GCSE Edexcel Foundation Paper 3 (2023)

💡 What are we being asked to do?

We need to convert the written words “three thousand one hundred and seven” into a numerical digit format.

Let’s break down the place values:

• “Three thousand” means 3 in the thousands column.
• “One hundred” means 1 in the hundreds column.
• “And seven” means 7 in the units column (and implies 0 in the tens column).

💡 Strategy:

Percent means “per 100”. To convert a fraction to a percentage, we can multiply the fraction by 100.

💡 Alternative Method:

Since the denominator is 10, we can convert it to a denominator of 100 by multiplying top and bottom by 10.

💡 What are we doing?

We are adding four identical terms together. This is the definition of multiplication.

Just like \( 2 + 2 + 2 + 2 = 4 \times 2 \), having \( m + m + m + m \) is the same as \( 4 \times m \).

💡 Knowledge Check:

We need to know how many grams are in a kilogram.

\( 1 \text{ kilogram (kg)} = 1000 \text{ grams (g)} \)

To go from grams (smaller unit) to kilograms (larger unit), we divide by 1000.

💡 Strategy:

When ordering numbers including negatives, imagine a number line.

• Negative numbers are smaller than positive numbers.
• The “bigger” the negative number looks (further from zero), the smaller it actually is.

💡 Method:

The area is the total number of square centimetres inside the shape. Since the shape is on a centimetre grid, we can simply count the squares.

💡 Method:

The perimeter is the distance around the outside of the shape. We start at one corner and walk around the edge, counting the grid units.

💡 Analysis:

The spinner has 4 sections. The numbers are: 1, 2, 2, 3.

The number 2 appears on 2 out of the 4 sections.

Probability = \( \frac{2}{4} = \frac{1}{2} \)

A probability of \( \frac{1}{2} \) (or 50%) is described as “evens”.

💡 Analysis:

The numbers on the spinner are 1, 2, 2, 3.

Are these numbers less than 4?

• 1 is less than 4.
• 2 is less than 4.
• 3 is less than 4.

Every possible outcome is less than 4.

If an event happens every time (100%), it is “certain”.

💡 Knowledge Check:

The sum of all probabilities for an event must add up to 1.

\( P(\text{not 5}) = 1 – P(5) \)

💡 Identification:

A shape with 4 right angles implies a rectangle or square.

A shape with 4 sides of equal length implies a rhombus or square.

A shape that has BOTH properties is a square.

💡 Identification:

The diagram shows a 3D box shape with rectangular faces.

This is known as a cuboid.

💡 Method:

The median is the middle number. To find it, we must first order the numbers from smallest to largest.

💡 Method:

Range = Biggest value – Smallest value

💡 Features Required:

• Linear scale on the vertical axis (0 to at least 9).
• Labels for the people (Ximena, Martha, Kezia, Tabby).
• Bars of correct heights (7, 9, 1, 5).
• Bars of equal width with gaps between them.

💡 Action: Add the first walking duration to the start time.

Start: 8:30 am

Duration: 1 hour 45 minutes

💡 Action: Add the rest time.

Rest: 15 minutes

💡 Action: Add 85 minutes to the current time.

Convert 85 minutes to hours and minutes first: \( 85 = 60 + 25 \), so 1 hour 25 minutes.

💡 Comparison:

Arrival time: 11:55 am

Target time: 12:00 noon

💡 Strategy:

Let Gabriel’s number be \( x \).

We can translate the English sentences into algebra:

• “Multiplies by 5” becomes \( 5x \)
• “Then adds 7” becomes \( 5x + 7 \)
• “Answer is 72” becomes \( 5x + 7 = 72 \)

💡 Method:

To find the original number, we need to work backwards (inverse operations).

The last thing he did was add 7, so we subtract 7.

The first thing he did was multiply by 5, so we divide by 5.

💡 Definition:

The “modal” grade is the mode—the most common grade. In a pie chart, this corresponds to the largest sector (biggest angle).

💡 Strategy:

We are told that 7 students represents the Distinction sector.

The Distinction sector is 75°.

We need to find the value of 360° (the whole circle/total students).

We can use a unitary method (find what 1 student represents) or a scaling method.

💡 Analysis:

The “stop” at the cafe is represented by the horizontal section of the line (where distance doesn’t change).

This horizontal section starts at 12:30 and distance 10 miles.

She started at 12:00.

💡 Strategy:

1. Stayed at beach: Horizontal line. Arrived at 13:30. Stayed 1.5 hours (1 hour 30 mins).
\( 13:30 + 01:30 = 15:00 \).
Draw horizontal line from 13:30 to 15:00 at distance 35.

2. Drove home: Diagonal line back to 0 distance.
Arrived home at 16:00.
Draw line from (15:00, 35) to (16:00, 0).

💡 Formula:

💡 Strategy:

We need to find the unit cost. Since the answer requires pence, let’s convert the money to pence first.

£6 = 600p

Cost per box = Total Cost ÷ Quantity

💡 Strategy:

£10 = 1000p

Cost per bag = Total Cost ÷ Quantity

💡 Strategy:

Subtract the cheaper price from the more expensive price.

💡 Requirement:

Round to 1 decimal place.

💡 Analysis:

Ratio R : G = 1 : 4.

We are told there are 35 red beads. This corresponds to the “1” part of the ratio.

💡 Method 1: Find Green then Total

Green = 4 parts = \( 4 \times 35 = 140 \).

Total = Red + Green = \( 35 + 140 = 175 \).

💡 Method 2: Total Parts

Total ratio parts = \( 1 + 4 = 5 \) parts.

Total beads = \( 5 \times 35 \).

