Worked Solution

We know that there are 60 minutes in 1 hour. To find out how many hours are in 180 minutes, we need to divide the total number of minutes (180) by 60.

$$180 \div 60 = 3$$

Answer: 3 hours (B1)

Worked Solution

“Per cent” means “out of 100”. To change a decimal to a percentage, you multiply the decimal by 100.

$$0.73 \times 100 = 73$$

Answer: 73% (B1)

Worked Solution

We must follow the order of operations, (which you might know as BODMAS or BIDMAS). This means we must do the operation inside the Brackets first.

$$3 + 5 = 8$$

Now we can do the multiplication.

$$10 \times 8 = 80$$

Answer: 80 (B1)

Worked Solution

A prime number is a whole number greater than 1 that has only two factors (divisors): 1 and itself. Let’s check the numbers between 20 and 30:

• 21: No (it can be divided by 3 and 7)
• 22: No (it can be divided by 2 and 11)
• 23: Yes (it can only be divided by 1 and 23)
• 24: No (it’s even, so divisible by 2)
• 25: No (it can be divided by 5)
• 26: No (it’s even, so divisible by 2)
• 27: No (it can be divided by 3 and 9)
• 28: No (it’s even, so divisible by 2)
• 29: Yes (it can only be divided by 1 and 29)

You only need to write one of the correct answers.

Answer: 23 (or 29) (B1)

Worked Solution

To find the number that is halfway between two numbers (also called the midpoint or mean), you add the two numbers together and then divide by 2.

Step 1: Add the numbers.

$$7 + 15 = 22$$

Step 2: Divide the result by 2.

$$22 \div 2 = 11$$

Answer: 11 (B1)

Worked Solution

We need to find the total cost for 4 people and 7 days. Let’s calculate each part separately and then add them up.

1. Total Travel Cost:

The cost is £150 per person. There are 4 people.

$$4 \times £150 = £600$$ (P1 for this step)

2. Total Hotel Cost:

The cost is £50 per day, per person. The holiday is for 7 days and 4 people.

First, find the hotel cost for one person for 7 days:

$$7 \text{ days} \times £50 = £350$$

Now, find the hotel cost for 4 people:

$$4 \text{ people} \times £350 = £1400$$ (P1 for a correct hotel step, e.g., $7 \times 50$ or $4 \times 7 \times 50$)

3. Total Spending Money:

The cost is £250 per person. There are 4 people.

$$4 \times £250 = £1000$$ (P1 for this step)

4. Total Holiday Cost:

Now, add all the costs together.

$$£600 \text{ (Travel)} + £1400 \text{ (Hotel)} + £1000 \text{ (Spending)} = £3000$$ (P1 for combining the costs)

Answer: £3000 (A1)

Worked Solution

(a) Number of blue flowers

First, let’s read the number of red, white, and yellow flowers from the bar chart.

• Red: 8
• White: 10
• Yellow: 5

Next, add these numbers together to find the total of red, white, and yellow flowers.

$$8 + 10 + 5 = 23$$

The question says the total number of all flowers is 30. The only remaining colour is blue. So, we subtract the 23 we found from the total of 30.

$$30 – 23 = 7$$ (P1 for this process)

Answer (a): 7 (A1)

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(b) The mode

The “mode” is the item that appears most often. In this case, it’s the colour with the highest bar on the chart.

• Red: 8
• White: 10 (This is the highest number)
• Yellow: 5
• Blue: 7 (from part a)

Answer (b): white (B1)

Worked Solution

To compare fractions, it’s easiest to give them all the same “bottom number” (a common denominator). Looking at the denominators (4, 4, 3, 12, 2), a number they all go into is 12. Let’s convert all the fractions to be “out of 12”.

• $\frac{1}{4}$: To get 12 on the bottom, we multiply 4 by 3. We must do the same to the top: $1 \times 3 = 3$. So: $\frac{3}{12}$
• $\frac{3}{4}$: Multiply top and bottom by 3: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$
• $\frac{1}{3}$: Multiply top and bottom by 4: $\frac{1 \times 4}{3 \times 4} = \frac{4}{12}$
• $\frac{7}{12}$: This is already out of 12.
• $\frac{1}{2}$: Multiply top and bottom by 6: $\frac{1 \times 6}{2 \times 6} = \frac{6}{12}$

(M1 for converting at least two fractions correctly)

Now we have: $\frac{3}{12}$, $\frac{9}{12}$, $\frac{4}{12}$, $\frac{7}{12}$, $\frac{6}{12}$. We can easily order these from smallest to largest by looking at the “top number” (numerator).

