). Divide the total volume by the number of parts to find the volume of one part: \(15 \div 6 = 2.5\) litres. Calculate the volume of each colour paint needed:
- Red: \(3 \times 2.5 = 7.5\) litres
- Blue: \(2 \times 2.5 = 5\) litres
- White: \(1 \times 2.5 = 2.5\) litres.

Next, calculate the number of tins needed for each, rounding up to the nearest whole tin where necessary:
- Red: \(7.5 \div 2.5 = 3\) tins
- Blue: \(5 \div 1 = 5\) tins
- White: \(2.5 \div 1 = 2.5 \implies 3\) tins.

Now, calculate the cost for each colour:
- Red: \(3 \times £18.50 = £55.50\)
- Blue: \(5 \times £8.20 = £41.00\)
- White: \(3 \times £5.50 = £16.50\).

Total cost = \(£55.50 + £41.00 + £16.50 = £113.00\).

M1: For dividing 15 by 6 or finding the volume of at least one colour (Red = 7.5, Blue = 5, White = 2.5)
M1: For correctly scaling up the tins needed to whole tins (Red = 3, Blue = 5, White = 3)
M1: For calculating the cost of all three colours based on their whole tins
A1: For the correct total cost of £113.00 (allow 113)

First, find the area of the rectangular section: \(\text{Area} = 12 \times 6 = 72\text{ m}^2\).
Next, find the area of the semicircle. The diameter is 6 metres, so the radius is \(6 \div 2 = 3\) metres. \(\text{Area of semicircle} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times \pi \times 3^2 = 4.5\pi \approx 14.137\text{ m}^2\).
Total area of the garden = \(72 + 14.137 = 86.137\text{ m}^2\).
Now, calculate the number of bags of seed required: \(86.137 \div 10 = 8.6137\) bags. Since you can only buy whole bags, this must be rounded up to 9 bags.
Finally, calculate the total cost: \(9 \times £4.99 = £44.91\).

M1: For calculating the area of the rectangle: \(12 \times 6 = 72\)
M1: For calculating the area of the semicircle: \(\frac{1}{2} \times \pi \times 3^2 \approx 14.14\)
M1: For adding the areas to get a total area (approx 86.14) and dividing by 10 to find the number of bags, rounding up to 9 bags
A1: For the correct final cost of £44.91 (allow 44.91)

First, find the total score of the original 8 friends: \(8 \times 74 = 592\).
Next, find the total score of the 9 friends after Sam's score is added: \(9 \times 76 = 684\).
Find Sam's score by subtracting the two totals: \(684 - 592 = 92\).
Now, find the total score of all 10 friends after Alex's score is added: \(10 \times 75 = 750\).
Find Alex's score by subtracting the 9-friend total from the 10-friend total: \(750 - 684 = 66\).
Finally, work out the difference between Sam's score and Alex's score: \(92 - 66 = 26\).

M1: For calculating the total score for 8 friends (592) or 9 friends (684)
M1: For calculating Sam's score: \(684 - 592 = 92\)
M1: For calculating Alex's score: \(750 - 684 = 66\)
A1: For the correct difference of 26

First, calculate the price at Shop A:
- 15% of \(£680 = 0.15 \times 680 = £102\)
- Reduced price = \(680 - 102 = £578\)
- Additional £20 off = \(578 - 20 = £558\).

Next, calculate the price at Shop B:
- \(\frac{1}{5}\) off \(£680 = 680 \div 5 = £136\)
- Sale price = \(680 - 136 = £544\)
- Extra 5% off the sale price = \(0.05 \times 544 = £27.20\)
- Final price at Shop B = \(544 - 27.20 = £516.80\).

Finally, compare the two prices:
- Shop B is cheaper.
- Difference = \(£558 - £516.80 = £41.20\).