💡 Observation:

• The shape has not changed size, so it is not an enlargement.
• The shape has changed orientation (it’s turned), so it is not a translation.
• It looks like it has turned around a point, which suggests a Rotation.

💡 Analysis:

Compare corresponding sides. The top edge of A (horizontal) becomes the left edge of B (vertical). This is a 90° turn.

Moving from Quadrant 4 (bottom right) to Quadrant 1 (top right) is moving Anti-Clockwise.

💡 Method:

We can use tracing paper or coordinate geometry. Let’s look at the coordinates.

Point on A: \( (2, -2) \)

Corresponding Point on B: \( (2, 2) \)

Point on A: \( (5, -2) \)

Corresponding Point on B: \( (2, 5) \)

The rule \( (x, y) \rightarrow (-y, x) \) describes a rotation of 90° anti-clockwise about the origin (0,0).

Let’s check: \( (2, -2) \rightarrow (-(-2), 2) = (2, 2) \). This matches.

💡 Scale:

1 cm represents 10 km.

We need to represent 55 km.

💡 Bearings:

Bearings are always measured clockwise from North (000°).

We need to measure 65° clockwise from the North line at T.

💡 Method:

Multiply the term outside (4) by each term inside the bracket.

💡 Method:

Add 12 to both sides to remove the -12.

💡 Method:

Divide by 8.

💡 Alternative Method (Division first):

Divide both sides by 4 first: \( 2x – 3 = 5 \). Then add 3: \( 2x = 8 \). Then divide by 2: \( x = 4 \).

💡 Strategy:

Since it is Simple Interest, the amount of interest earned is the same every year.

Total interest is £450 over 6 years.

💡 Method:

We need to find what percentage £75 is of the original £3000 investment.

💡 Rule: \( (a^m)^n = a^{m \times n} \)

When raising a power to another power, multiply the indices.

💡 Rule: \( a^m \times a^n = a^{m + n} \)

When multiplying terms with the same base, add the indices.

💡 Method:

Multiply the term outside the bracket by every term inside.

💡 Strategy:

1. Find the number of people drinking coffee (68% of 800).
2. Find the total cups needed (2 per person).
3. Find the total mass of coffee (10.6 g per cup).

💡 Logic:

The original assumption was 68%. The actual value is 72%.

Since the percentage of people is higher (72 > 68), more people will drink coffee.

Therefore, he will need more coffee.

💡 Reasoning:

We need to relate the angles on the bottom line to the top line. Lines \( BCD \) and \( EFG \) are parallel.

Angles \( \angle BCF \) and \( \angle EFC \) are co-interior (allied) angles, which add up to 180°.

💡 Reasoning:

Angles on a straight line add up to 180°. The angle \( 110^\circ \) is outside the triangle on the straight line ADG? No, the diagram shows the angle between line AD and the extension of BCD.

Wait, let’s verify position. If the angle marked 110° is \( \angle CDG \) (exterior to the parallel line segment CD), then \( \angle ADC \) and \( \angle CDG \) are on a straight line.

However, if the angle is \( \angle AD(\text{right}) \), we simply use angles on a straight line.

💡 Reasoning:

Angles in a triangle add up to 180°.

💡 Proof:

We have found the angles of triangle \( ACD \) to be:

• \( \angle ACD = 55^\circ \)
• \( \angle CAD = 55^\circ \)
• \( \angle ADC = 70^\circ \)

Since two angles are equal (\( 55^\circ \)), the triangle is isosceles.

💡 Logic (Inverse Proportion):

If you have fewer pumps, it will take more time. We calculate the total “pump-hours” needed.

Total Work = Pumps \( \times \) Hours.

💡 Logic:

We have 70 hours of work for 1 pump. We share this among 4 pumps.

💡 Method:

The Highest Common Factor consists of the prime factors that are in both numbers, raised to the lowest power appearing in either.

• Factor 2: In A ($2^2$) but not B ($2^0$). Min power is 0. (Ignore).
• Factor 3: In A ($3^4$) and B ($3^2$). Min power is 2. Use \( 3^2 \).
• Factor 7: In A ($7^1$) and B ($7^2$). Min power is 1. Use \( 7 \).

💡 Method:

The Lowest Common Multiple includes all prime factors involved, raised to the highest power appearing in either.

• Factor 2: Highest is \( 2^2 \).
• Factor 3: Highest is \( 3^4 \).
• Factor 7: Highest is \( 7^2 \).

💡 Method:

Time = Volume ÷ Rate

💡 Knowledge Check:

• 60 seconds in a minute.
• 60 minutes in an hour.
• 24 hours in a day.

Seconds in a day = \( 60 \times 60 \times 24 = 86\,400 \).

💡 Requirement:

Round to the nearest day.

65.36 rounds down to 65.

💡 Method:

The turning point is the minimum point (the lowest point) on the curve.

Looking at the graph, the lowest point aligns with:

• \( x = 1 \) on the horizontal axis.
• \( y = -3 \) on the vertical axis.

💡 Method:

The roots of the equation \( x^2 – 2x – 2 = 0 \) are the values of \( x \) where the graph crosses the x-axis (where \( y = 0 \)).

Looking at the graph, it crosses the x-axis at two points.

• Between -0.6 and -0.8 (Negative root)
• Between 2.6 and 2.8 (Positive root)

The exact values are \( 1 \pm \sqrt{3} \approx -0.73, 2.73 \).

💡 Formula:

Rearranging for Mass:

💡 Method:

To get the ratio in the form \( 1 : n \), we must divide both sides by the left-hand number.

💡 Strategy:

Convert all numbers to ordinary numbers (decimals) to compare them easily.