Order (converted): $\frac{3}{12}$, $\frac{4}{12}$, $\frac{6}{12}$, $\frac{7}{12}$, $\frac{9}{12}$

Finally, write down the original fractions in this order.

Answer: $\frac{1}{4}, \frac{1}{3}, \frac{1}{2}, \frac{7}{12}, \frac{3}{4}$ (A1)

Worked Solution

(a) Distance

We can use the formula: Distance = Speed $\times$ Time.

Speed: 4 mph

Time: From 9 am to 10 30 am is 1 hour and 30 minutes. To use the formula, we must write this in hours. 30 minutes is half (0.5) an hour. So the time is 1.5 hours.

Now, calculate the distance:

$$Distance = 4 \times 1.5$$

To work this out: $4 \times 1 = 4$. $4 \times 0.5 = 2$. Add them: $4 + 2 = 6$.

(M1 for $4 \times 1.5$ or $4 \times 1\frac{1}{2}$)

Answer (a): 6 miles (A1)

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(b) Time she got home

We need to start from when she got to the library and add on the time she spent there and the time she took to walk home.

Start time: Arrived at library at 10 30 am.

Add library time: She stayed for 50 minutes.

10 30 am + 50 minutes:

10 30 am + 30 minutes = 11 00 am

11 00 am + remaining 20 minutes = 11 20 am. (This is when she left the library).

Add walk home time: She took $1\frac{1}{4}$ hours. $\frac{1}{4}$ of an hour is 15 minutes. So she took 1 hour and 15 minutes.

Now add this to her leaving time (11 20 am).

$$11:20 \text{ am} + 1 \text{ hour} = 12:20 \text{ pm}$$ (M1 for adding 50 mins and 75 mins, or for getting 11 20) $$12:20 \text{ pm} + 15 \text{ minutes} = 12:35 \text{ pm}$$

Answer (b): 12 35 pm (A1)

Worked Solution

(a) Solve $t + t + t = 12$

First, simplify the left side. $t + t + t$ is 3 lots of $t$, which we write as $3t$.

$$3t = 12$$

To find $t$, we divide both sides by 3.

$$t = 12 \div 3$$ $$t = 4$$

Answer (a): $t = 4$ (B1)

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(b) Solve $x – 2 = 6$

To get $x$ on its own, we need to get rid of the “-2”. We do the opposite operation, which is to “+2”, to both sides of the equation.

$$x – 2 + 2 = 6 + 2$$ $$x = 8$$

Answer (b): $x = 8$ (B1)

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(c) Solve $6w + 2 = 20$

This is a two-step equation. First, we get rid of the “+2” by subtracting 2 from both sides.

$$6w + 2 – 2 = 20 – 2$$ $$6w = 18$$ (M1 for this correct first step)

Now, $6w$ means $6 \times w$. To get $w$ on its own, we do the opposite and divide both sides by 6.

$$w = 18 \div 6$$ $$w = 3$$

Answer (c): $w = 3$ (A1)

Worked Solution

Since this is a non-calculator paper, we must use a written method like long multiplication or the grid method.

Method 1: Long Multiplication

    74
  x 58
  ----
   592  (This is 8 x 74)
  3700  (This is 50 x 74)
  ----
  4292  (This is 592 + 3700)

Method 2: Grid Method

We split 74 into 70 and 4, and 58 into 50 and 8.

• $50 \times 70 = 3500$
• $50 \times 4 = 200$
• $8 \times 70 = 560$
• $8 \times 4 = 32$

Now add all these answers together:

$$3500 + 200 + 560 + 32 = 4292$$ (M1 for a complete, correct method)

Answer: 4292 (A1)

Worked Solution

(a) Find the value of x

“Perpendicular lines” means they meet at a right angle, which is 90°. The diagram shows the 90° angle is made up of three parts: 25°, x°, and 25°.

First, add up the parts we know:

$$25 + 25 = 50^{\circ}$$

To find x, subtract this from the total 90°.

$$x = 90 – 50 = 40^{\circ}$$ (M1 for $90-25-25$ or $90-50$)

Answer (a): $x = 40$ (A1)

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(b)(i) Find another 125° angle

There are two possible answers here.

