GCSE June 2019 Edexcel Foundation Paper 2 (Calculator)

💡 What are we being asked?

We need to convert a decimal into a fraction. The digits \(7\) and \(5\) are in the tenths and hundredths columns.

🔍 Calculator Check:

Type 0.75 and press = on your calculator. It will automatically simplify it.

💡 Why we simplify:

It’s best practice to give fractions in their simplest form. We can divide the top and bottom by the same number (25).

💡 Strategy:

Negative numbers are smaller than zero. The “bigger” the negative number looks, the smaller its actual value (further left on the number line).

Our negatives are: \( -3 \) and \( -1 \).

\( -3 \) is colder/lower than \( -1 \), so \( -3 \) is the smallest.

💡 Logic:

After the negatives, we have \( 0 \), then positive numbers increasing in value.

• Smallest: \( -3 \)
• Next: \( -1 \)
• Middle: \( 0 \)
• Next: \( 2 \)
• Largest: \( 4 \)

💡 What is a factor?

A factor is a whole number that divides into another number exactly without leaving a remainder.

Let’s find pairs of numbers that multiply to make 15:

💡 The List:

The factors of 15 are: \(1, 3, 5, 15\).

You can write down any two of these.

💡 The Rule:

There are \( 1000 \) grams in \( 1 \) kilogram.

To convert from grams (small unit) to kilograms (large unit), we divide by 1000.

💡 Quick Tip:

Dividing by 1000 moves the decimal point 3 places to the left.

\( 1756.0 \to 1.756 \)

💡 The Logic:

One million is written as \( 1,000,000 \) (a one followed by six zeros).

So, two million is a two followed by six zeros.

💡 Strategy:

First, we find the cost of 2 coffees, then add the cost of the cake.

💡 Strategy:

Subtract the total cost from the £10 note.

💡 Comparison:

Dave thinks he gets more than £1.50.

He gets £1.60.

Is 1.60 > 1.50? Yes.

💡 Reasoning:

If we have 1 boat, we have \( 1 \times 7 \) people.

If we have 2 boats, we have \( 2 \times 7 \) people.

If we have \( y \) boats, we have \( y \times 7 \) people.

In algebra, we write numbers before letters and remove the multiplication sign.

💡 Rule: Number first, then letters in alphabetical order. Remove multiplication signs.

💡 Rule: When we multiply a number by itself, we use indices (powers).

\( y \) appears 3 times.

💡 Strategy: Cancel out terms that appear on both the top and bottom.

💡 Key Information:

One full circle = 8 records.

💡 We know:

1. Total (Wed + Thu) = 36
2. Thu = 8 times Wed

Let’s use algebra or trial and error.

Let Wednesday = \( x \).
Thursday = \( 8x \).

💡 Conversion to Symbols:

Full circle = 8.

Wednesday (4): Half of 8, so draw a half circle.

Thursday (32): \( 32 \div 8 = 4 \), so draw 4 full circles.

Statement 1: \( 14 \) vs \( 21 \). 14 is smaller.

Statement 2: \( 4+7 = 11 \). \( 103-92 = 11 \). They are equal.

Statement 3: \( 2^2 = 4 \). \( 2 \times 2 = 4 \). They are equal.

Statement 4: \( -3 \) vs \( -5 \). On a thermometer, -3 is warmer (higher) than -5. So -3 is bigger.

💡 Strategy:

Replace the letters \( r \) and \( q \) with their values. Always use brackets when substituting, especially with negative numbers.

\( 7r \) means \( 7 \times r \).

💡 Watch out:

A positive number times a negative number gives a negative result.

\( 3 \times -4 = -12 \)

💡 Strategy:

He arrives at 07:15. The train leaves at 07:22.

Subtract the arrival time from the departure time.

💡 Route Map:

The first column shows the train leaves Brighton at 07 22 and arrives in London at 09 00.

Method: Bridging

• 07:22 to 08:00 = 38 minutes (60 – 22)
• 08:00 to 09:00 = 1 hour

💡 Count the squares across:

Look at the diagram horizontally (left to right). The squares are arranged such that they span a total width of 6 squares.

