GCSE November 2023 Edexcel Foundation Paper 3 (Calculator)

💡 What does “percent” mean?

The word “percent” means “per 100” or “out of 100”. To convert a percentage to a decimal, we divide the number by 100.

🔢 Calculator Method:

Type 35 ÷ 100 =

💡 Strategy:

We need to round to the nearest hundred. Let’s look at the place value of the digits.

• 8 is in the Thousands column.
• 0 is in the Hundreds column (this is the digit we are rounding).
• 6 is in the Tens column (this is the “decider”).
• 1 is in the Units column.

What this tells us:

The 0 becomes a 1, and everything after it becomes zero.

8061 is closer to 8100 than to 8000.

💡 Thinking about the number line:

On a number line, “less than” means further to the left.

For negative numbers, a number is smaller (less) if the digit part looks bigger. For example, -10 is colder (less) than -5.

💡 Strategy:

A fraction is written as: \[ \frac{\text{Number of shaded squares}}{\text{Total number of squares}} \]

Observation: You can also see from the diagram that 2 out of the 7 full columns are shaded, which gives \( \frac{2}{7} \) directly.

💡 Why we do this:

We need to identify the position of the digit relative to the decimal point.

The 9 is in the first column after the decimal point.

This is the tenths column.

The value can be written as:

• 9 tenths
• \( \frac{9}{10} \)
• 0.9

Any of these forms are correct.

💡 The Key: The key tells us that one full circle represents 12 cakes.

Target: We need to represent 24 banana cakes.

💡 Strategy: We know Year 8 sold 150 cakes. We need to calculate the total for Year 7 by interpreting the pictogram rows and adding them up.

💡 Comparison:

Year 7 Total: 141

Year 8 Total: 150

150 is greater than 141.

💡 Strategy: We have two options:

1. Find the total number of lengths needed for 8050 m, then subtract the ones he has already done.
2. Find the distance he has already swum, find the remaining distance, and convert that to lengths.

Let’s use Method 1 as it deals with smaller numbers sooner.

Distance swum = \( 178 \times 25 = 4450 \) m.

Distance remaining = \( 8050 – 4450 = 3600 \) m.

Lengths remaining = \( 3600 \div 25 = 144 \).

The sequence goes down by 6 each time.

Alternatively, the difference is two “steps” of 6, which is \( 2 \times 6 = 12 \).

💡 Strategy: We need to check the properties of numbers in this sequence.

• The numbers are: 97, 91, 85, 79, 73, 67, 61, 55, 49…

Observation 1 (Parity):

All terms are odd numbers. (Odd – Even = Odd).

52 is an even number.

⚠️ Critical Step: The bag is in kilograms (kg), but the meals are in grams (g). We must match them.

1 kg = 1000 g

The shape has 5 straight sides.

A 5-sided polygon is called a Pentagon.

💡 Strategy: We need to count the total number of edges on a pentagonal prism and multiply by the length of one edge.

Counting Edges:

• Front face (Pentagon): 5 edges
• Back face (Pentagon): 5 edges
• Connecting edges (Rectangular faces): 5 edges

Total Edges: \( 5 + 5 + 5 = 15 \)

💡 Strategy: To find a fraction, we need the “part” we are interested in (green) and the “whole” (total counters).

We want the fraction of green counters.

Green parts = 7

💡 Strategy: It’s easier to subtract time if we know the minutes.

\( \frac{1}{4} \) of an hour is \( 60 \div 4 = 15 \) minutes.

So, the match lasted 3 hours and 15 minutes.

💡 Rule: All terms have \( h^3 \), so we can just add/subtract the coefficients (the numbers in front).

💡 Order of Operations: Simplify inside the brackets first.

💡 Strategy: To compare different formats (fractions, percentages, decimals), convert them all to decimals with 3 decimal places.

💡 Analyze the Clues:

1. “64 sold to families” → Top middle circle = 64.
2. “Rest sold to couples” → \( 100 – 64 = 36 \) → Bottom middle circle = 36.
3. “11 of the 18 holidays abroad were sold to couples”
   This tells us two things:
   • Couples/Abroad = 11
   • Total Abroad = 18

💡 Identify the Sample Space:

The question says “One of the holidays sold to a family is chosen”.