Answer 1:

Angle b is 125°. (B1)

Reason: Because vertically opposite angles are equal. (C1)

Answer 2:

Angle d is 125°. (B1)

Reason: Because corresponding angles are equal (they are in the same ‘F’ shape). (C1)

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(b)(ii) Explain why $a + b + c = 235^{\circ}$

The angles $a$, $b$, $c$ and the $125^{\circ}$ angle are all around a single point. Angles around a point add up to 360°.

So, $a + b + c + 125^{\circ} = 360^{\circ}$.

To find just $a + b + c$, we subtract $125^{\circ}$ from $360^{\circ}$.

$$360 – 125 = 235^{\circ}$$

Explanation: Angles around a point sum to 360°. $360 – 125 = 235$. (C1)

Worked Solution

We need to know the conversion: 1 centimetre (cm) = 10 millimetres (mm)

To convert from cm to mm, we must multiply by 10.

The length is $x$ cm. So, in mm, the length is $x \times 10$.

Answer: 10x (B1)

Worked Solution

(a) Work out $\frac{1}{5}$ of 70

Finding $\frac{1}{5}$ (one fifth) of a number is the same as dividing that number by 5.

$$70 \div 5 = 14$$

(You can work this out by: $50 \div 5 = 10$ and $20 \div 5 = 4$. Then $10 + 4 = 14$).

Answer (a): 14 (B1)

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(b) Explain what is wrong

Fiona’s answer $48 \div \frac{1}{2} = 24$ is wrong. She has actually worked out $48 \times \frac{1}{2}$ or $48 \div 2$.

The question $48 \div \frac{1}{2}$ is asking “How many halves fit into 48?”. Her reason “There are 2 halves in 1” is correct, but she doesn’t follow it through. If there are 2 halves in 1, then in 48 there must be $48 \times 2 = 96$ halves.

Explanation: She has divided by 2, but dividing by $\frac{1}{2}$ is the same as multiplying by 2. The correct answer is $48 \times 2 = 96$. (C1 for a correct explanation, e.g., “she divided by 2 but should have multiplied by 2”)

Worked Solution

(a) Value of $\sqrt{64}$

$\sqrt{64}$ means the “square root of 64”. This asks “what number, when multiplied by itself, gives 64?”.

$$8 \times 8 = 64$$

Answer (a): 8 (B1)

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(b) Value of $5^3$

$5^3$ means “5 to the power of 3”, or “5 cubed”. This means 5 multiplied by itself 3 times.

$$5^3 = 5 \times 5 \times 5$$

First, $5 \times 5 = 25$.

Then, $25 \times 5 = 125$.

Answer (b): 125 (B1)

Worked Solution

(a) Expand $5(2m – 3)$

“Expand” means to multiply the term on the outside of the bracket by everything inside the bracket.

First, multiply 5 by $2m$:

$$5 \times 2m = 10m$$

Next, multiply 5 by $-3$:

$$5 \times -3 = -15$$

Put them together.

Answer (a): $10m – 15$ (B1)

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(b) Factorise $3n + 12$

“Factorise” is the opposite of expanding. We need to find the highest common factor (HCF) of $3n$ and $12$.

The HCF of 3 and 12 is 3.

We put the 3 outside a bracket. Then we work out what goes inside.

$3 \times ? = 3n$. The answer is $n$.

$3 \times ? = +12$. The answer is $+4$.

Answer (b): $3(n + 4)$ (B1)

Worked Solution

(i) Whose estimate is better?

In probability, more experiments (or “trials”) give a more reliable and accurate estimate.

Maxine threw the coin 50 times. Stuart only threw it 10 times.

Answer: Maxine (C1)

Reason: Because she threw the coin more times (50 is more than 10). A larger number of trials gives a better estimate. (C1)

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(ii) Combined estimate

To get the best possible estimate, we should combine all the results.

Total number of throws = (Stuart’s throws) + (Maxine’s throws)

$$10 + 50 = 60 \text{ throws}$$

Total number of Tails = (Stuart’s tails) + (Maxine’s tails)

$$7 + 30 = 37 \text{ tails}$$

The probability is the number of tails out of the total number of throws.

Answer: $\frac{37}{60}$ (B1)

Worked Solution

This is a “show that” question, so we must show every step of our working clearly. The plan is: 1. Find the area of the path. 2. Find the area of one paving stone. 3. Find out how many stones are needed. 4. Find the total cost of the stones. 5. Compare the cost to the £220 Wasim has.

