GCSE Mathematics 2024 – Edexcel Foundation Paper 1 (Non-Calculator)

Concept: The angles meet at a single point \( O \). The sum of all angles around a point is a full circle.

Rule: Angles around a point add up to \( 360^\circ \).

We have two known angles: \( 220^\circ \) and \( 90^\circ \) (indicated by the square symbol).

We need to subtract these from \( 360^\circ \) to find \( x \).

Method: Follow the machine steps from left to right.

Input \( \rightarrow \times 2 \rightarrow – 10 \rightarrow \) Output

Method: To find the input, we work backwards from the output and do the inverse operations.

• Inverse of \( – 10 \) is \( + 10 \)
• Inverse of \( \times 2 \) is \( \div 2 \)

Method: Let the input be \( x \). The output is also \( x \).

We can write this as an equation based on the machine:

Input \( \times 2 – 10 = \) Output

Order Matters: The question asks for “cows to sheep”.

Cows : Sheep

How: Find the highest common factor (HCF) that divides both numbers.

Both divide by 12.

Rule: A negative number divided by a negative number gives a positive result.

Meaning: \( 2^5 \) means multiplying 2 by itself 5 times.

Trial: We need the result to be 2. Let’s try placing brackets around addition first, as usually division happens before addition (BIDMAS).

If we bracket \( (3 + 2) \):

Why: The perimeter is the total distance around the outside of the shape. We add all three side lengths together.

Why: The question states the rectangle’s perimeter is a “quarter” of the triangle’s perimeter.

Method: The perimeter of a rectangle is \( 2 \times (\text{length} + \text{width}) \) or just adding all 4 sides. We know the width is \( 4 \text{ cm} \).

Two widths = \( 4 + 4 = 8 \text{ cm} \).

The remaining perimeter must be the two lengths.

Reasoning: The wallet contains £10 and £20 notes. Both £10 and £20 are greater than £5. Therefore, it is certain he picks a note worth more than £5.

Step 1: Split the value. The total value is 40p. The value is split equally between 1p coins and 2p coins.

Value of 1p coins = \( 40 \div 2 = 20\text{p} \)

Value of 2p coins = \( 40 \div 2 = 20\text{p} \)

Formula: \( \text{Probability} = \frac{\text{Number of desired outcomes}}{\text{Total number of outcomes}} \)

Method: We will multiply \( 273 \) by \( 4 \), then by \( 50 \), and add the results.

Definition: The median is the middle value. Since there are \( 15 \) adults, we use the formula \( \frac{n+1}{2} \) to find the position.

Position = \( \frac{15+1}{2} = 8 \text{th} \) value.

Formula: Range = Highest Value – Lowest Value.

Requirement: Compare the medians. We need to say who was faster (took less time).

• Adults Median: \( 64 \) mins
• Children Median: \( 75 \) mins

Why: To compare values, we need to compare like with like. We can either:

• Find the cost of 1 battery (Unit Cost) for each pack.
• Or find the cost for a common number of batteries (e.g., 24).

Let’s find the cost per battery (Unit Cost) as it’s a standard method.

Compare: 45p vs 40p vs 50p.

The lowest cost per battery is 40p.

Method: Multiply the term outside the bracket (\( 2 \)) by each term inside.

Method: Add 10 to both sides to remove the -10.

Method: Divide by 8. Note that 28 doesn’t divide by 8 perfectly into a whole number, so we expect a decimal or fraction.

Rule: Any non-zero number raised to the power of 0 is always 1.

\( a^0 = 1 \)

Observation:

• Is it rotated? No, the orientation is identical.
• Is it a reflection? No, it hasn’t flipped.
• Is it resized? No, same size.

Conclusion: It has simply moved position. This is a Translation.

Method: Pick one corner on A and find the path to the matching corner on B.

Let’s use the top-left corner (the right angle).

• Triangle A corner: \( (-3, 3) \)
• Triangle B corner: \( (2, -1) \)

Why: In an arithmetic sequence, the difference between consecutive terms is constant.

Method: Compare our sequence to the 4 times table.

Method: Convert mixed numbers to improper fractions first, then find a common denominator.

Check his conversion: \( \frac{35}{24} \rightarrow 1 \frac{9}{24} \). Is this correct?

\( 35 – 24 = 11 \). The remainder is 11, not 9.

Method: Split the compound shape into two rectangles.

Option A: Split vertically.

• Rectangle 1 (Left): Height \( 8\text{m} \), Width \( 6\text{m} \) (Wait, if bottom is 6m and top is 10m…). Let’s re-read diagram logic.
• If Left is 8m and Right is 5m, the missing vertical is 3m.
• If Top is 10m and the shape closes… usually the bottom width is derived. If the label “6m” is on the bottom edge, then the shape is likely an L-shape where the bottom leg is 6m wide?
• Let’s assume standard splitting: Top rectangle \( 10 \times 5 \) and bottom rectangle?
• Let’s look at the labels again. Top=10, Right=5. Left=8. Bottom=6.
• Area = Area of large rectangle minus missing piece? Or sum of two parts.
• Let’s split horizontally at the 5m mark.
• Top Rectangle: Width \( 10\text{m} \), Height \( 5\text{m} \). Area = \( 50\text{m}^2 \).
• Bottom Rectangle: Total height is \( 8\text{m} \), top part used \( 5\text{m} \), so remaining height is \( 3\text{m} \). The label says \( 6\text{m} \) on bottom. So area = \( 6 \times 3 = 18\text{m}^2 \).
• Total Area: \( 50 + 18 = 68\text{m}^2 \).

Rule: 1 litre covers \( 10\text{m}^2 \).

Compare what she has vs what she needs.

Scenario: Paint covers MORE area (\( 11\text{m}^2 \)) per litre.

This means the paint is more efficient, so she will need LESS paint than calculated in part (a).

Definition: \( P’ \) means the “complement of P”. It includes everything NOT inside the circle P.

Definition: \( P \cup Q \) means “P union Q”, or everything inside EITHER circle P OR circle Q (or both).

Rule: To estimate, we round numbers to 1 significant figure.

Analyze the rounding:

• We rounded \( 513 \) down to \( 500 \).
• We rounded \( 0.81 \) down to \( 0.8 \).

Since we rounded both numbers down, our result will be lower than the actual cost.

Step 1: Find the y-intercept (\( c \)).

Look where the line crosses the vertical y-axis. It crosses at \( 3 \).

So, \( c = 3 \).

Rule: Parallel lines have the same gradient.

The line \( y = 5x \) has a gradient of 5.

Any line with \( y = 5x + c \) is correct (as long as \( c \) is any number).

Method: Find \( \frac{3}{8} \) of 400.

We have two separate ratios for the empty jars:

• Small : Medium = \( 3 : 4 \)
• Medium : Large = \( 1 : 2 \)

The “Medium” part must be the same number to combine them. Currently, it is 4 in the first ratio and 1 in the second.

Multiply the second ratio by 4:

Medium : Large = \( 4 : 8 \)

Total parts in ratio: \( 3 + 4 + 8 = 15 \) parts.

Value of one part: Total empty jars (\( 150 \)) divided by 15.

Question asks: Percentage of Kasim’s jars (all 400) that are empty small jars (30).

Logic: If the price is reduced by \( 30\% \), the sale price represents \( 100\% – 30\% = 70\% \) of the original price.

So, \( 70\% = £280 \).

Method: Multiply every term by 2 to get rid of the \( \frac{1}{2} \). This makes it easier to solve.

Add \( x \) to both sides.