So, our denominator is the total number of families: 64.

💡 Identify the Success criteria:

We want “not abroad”, which means UK.

Number of families going to UK: 57.

💡 Strategy: We need to get \( x \) on its own. First, move the \( +9 \) to the other side by subtracting 9.

Since \( x \) is divided by 7, we multiply both sides by 7.

💡 Strategy: To compare, we need the hourly rate for both jobs in the same currency.

We can either convert the UK rate to Dollars OR the Australian rate to Pounds.

Method A: Convert UK rate (£) to Dollars ($)

💡 Using Symmetry: Since the shape has a line of symmetry, the gaps on the left and right of the triangle must be equal.

Left gap = 3.8 cm

Right gap = 3.8 cm

Total width = 14 cm

We are told: “Area of the rectangle is 3.5 times the area of the triangle”.

The front elevation shows the Length and Height.

Counting the squares (1 square = 1 cm):

• Length = 8 cm
• Height = 4 cm

The Plan is the view from the top.

It shows the Length (8 cm) and the Depth (7 cm).

We need to draw a rectangle that is 8 squares wide and 7 squares high.

💡 Rule: \( A \times 10^n \) where \( 1 \le A < 10 \).

Decimal point moves from the end (468000.) to between the 4 and 6 (4.68).

It moves 5 places to the left.

💡 Rule: Negative power means the number is small (less than 1).

Move the decimal point 4 places to the left.

💡 Formula: To find the expected number of times an event happens, we multiply the probability of the event by the number of trials.

💡 Strategy:

1. Find the Midpoint of each group (because we don’t know exact temperatures).
2. Multiply Midpoint by Frequency (\( f \times x \)).
3. Sum these totals and divide by the total frequency (50).

💡 Determining the Median Position:

Total data points = 50.

The median is at position \( \frac{50+1}{2} = 25.5 \), so between the 25th and 26th values.

Counting Cumulative Frequency:

• 1st group: 2 values (Total 2)
• 2nd group: 8 values (Total 10)
• 3rd group: 13 values (Total 23) ← Still not reached 25.5
• 4th group (\( 25 < T \le 30 \)): 21 values (Total 44) ← The 25th and 26th values are in here!

💡 Compare:

Correct Inequality: \( -3 < x \le 4 \)

• Starts at -3.
• End point 4 is inclusive (\( \le \)), so circle should be filled.

Jenna’s Drawing:

• Starts at -2.
• End point 4 has an empty circle (not inclusive).

We need the greatest integer (whole number) that is less than 4.6.

The integers less than 4.6 are 4, 3, 2…

The largest is 4.

💡 Concept: We need a number that both 30 and 24 divide into perfectly.

This is a Lowest Common Multiple (LCM) problem.

We need 120 balloons and 120 pencils.

💡 Logic: If we want to finish faster (fewer hours), we need more machines.

First, calculate the total “machine-hours” needed to do the job.

💡 Formula: \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)

We have 2 full hours and 0.6 of an hour.

To convert 0.6 hours to minutes, multiply by 60.

💡 Strategy: Rearrange the formula: \( \text{Area} = \frac{\text{Force}}{\text{Pressure}} \)

This gives us the area of the base of the cube (the part touching the table).

A cube has 6 identical square faces.

We know the area of one face is 144 cm².

The equation of a straight line is \( y = mx + c \)

• \( m \) is the gradient (slope).
• \( c \) is the y-intercept (where it crosses the y-axis).

Look at the graph. The line crosses the vertical y-axis at 3.

💡 Strategy: Pick two points on the line and check the “Rise over Run”.

Point 1: \( (0, 3) \)

Point 2: \( (1, 1) \)

💡 Key Insight: The diagonal of the square is equal to the diameter of the circle.

We can use Pythagoras’ Theorem to find the diagonal.

💡 Formula: \( C = \pi \times d \)