Step 1: Area of the path

The path is a rectangle 600 cm by 120 cm.

$$Area = 600 \times 120 = 72000 \text{ cm}^2$$ (M1 for a correct first step, e.g., $600 \times 120$)

Step 2: Area of one paving stone

The stone is a square of side 30 cm.

$$Area = 30 \times 30 = 900 \text{ cm}^2$$ (M1 for this step or $600 \div 30$ or $120 \div 30$)

Step 3: Number of stones needed

We divide the total path area by the area of one stone.

$$Number\ of\ stones = \frac{72000}{900} = \frac{720}{9} = 80 \text{ stones}$$

(Alternatively, $600 \div 30 = 20$ stones long. $120 \div 30 = 4$ stones wide. $20 \times 4 = 80$ stones.)

Step 4: Total cost of the stones

Each of the 80 stones costs £2.50.

$$Total\ cost = 80 \times £2.50$$

To work this out: $80 \times £2 = £160$ $80 \times £0.50 = £40$ (This is half of 80) $£160 + £40 = £200$

(M1 for full method, e.g., $80 \times 2.5$)

Step 5: Compare the cost

Wasim has £220. The stones cost £200. (A1 for 200 or 80 and 88)

Conclusion:

£200 is less than £220, so Wasim has enough money.

Worked Solution

(a) $\frac{2}{3} – \frac{1}{5}$

To add or subtract fractions, we need a common denominator. The lowest common multiple of 3 and 5 is 15.

Convert $\frac{2}{3}$: Multiply top and bottom by 5: $\frac{2 \times 5}{3 \times 5} = \frac{10}{15}$

Convert $\frac{1}{5}$: Multiply top and bottom by 3: $\frac{1 \times 3}{5 \times 3} = \frac{3}{15}$

(M1 for correct common denominator with at least one fraction correct)

Now subtract the new fractions:

$$\frac{10}{15} – \frac{3}{15} = \frac{10 – 3}{15} = \frac{7}{15}$$

Answer (a): $\frac{7}{15}$ (A1)

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(b) $\frac{2}{3} \times \frac{3}{4}$

To multiply fractions, you just multiply the numerators (tops) together and multiply the denominators (bottoms) together.

$$\frac{2 \times 3}{3 \times 4} = \frac{6}{12}$$ (M1 for this step)

The question asks for the answer in its simplest form. We can divide both 6 and 12 by their highest common factor, which is 6.

$$6 \div 6 = 1$$ $$12 \div 6 = 2$$

Answer (b): $\frac{1}{2}$ (A1)

Worked Solution

This looks hard, but it’s easier if we just pick a number for the side length of square B.

Step 1: Pick a length for Square B.

Let’s say the side of Square B is 10 cm.

Step 2: Find the Area of Square B.

$$Area\ of\ B = 10 \times 10 = 100 \text{ cm}^2$$

(Picking 10 is smart, because 100 cm² makes percentages easy to find!)

Step 3: Find the side length of Square A.

The side of A is 50% (half) of the side of B.

$$Side\ of\ A = 50\% \text{ of } 10 \text{ cm} = 5 \text{ cm}$$ (P1 for finding the ratio of areas, e.g., 1:4, or for finding correct areas for A and B)

Step 4: Find the Area of the shaded region.

The shaded region in square A is a triangle. Its base is the side of square A (5 cm) and its height is also the side of square A (5 cm).

$$Area\ of\ Triangle = \frac{1}{2} \times base \times height$$ $$Area\ of\ Shaded\ Region = \frac{1}{2} \times 5 \times 5$$ $$Area\ of\ Shaded\ Region = \frac{1}{2} \times 25 = 12.5 \text{ cm}^2$$ (P1 for process to find shaded area, e.g., 1/8)

Step 5: Express as a percentage.

We want the (Area of Shaded Region) as a percentage of the (Area of Square B).

$$\frac{\text{Area of Shaded}}{\text{Area of B}} \times 100 = \frac{12.5}{100} \times 100 = 12.5\%$$

Answer: 12.5% (A1)

Worked Solution

This is a problem that is much easier to solve using a two-way table.

Step 1: Set up the table.

	Walk	Cycle	Bus	Total
Boys
Girls
Total

Step 2: Fill in the numbers from the question.

• Total students = 40.
• Total girls = 22.
• Girls who walk = 9.
• Boys who cycle = 7.
• Total bus = 10.
• Boys on bus = 6.

	Walk	Cycle	Bus	Total
Boys		7	6
Girls	9			22
Total			10	40

Step 3: Fill in the missing gaps.