Since the total length is 12 cm, we can find the side length of one square.

💡 Count the squares down:

Now look at the diagram vertically (top to bottom). The rectangle’s height (labelled width) spans 5 squares.

We know the side of one square is 2 cm.

💡 Strategy:

To get the form \( n : 1 \), the right-hand side must become 1.

We divide both sides of the ratio by the number on the right (2.25).

💡 Check: \( 2.25 \times 2 = 4.5 \). Correct.

💡 Formula: Area = Length × Width

💡 Logic:

The whole garden is 100%.

Flowers are 40%.

So grass is \( 100\% – 40\% = 60\% \).

💡 Calculation:

Find 60% of 5400.

60% as a decimal is 0.6.

💡 Strategy: Convert all values to decimals to compare them easily.

• A: \( 0.33 \)
• B: \( 0.033 \)
• C: \( \frac{1}{3} = 0.3333… \)
• D: \( 30\% = 0.30 \)

💡 Logic:

If probability of Heads = probability of Tails, both must be \( 0.5 \) (or 50%).

For Coin C, Prob(Heads) = \( \frac{1}{3} \).

Prob(Tails) = \( 1 – \frac{1}{3} = \frac{2}{3} \).

💡 Formula: Expected Number = Probability × Number of Trials

For Coin B, Prob(Heads) = 0.033. Trials = 4000.

💡 Strategy:

We know the value for 100 g. We need to find the value for 60 g.

Method: Find 1 g first, then multiply by 60.

\( 1 \text{ g} = 84 \div 100 = 0.84 \text{ cal} \)

💡 Strategy:

We have 150 g of yogurt. This is \( 1.5 \) times 100 g.

So we multiply the calories by \( 1.5 \).

💡 Strategy:

Find the rate per hour or calculate total minutes.

12 hours = \( 12 \times 60 = 720 \) minutes.

Rate: 6 parts per 10 mins = 0.6 parts per minute.

💡 Strategy:

Machine B works for 10 hours = 600 minutes.

It makes 13 parts every 15 minutes.

How many 15-minute blocks in 600 minutes?

\( 600 \div 15 = 40 \).

💡 Scale 1:30: This means 1 cm on the paper represents 30 cm in real life.

To find the drawing length, divide real length by 30.

1. From Point A:

Draw an arc (part of a circle) with centre \( A \) and radius 6 cm.

The region “more than” is outside this arc.

2. From Line BC:

Draw a straight vertical line parallel to \( BC \), exactly 5 cm away to the left.

The region “more than” is to the left of this line.

💡 Final Region:

Shade the area that satisfies BOTH conditions:

• Outside the quarter-circle arc.
• To the left of the vertical line.

(See the shaded area in the diagram above)

💡 Strategy: Treat the \( > \) sign like an equals sign. Get \( n \) on one side.

Subtract \( 11n \) from both sides.

💡 Strategy: Isolate \( x \) in the middle.

We have \( -2 < x + 3 \le 4 \).

Subtract 3 from all three parts.

💡 Rules:

• \( < \) means an open circle (value not included).
• \( \le \) means a filled circle (value included).

Action:

Draw an open circle at \( -5 \).

Draw a filled circle at \( 1 \).

Draw a line connecting them.

💡 Strategy:

Pick values for \( x \) and calculate \( y \) using the formula \( y = 2x – 3 \).

💡 Action:

Plot the points: \( (-2, -7), (0, -3), (2, 1), (4, 5) \).

Draw a straight line through them with a ruler.

(See the graph above)

💡 Strategy:

We know the value for 100 g. We need to find the value for 60 g.

Method: Find 1 g first, then multiply by 60.

\( 1 \text{ g} = 84 \div 100 = 0.84 \text{ cal} \)

💡 Strategy:

We have 150 g of yogurt. This is \( 1.5 \) times 100 g.

So we multiply the calories by \( 1.5 \).

💡 Strategy:

Find the rate per hour or calculate total minutes.