M1: For calculating the final price at Shop A: £558
M1: For calculating the intermediate sale price at Shop B: £544
M1: For calculating the final price at Shop B after the loyalty discount: £516.80
A1: For concluding 'Shop B' is cheaper by '£41.20' (or 41.2)

First, set up an expression for the perimeter: \(\text{Perimeter} = 2(3x + 5) + 2(2x - 1) = 6x + 10 + 4x - 2 = 10x + 8\).
We are given the perimeter is 68, so:
\(10x + 8 = 68\)
\(10x = 60\)
\(x = 6\).

Now find the dimensions of the field:
- Length = \(3(6) + 5 = 23\) metres
- Width = \(2(6) - 1 = 11\) metres.

Finally, calculate the area of the field:
\(\text{Area} = \text{length} \times \text{width} = 23 \times 11 = 253\text{ m}^2\).

M1: For setting up a correct expression or equation for the perimeter, e.g. \(10x + 8 = 68\)
M1: For solving the equation to find \(x = 6\)
M1: For substituting \(x = 6\) to find the length (23) and width (11)
A1: For the correct area of 253 (allow 253 m²)

First, use the ratio of blue to yellow counters (\(2 : 3\)) to find the number of yellow counters.
Since 2 parts represent 24 blue counters, 1 part represents \(24 \div 2 = 12\) counters.
So, the number of yellow counters is \(3 \times 12 = 36\).

The total number of blue and yellow counters is \(24 + 36 = 60\).

Since the probability of choosing a red counter is 0.5, the probability of choosing either a blue or yellow counter is \(1 - 0.5 = 0.5\).
Therefore, 60 counters represent 50% (or 0.5) of the total counters in the bag.

Total number of counters = \(60 \div 0.5 = 120\).

M1: For finding the number of yellow counters: 36
M1: For finding the total number of blue and yellow counters: \(24 + 36 = 60\)
M1: For recognizing that \(P(\text{blue or yellow}) = 0.5\) or writing an equation like \(0.5x = 60\)
A1: For the correct total number of counters of 120

To find the total revenue, we first need to express the ratios of tea, coffee, and hot chocolate as a single combined ratio.

Let \(T\) = Tea, \(C\) = Coffee, and \(H\) = Hot chocolate.
We are given:
\(T : C = 3 : 5\)
\(C : H = 4 : 3\)

To combine these, find the lowest common multiple of the parts for coffee (5 and 4), which is 20.
* Multiply the first ratio by 4: \(T : C = 12 : 20\)
* Multiply the second ratio by 5: \(C : H = 20 : 15\)

This gives the combined ratio:
\(T : C : H = 12 : 20 : 15\)

Next, calculate the total number of parts in the ratio:
\(12 + 20 + 15 = 47\) parts

Find the value of one part:
\(376 \div 47 = 8\) drinks

Now, calculate the quantity of each drink sold:
* Tea: \(12 \times 8 = 96\) cups
* Coffee: \(20 \times 8 = 160\) cups
* Hot chocolate: \(15 \times 8 = 120\) cups

Finally, calculate the money made from each drink type and sum them up:
* Tea revenue: \(96 \times £1.80 = £172.80\)
* Coffee revenue: \(160 \times £2.50 = £400.00\)
* Hot chocolate revenue: \(120 \times £2.20 = £264.00\)

Total revenue = \(£172.80 + £400.00 + £264.00 = £836.80\).

M1: For scaling the ratios to find a common term for coffee, e.g. showing \(12 : 20\) and \(20 : 15\) or an equivalent method to link all three quantities.
A1: For the correct combined ratio \(12 : 20 : 15\) (or equivalent).
M1: For dividing 376 by their sum of parts (e.g. \(376 \div 47\)) and multiplying by at least one of their parts, or finding at least one correct quantity of drink (96, 160 or 120).
A1: For finding the correct quantities of all three drinks: 96 teas, 160 coffees and 120 hot chocolates.
A1: For the correct total revenue of £836.80 (accept 836.8 or 836.80).