Total Boys = Total Students – Total Girls = 40 – 22 = 18. (P1)

Girls on Bus = Total Bus – Boys on Bus = 10 – 6 = 4. (P1)

Boys who Walk = Total Boys – Boys (Cycle) – Boys (Bus) = 18 – 7 – 6 = 5. (P1)

The table now looks like this:

	Walk	Cycle	Bus	Total
Boys	5	7	6	18
Girls	9		4	22
Total			10	40

Step 4: Answer the question.

The question asks for “the number of these students who walk to school”. This is the Total for Walk.

$$Total\ Walk = \text{Boys who Walk} + \text{Girls who Walk}$$ $$Total\ Walk = 5 + 9 = 14$$ (P1 for full process)

Answer: 14 (A1)

Worked Solution

(a) Complete the table

All probabilities must add up to 1. First, let’s find the total probability for red and yellow combined.

$$1 – 0.2 \text{ (blue)} = 0.8$$ (P1 for $1 – 0.2$)

This 0.8 is the combined probability for red and yellow. The question says the number of red cubes is the same as the number of yellow cubes, which means their probabilities must also be the same.

So, we split the 0.8 evenly between red and yellow.

$$0.8 \div 2 = 0.4$$

Answer (a):

Colour	blue	red	yellow
Probability	0.2	0.4 (A1)	0.4 (A1)

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(b) Total number of cubes

We are told that the probability of blue is 0.2, and the number of blue cubes is 12.

This means 0.2 (or 20%) of the total cubes is equal to 12 cubes.

$$0.2 \times \text{Total} = 12$$

To find the total, we divide 12 by 0.2.

$$Total = 12 \div 0.2$$

Dividing by 0.2 is the same as dividing by $\frac{1}{5}$, which is the same as multiplying by 5.

$$Total = 12 \times 5 = 60$$ (P1 for $12 \div 0.2$)

Answer (b): 60 (A1)

Worked Solution

(a) Amount of flour for 60 biscuits

First, let’s find the “scale factor”. Deon wants to make 60 biscuits, but the recipe is for 15 biscuits.

$$Scale\ factor = 60 \div 15 = 4$$ (P1 for finding the scale factor 4)

This means she needs 4 times the amount of every ingredient.

Sugar needed: $50\ g \times 4 = 200\ g$.

Flour needed: The recipe needs “three times as much flour as sugar”.

$$Flour = 3 \times \text{Sugar}$$ $$Flour = 3 \times 200\ g = 600\ g$$ (P1 for complete process)

Answer (a): 600 g (A1)

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(b) Packs of butter

First, let’s find the total amount of butter needed for 60 biscuits. The recipe needs “two times as much butter as sugar”.

We already know she needs 200g of sugar for 60 biscuits.

$$Butter\ needed = 2 \times \text{Sugar}$$ $$Butter\ needed = 2 \times 200\ g = 400\ g$$ (P1 for calculating 400g)

The butter is sold in 250g packs. We need to find how many packs she must buy to get 400g.

1 pack = 250 g (Not enough)

2 packs = $2 \times 250\ g = 500\ g$ (This is enough)

She can’t buy 1.something packs, so she must buy 2.

Answer (b): 2 (A1)

Worked Solution

The “highest common factor” is the biggest number that divides into both 72 and 90. We can find this by listing the factors of both numbers.

Factors of 72:

1, 72

2, 36

3, 24

4, 18

6, 12

8, 9

List: {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72}

Factors of 90:

1, 90

2, 45

3, 30

5, 18

6, 15

9, 10

List: {1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90}

(M1 for listing at least 4 factors of each number)

Now we look for the highest number that is in both lists.

The common factors are {1, 2, 3, 6, 9, 18}.

The highest one is 18.

Answer: 18 (A1)

Worked Solution

We need to work out the 3D shape from its 2D views.

• Plan (view from above): Is a circle.
• Front Elevation (view from front): Is a rectangle.
• Side Elevation (view from side): Is a rectangle.

A shape that looks like a circle from above and a rectangle from the front and side is a cylinder. (M1 for sketch of a cylinder)

Now we need to add the dimensions from the grid. Each square is 1 cm.

• From the Plan, the circle’s diameter is 4 squares, so diameter = 4 cm (or radius = 2 cm).
• From the Front/Side elevations, the height is 5 squares, so height = 5 cm.

Sketch:

(A sketch of a cylinder with the correct dimensions labelled) (A1 for correct dimensions)

Worked Solution

This is a two-step transformation. Let’s do step 1 first.

Step 1: Reflect Shape A in the x-axis.