12 hours = \( 12 \times 60 = 720 \) minutes.

Rate: 6 parts per 10 mins = 0.6 parts per minute.

💡 Strategy:

Machine B works for 10 hours = 600 minutes.

It makes 13 parts every 15 minutes.

How many 15-minute blocks in 600 minutes?

\( 600 \div 15 = 40 \).

💡 Scale 1:30: This means 1 cm on the paper represents 30 cm in real life.

To find the drawing length, divide real length by 30.

1. From Point A:

Draw an arc (part of a circle) with centre \( A \) and radius 6 cm.

The region “more than” is outside this arc.

2. From Line BC:

Draw a straight vertical line parallel to \( BC \), exactly 5 cm away to the left.

The region “more than” is to the left of this line.

💡 Final Region:

Shade the area that satisfies BOTH conditions:

• Outside the quarter-circle arc.
• To the left of the vertical line.

(See the shaded area in the diagram above)

💡 Strategy: Treat the \( > \) sign like an equals sign. Get \( n \) on one side.

Subtract \( 11n \) from both sides.

💡 Strategy: Isolate \( x \) in the middle.

We have \( -2 < x + 3 \le 4 \).

Subtract 3 from all three parts.

💡 Rules:

• \( < \) means an open circle (value not included).
• \( \le \) means a filled circle (value included).

Action:

Draw an open circle at \( -5 \).

Draw a filled circle at \( 1 \).

Draw a line connecting them.

💡 Strategy:

Pick values for \( x \) and calculate \( y \) using the formula \( y = 2x – 3 \).

💡 Action:

Plot the points: \( (-2, -7), (0, -3), (2, 1), (4, 5) \).

Draw a straight line through them with a ruler.

(See the graph above)

💡 Strategy:

Find the fraction of the sample that chose Theme Park, then apply it to the whole population.

Sample size = 30.

Theme Park votes = 10.

Fraction = \( \frac{10}{30} = \frac{1}{3} \).

💡 Concept:

For a sample to accurately predict the population, it must be representative and unbiased.

💡 Formula: \( \text{Volume} = \text{Length} \times \text{Width} \times \text{Height} \)

💡 Info: The container is only \( \frac{2}{3} \) full.

We need to find \( \frac{2}{3} \) of the total volume.

💡 Strategy: Divide the total water by the size of one cup (275 ml).

💡 Identify sides:

• Hypotenuse (H): The side opposite the right angle (\( 16 \text{ cm} \)).
• Opposite (O): The side opposite the angle 38° (\( AB \)). This is what we want to find.
• Adjacent (A): The side next to the angle (\( BC \)).

We have \( H \) and want \( O \). We use sin (SOH).

💡 Algebra:

Multiply both sides by 16 to get \( AB \) on its own.

💡 Concept:

The display began with 8.3. This means the number was not rounded, but cut off (truncated) after the 3.

Possible numbers:

• 8.3 (exactly)
• 8.30001
• 8.39999…

Impossible numbers:

• 8.29 (starts with 8.2)
• 8.40 (starts with 8.4)

💡 Strategy: Let’s define the ratio for all four people based on Abby’s shares.

Abby (A) = 2 parts.

Ben (B) = 7 parts.

Chloe (C) gets 1.5 times Abby.

💡 Total Parts:

Add all the parts together.

💡 Goal: Find Ben’s share (7 parts).

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

Move the decimal point so it sits after the first non-zero digit (5).

💡 Rule: The power is positive 3, so multiply by 1000 (move decimal point 3 places to the right).

💡 Rule: In a Fibonacci sequence, the next term is the sum of the previous two terms.

Given: 3, 3, 6, 9, 15…

💡 Strategy: Keep adding the previous two terms to generate the list up to the 6th term.

Term 1: \( a \)

Term 2: \( a \)

Term 3: \( a + a = 2a \)

💡 Strategy: Multiply each number inside the vector \(\mathbf{b}\) by 2.

💡 Strategy: Subtract the top numbers (x-components) and subtract the bottom numbers (y-components).