GCSE Mathematics – Edexcel Foundation Paper 2 (Calculator) – June 2018

What are we being asked to find?

We need to convert a fraction into a percentage. Remember that “percent” literally means “out of 100”. So, we want to find out what fraction out of 100 this is equal to.

Why we do this:

Since the denominator is currently 50, we can easily turn it into 100 by multiplying it by 2. Whatever we do to the bottom of the fraction, we must also do to the top to keep the fraction equivalent.

What this tells us:

The fraction \(\frac{8}{100}\) is exactly the definition of 8%.

What are we being asked to find?

We need to round the number so it only has one digit after the decimal point.

Why we do this:

Identify the 1st decimal place (the tenths column). Then look at the next digit to the right (the hundredths column) to decide whether to round up or keep the digit the same. If the next digit is 5 or more, we round up.

What this tells us:

1.59 is much closer to 1.6 than it is to 1.5, so 1.6 is the correct rounded value.

What are we being asked to find?

We need to calculate 3 to the power of 5. The small 5 (the index or power) tells us how many times to multiply the base number (3) by itself.

Why we do this:

We can use a calculator since this is a Calculator Paper, but it’s good to understand the manual multiplication behind it.

What this tells us:

Our manual calculation matches the calculator result, so we can be confident the answer is correct.

What are we being asked to find?

We need to create a number with exactly 6 digits. We must place the digit 4 in the “thousands” place value position, and we cannot use the digit 4 anywhere else in the number.

Why we do this:

Let’s map out the 6 place values: Hundred-Thousands, Ten-Thousands, Thousands, Hundreds, Tens, and Units.

What this tells us:

There are many correct answers! As long as the number looks like XX4XXX (where X is any number but 4, and the first X is not 0), you will get the mark. e.g. 124356, 984000, etc.

Why we do this:

There are 10 millimetres (mm) in 1 centimetre (cm). To go from a larger unit (cm) to a smaller unit (mm), we need more of them, so we multiply by the conversion factor 10.

Why we do this:

There are 1000 millilitres (ml) in 1 litre (L). To go from a smaller unit (ml) to a larger unit (L), we need fewer of them, so we divide by 1000.

Why we do this:

The prefix “kilo” means 1000. So there are 1000 grams (g) in 1 kilogram (kg). To go from kg to g, we multiply by 1000.

What this tells us:

Checking mentally: 0.32kg is about a third of a kilogram. A kilogram is 1000g, and a third of 1000 is about 333, so 320g is the correct answer.

What are we being asked to find?

We need to find two numbers that satisfy three rules at the same time[cite: 193, 195]:

• Rule 1: It must be an odd number (ends in 1, 3, 5, 7, or 9).
• Rule 2: It must be a factor of 36 (a number that divides exactly into 36 with no remainder).
• Rule 3: It must be a multiple of 3 (in the 3 times table).

Why we do this:

The best strategy is to start by listing all the numbers that fit Rule 2 (factors of 36), and then eliminate the ones that don’t fit Rules 1 and 3[cite: 101].

What this tells us:

We found exactly two numbers, 3 and 9. Let’s verify: Both are odd, both divide into 36, and both are in the 3 times table. They meet all criteria.

What are we being asked to find?

We need to list all the possible orderings (permutations) for the 3 players: Mohsin (M), Yusuf (Y), and Luke (L). We must ensure we don’t repeat any combinations and we don’t miss any[cite: 101].

Why we do this:

If we list them randomly, we might miss one or write one twice. The best method is to lock in the person who comes First, and then list the combinations for Second and Third. Then switch who comes First, and repeat.

What this tells us:

There are exactly 6 possible outcomes. This makes sense mathematically because \(3 \times 2 \times 1 = 6\).

What are we being asked to find?

We need to calculate the total cost for exactly 30 of each of the four items. The items are sold in different batch sizes, so we must first figure out how many ‘batches’ of each item Neil needs to buy to get exactly 30. Finally, we must ensure all prices are in the same units (e.g., pounds £) before adding them up[cite: 102].

Why we do this:

We’ll tackle each item one by one. We divide the total needed (30) by the number in a batch to find the number of batches required, then multiply by the batch price[cite: 102].

Why we do this:

Now that we have the individual totals for all four items in pounds (£), we simply add them together[cite: 102].

What this tells us:

The total amount Neil spends on stationery for 30 sets is £19.85. We can quickly estimate to check: £4 + £1 + £4 + £11 = £20. Our exact answer of £19.85 is very close, meaning it is correct.

Why we do this:

The relationship between Speed, Distance, and Time is defined by the formula: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}}\)[cite: 102]. We have the Distance (186 miles) and the Time (3 hours).

Why we do this:

By rearranging our formula, we get: \(\text{Distance} = \text{Speed} \times \text{Time}\)[cite: 102]. We have Sarah’s Speed (58 mph) and her Time (4 hours).

What this tells us:

Emily drove at 62 mph and covered 186 miles in 3 hours. Sarah drove slightly slower at 58 mph, but for an extra hour, meaning she covered a further distance overall (232 miles).