The x-axis is the horizontal line at y=0. A reflection in the x-axis “flips” the shape upside down. The x-coordinates stay the same, and the y-coordinates become their negative.

Vertices of Shape A:

(3, 2), (3, 5), (5, 2)

Vertices of the new reflected shape (let’s call it Shape X):

(3, -2), (3, -5), (5, -2)

(M1 for showing this reflection)

Step 2: Translate Shape X to Shape B.

Now we find the translation $\begin{pmatrix} c \\ d \end{pmatrix}$ that moves Shape X to Shape B. We can do this by seeing how one point moves.

Vertices of Shape X: (3, -2), (3, -5), (5, -2)

Vertices of Shape B: (-3, -3), (-3, -6), (-1, -3)

Let’s pick the point (3, -2) from Shape X. It maps to the point (-3, -3) on Shape B.

To find c (the x-movement):

How do we get from x=3 to x=-3? We must subtract 6.

So, $c = -6$. (A1)

To find d (the y-movement):

How do we get from y=-2 to y=-3? We must subtract 1.

So, $d = -1$. (A1)

Answer: $c = -6$, $d = -1$

Worked Solution

This is a ratio problem. The ratio 7:3:4 is for *packs*, not *pens*.

Step 1: Find the ratio of pens sold.

Let’s find the number of pens sold for each “part” of the ratio.

Black: 7 (parts) $\times$ 2 (pens/pack) = 14 parts

Red: 3 (parts) $\times$ 5 (pens/pack) = 15 parts

Green: 4 (parts) $\times$ 6 (pens/pack) = 24 parts

So, the ratio of pens sold (Black:Red:Green) is 14 : 15 : 24. (P1)

Step 2: Find the total number of parts.

A total of 212 pens were sold. Let’s find the total number of parts in our new ratio.

$$Total\ parts = 14 + 15 + 24 = 53\ parts$$ (P1 for this step)

Step 3: Find the value of one part.

The 53 parts must be equal to the 212 pens sold.

$$53 \times ? = 212$$

Let’s try multiplying 53 by a small number. $53 \times 2 = 106$. $53 \times 4 = 212$. So, 1 part = 4 pens.

Step 4: Find the number of green pens.

The question asks for the number of green pens.

From Step 1, green pens = 24 parts.

$$Number\ of\ green\ pens = 24 \times (\text{value of 1 part})$$ $$Number\ of\ green\ pens = 24 \times 4 = 96$$ (P1 for this complete process)

Answer: 96 (A1)

Worked Solution

We need to find the length of AB. We can see that AB is part of the perimeter of rectangle ABCD. To find AB, we first need to know the length of the other side, BC.

Step 1: Find the length of PQ.

We are given the area of PQRS ($45 \text{ cm}^2$) and the length of QR (10 cm).

$$Area\ of\ PQRS = PQ \times QR$$ $$45 = PQ \times 10$$

To find PQ, we divide 45 by 10.

$$PQ = 45 \div 10 = 4.5 \text{ cm}$$ (P1 for this step)

Step 2: Find the length of BC.

The question tells us that $BC = PQ$.

Since $PQ = 4.5 \text{ cm}$, then $BC = 4.5 \text{ cm}$.

Step 3: Find the length of AB.

We are given the perimeter of ABCD is 26 cm.

The formula for the perimeter of a rectangle is $2 \times (length + width)$, or $2 \times (AB + BC)$.

$$Perimeter = 2 \times (AB + BC)$$ $$26 = 2 \times (AB + 4.5)$$ (P1 for using the perimeter formula)

Divide both sides by 2:

$$13 = AB + 4.5$$

Now, subtract 4.5 from both sides to find AB.

$$AB = 13 – 4.5 = 8.5 \text{ cm}$$ (P1 for this step)

Answer: 8.5 cm (A1)

Worked Solution

(a) Coordinates of the turning point

The “turning point” of a parabola (a ‘U’ shape) is the minimum point at the very bottom of the curve.

Looking at the graph, the lowest point of the curve is at:

$x = 1$ and $y = -4$.

As coordinates, we write this as (x, y).

Answer (a): (1, -4) (B1)

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(b) Roots of the equation

The “roots” of the equation $x^2 – 2x – 3 = 0$ are the x-values where the graph $y = x^2 – 2x – 3$ is equal to 0.

This happens where the graph crosses the x-axis (the line where y=0).

Looking at the graph, the curve crosses the x-axis at two points:

$x = -1$ and $x = 3$.

Answer (b): -1 and 3 (B2 for both, B1 for one correct)