Why we do this:

A prime number is a number that has exactly two distinct factors: 1 and itself. We need to check all whole numbers from 21 up to 29 and eliminate any number that can be divided by something else.

Why we do this:

We need to think about the definition of an “even” number. An even number is any number that can be divided exactly by 2[cite: 103]. If it’s larger than 2, it will have at least three factors: 1, 2, and itself.

What this tells us:

Because all other even numbers divide by 2, 2 is truly unique as the only even prime number in all of mathematics.

Why we do this:

We have three identical unknown values added together. We can simplify this by grouping them into a single term, then dividing to find the value of one \(x\).

Why we do this:

The equation says “\(y\) divided by 4 equals 3”. To undo a division, we perform the inverse operation, which is multiplication.

Why we do this:

We need to isolate \(f\). We do this using inverse operations in reverse BIDMAS order. First, we remove the \(+7\) by subtracting 7, then we deal with the multiplication by dividing.

What this tells us:

We can check our answer by substituting \(f=5.5\) back into the original equation: \(2 \times 5.5 = 11\), and \(11 + 7 = 18\). It works perfectly!

What are we being asked to find?

A full pie chart represents all the fans in a full \(360^\circ\) circle. We need to find the total number of fans, and then determine how many degrees on the pie chart represents a single fan.

Why we do this:

First, find the total frequency (total fans). Then divide the \(360^\circ\) of the circle by this total to find the “multiplier” – the angle for just 1 fan.

Why we do this:

Now we multiply the number of fans for each snack by \(10^\circ\) to find out how large to draw each sector using a protractor.

What this tells us:

Check the total: \(110 + 170 + 80 = 360^\circ\). Since it adds up perfectly, our angles are correct. We now draw these on the pie chart using a protractor and label each sector.

What are we being asked to find?

We need to find the probability of Laura not winning. There are two ways to do this: find the number of losing tickets and divide by the total, OR find the probability of winning and subtract it from 1 (the total probability).

Why we do this:

Let’s use the first method. Any ticket Laura didn’t buy is a ticket that causes her to lose. So, we find the number of tickets someone else bought.

What this tells us:

The fraction \(\frac{338}{350}\) is the correct probability. You do not strictly need to simplify it for full marks, but it can be simplified to \(\frac{169}{175}\). Converting it to a percentage gives approximately 96.6%.

What are we being asked to find?

We need to fill in all the blank ovals. The key rule of a frequency tree is that the sum of the numbers on the branches splitting off from a node must equal the number in that node.

Why we do this:

We use the text to place the known values, and then subtraction to find the missing ones.

What this tells us:

We can check everything: Males (4+18=22), Females (7+16=23). Total (22+23=45). Total right-handed (18+16=34). All conditions from the question are perfectly met!

Why we do this:

We look at the marks on the triangle lines. The small tick marks on lines AC and AB tell us those two sides are the equal ones. The “base angles” of an isosceles triangle are located where the two equal sides meet the third side (the base). Here, the equal angles are at A and B, not B and C.

Why we do this:

We need to check William’s geometry definitions. He claims angle DEG and angle EGH are “corresponding”. Corresponding angles form an “F” shape. These two angles form a “Z” shape across the transversal line.

What this tells us:

William’s math and logic were fine to find the answer, but in geometry exams, using the exact correct terminology (alternate instead of corresponding) is strictly required to earn the reasoning marks.

What are we being asked to find?

We need to find the missing “Frequency” for the bags that contain exactly 19 buttons. The “Frequency” tells us how many bags there are of each size. The total number of all individual buttons combined is 320.

Why we do this:

By multiplying the number of buttons in a bag by the frequency of that bag, we can find out how many buttons are accounted for in the known rows.

Why we do this:

Subtract the known buttons from the grand total (320) to find how many buttons must be in the “19” bags. Then divide by 19 to find out how many bags that is.

What this tells us:

There are exactly 5 bags that contain 19 buttons. If we plug this back into the table, the total buttons will perfectly equal 320.

What are we being asked to find?

We need to figure out how many times Lucas can make the recipe (which is for 30 biscuits) based on the ingredients he has in his cupboard. The ingredient that runs out first will limit the total number of biscuits he can make.

Why we do this:

By dividing the amount he has by the amount needed for one recipe batch, we find out how many batches he can make with each ingredient.

Why we do this:

Lucas can only make as many batches as his most limited ingredient allows. Looking at the numbers, he only has enough chocolate chips for 3 batches. Once he makes 3 batches, he runs out of chocolate chips, so he must stop.

What this tells us:

Even though he has enough sugar to make over 9 batches, he can only make 90 biscuits in total because he will have used up all 225g of his chocolate chips.

What are we being asked to find?

We need to name the exact type of transformation that turned Shape A into Shape B, and provide the specific geometric detail that defines it (e.g., center of rotation, line of reflection, or vector of translation).

Why we do this:

First, we look at the orientation. Shape B is “flipped” backwards compared to Shape A. A flipped shape always indicates a reflection.

What this tells us:

To get full marks, we must state both the transformation type (“reflection”) AND the line (“in the y-axis”). If you give more than one transformation (like saying it was translated AND reflected), you will lose marks.

What are we being asked to find?

To find the total cost of the fence, we first need to know exactly how long the fence is. The fence goes completely around the “edge of the field”, which means we need the full perimeter: the curved arc AND the straight straight line at the bottom.

Why we do this:

The fence costs £29.86 for every metre. We multiply our total length by this cost. We should keep the unrounded number in our calculator to maintain accuracy.

Why we do this:

Jim charges £180 per day, and he works for 3 days. We calculate this labour cost and add it to the materials cost.

What this tells us:

Since this is money, we round our final answer to 2 decimal places (pence).

Why we do this:

When multiplying algebraic terms with the same base (the letter \(m\)), the rule of indices says we add the powers together.

Why we do this:

When a whole bracket is raised to a power, we must apply that power to every single item inside the bracket. Remember that \(5\) is \(5^1\) and \(n\) is \(n^1\).

Why we do this:

We handle this in three separate pieces: the regular numbers, the \(q\) terms, and the \(r\) terms. When dividing terms with the same base, the index rule is to subtract the powers.

Why we do this:

The LCM is the smallest number that is in both the 40 times table and the 56 times table. We can find this by listing out the multiples of both numbers until we find a match.

Why we do this:

The numbers A and B are already given in prime factor form. To find the HCF, we look for the common prime factors in both numbers and take the lowest power of each.

What this tells us:

The largest whole number that divides perfectly into both A (120) and B (300) is 60.

What are we being asked to find?

The equation of a straight line is written in the form \(y = mx + c\), where \(m\) is the gradient (steepness) and \(c\) is the y-intercept (where the line crosses the y-axis).

Why we do this:

The y-intercept is the easiest part to find. We just look at the graph to see exactly where the line crosses the vertical y-axis.

Why we do this:

Gradient is the “change in \(y\)” divided by the “change in \(x\)”. We need to pick two easy-to-read points on the line to calculate this.

What this tells us:

The gradient is 3, which means for every 1 square we move right, the line goes up 3 squares. Combining this with the intercept gives us the final equation.

What are we being asked to find?

We first need to work out the exact full price of the van including the VAT. “VAT at 20%” means we add 20% to the base price.

Why we do this:

We know how much she pays monthly, but we need the total of all 12 payments to build our ratio. Once we know how much she paid via monthly payments, whatever is left from the Total Cost must have been her initial deposit.

Why we do this:

The question asks for the ratio of “deposit” to “total of the 12 payments”. We put the numbers in that order, then divide them both by common factors to simplify the ratio completely.

What this tells us:

For every £3 Raya paid in deposit, she paid £5 in monthly payments.

Why we do this:

To find the missing \(y\) values, substitute each given \(x\) value into the equation \(y = x^2 – x – 6\). Be very careful with negative numbers squared.

Why we do this:

We take each pair of \((x, y)\) coordinates from our completed table and plot them as points on the grid. Because there’s an \(x^2\) term, this is a quadratic equation, which means it will form a smooth U-shaped curve (a parabola).

Why we do this:

We are asked to solve \(x^2 – x – 6 = -2\). Since our drawn curve represents \(y = x^2 – x – 6\), we need to find where the curve crosses the horizontal line \(y = -2\). At those intersection points, we read down to the x-axis to get our solutions.

What this tells us:

Graphical solutions are estimates. The examiners will accept any answer in the range of 2.5 to 2.7, and -1.5 to -1.7[cite: 108].

What are we being asked to find?

We need to test Helen’s claim about a percentage decrease. First, we must use the given formula to calculate the exact pressure before the changes, and the exact pressure after the changes.

Why we do this:

Helen claims the decrease is “less than 20%”. We can test this by calculating exactly what a 20% decrease from the original pressure would be, and comparing it to our actual new pressure.

What this tells us:

Because the pressure dropped further than the 20% mark, Helen’s statement (“decreases by less than 20%”) is false.

What are we being asked to find?

The volume of any prism is calculated by finding the Area of the cross-section and multiplying it by the length. The cross-section here is the right-angled triangle. We know its height (7.2 cm) and its length (18 cm), but we don’t know the base of the triangle to find its area.

Why we do this:

To find the missing base of a right-angled triangle when we know the other two sides, we use Pythagoras’ theorem: \(a^2 + b^2 = c^2\). Here, the longest side (the hypotenuse, \(c\)) is 8.4 cm.

Why we do this:

Now that we have the base (\(4.32666…\)) and the height (\(7.2\)), we can find the area of the triangle (\(\frac{1}{2} \times \text{base} \times \text{height}\)). Finally, we multiply that area by the length of the prism (18) to find the full volume.

What this tells us:

The exact volume is \(280.367…\) cm³. The question asks for 3 significant figures. The first three important digits are 2, 8, and 0. The next digit is a 3, which is less than 5, so we round down (keep it as 280).