GCSE Mathematics – Edexcel Foundation Paper 3 (June 2018)

What are we being asked to find?

We need to convert a fraction into a decimal number. A fraction out of 10 simply tells us the value of the “tenths” column after the decimal point.

Why we do this:

The first column after a decimal point is the tenths column. Because our fraction is literally “9 tenths”, we just put a 9 in that exact spot.

What this tells us:

The fraction \(\frac{9}{10}\) is exactly equivalent to the decimal 0.9.

What are we being asked to find?

We need to change a decimal into a percentage. “Percent” literally means “out of 100”.

Why we do this:

To convert any decimal or fraction to a percentage, we multiply it by 100. This is because 1 whole unit equals 100%. Multiplying by 100 shifts the digits two places to the left (or moves the decimal point two places to the right).

What this tells us:

0.3 represents 3 tenths, which is the same as 30 hundredths, or 30%.

What are we being asked to find?

We need to round 2538 to the nearest multiple of 100. Is it closer to 2500 or 2600?

Why we do this:

To round to the nearest hundred, we locate the digit in the hundreds column, then look at the digit directly to its right (the tens column) to decide whether to round up or stay the same.

What this tells us:

2538 is closer to 2500 than to 2600, so 2500 is the correct rounded value.

What are we being asked to find?

We need to figure out the pattern of the numbers, then use that pattern to find the very next number (5th term), write down the rule, and finally jump ahead to find the 10th term.

Why we do this:

To find the next term, let’s see how much the sequence increases from one term to the next.

What this tells us:

This is a linear (or arithmetic) sequence because it has a constant common difference of 7.

Why we do this:

We need the 10th term. We can either continue adding 7 until we reach the 10th position, or we can use an \(n\)th term formula. Let’s show both methods!

What this tells us:

Both methods give us the same reliable result. The 10th number in the pattern is 65.

Why we do this:

To make the largest 3-digit number possible, we need to pick the three largest digits from {7, 3, 4, 9} and place the biggest ones in the highest place value positions (the Hundreds column, then the Tens column, then the Units).

Why we do this:

We are creating an addition sum of two 2-digit numbers: [Tens][Units] + [Tens][Units].

To make the total as small as possible, the digits in the Tens columns must be the smallest available numbers, because the Tens column affects the total value much more than the Units column.

What this tells us:

Place value strategy is key: biggest digits at the front for maximum size, smallest digits at the front for minimum sums.

What are we being asked to find?

We need to list every whole number that divides exactly into 30 without leaving a remainder. These numbers are called “factors”.

Why we do this:

It is best to find factors in pairs (numbers that multiply together to make 30). This ensures we don’t accidentally miss any numbers out. We start from 1 and work our way up.

What this tells us:

Writing them in numerical order makes it easy to read: 1, 2, 3, 5, 6, 10, 15, 30.

What are we being asked to find?

We need to find the number of cousins David has, based on a chain of relationships starting from Nishat’s known number of cousins.

Why we do this:

We start with the person whose value we definitely know (Nishat = 6). Then we use the rules provided to find the next person’s value (Becky), and finally use that to find the last person (David).

What this tells us:

By following the logical chain backwards, we’ve accurately calculated that David must have 24 cousins.

What are we being asked to find?

This is a calculator paper! We need to carefully input these exact expressions into our scientific calculator to find the numeric values. The key is understanding how to use brackets and roots correctly on the calculator.

Why we do this:

We must ensure everything under the square root symbol is multiplied *before* taking the root. Typing it exactly as written on a modern scientific calculator achieves this.

Why we do this:

We must follow the order of operations (BIDMAS/BODMAS). The brackets happen first, then the squaring, then the subtraction. Our calculator handles this perfectly if we use the bracket keys.

What this tells us:

Modern calculators are very smart, but typing the expression exactly as it appears on the paper (including brackets and roots over everything) guarantees accuracy.

What are we being asked to find?

We need to look at the first column of the timetable. We want the time difference between leaving Bury at 0825 and arriving in Manchester at 0905.

Why we do this:

We need to build a chronological timeline of Daniel’s morning. We’ll start at his house, figure out what time he arrives at the bus stop, identify which bus he catches, and then add his walking time in Manchester to find his final arrival time at work.

What this tells us:

Because 09:50 is before 10:00 am, Daniel makes it to work with 10 minutes to spare. A clear “Yes” is needed for full marks!

What are we being asked to find?

We need to calculate Bronwin’s total earnings for the week. Since there are two different rates of pay (weekend vs weekday), we must split her total hours correctly, calculate the pay for each part, add them together, and prove it is under £200.

Why we do this:

We know she worked 8 hours at the weekend. We can subtract this from her 20 total hours to find how many hours she worked during the week. Then, we multiply the hours by the relevant pay rates.

What this tells us:

Bronwin earned exactly £190.40 last week. Because 190.40 is less than 200, we have successfully shown (“proved”) what the question asked.

What are we being asked to find?

We need to create a fraction. The top of the fraction (numerator) must be the increase in the cost, and the bottom of the fraction (denominator) must be last year’s cost.

Why we do this:

First, we need to find out exactly how much the price went up by subtracting last year’s price from this year’s price. Then we can build our fraction.

What this tells us:

The ticket price went up by \(\frac{1}{14}\) of its original price. Writing \(\frac{40}{560}\) is perfectly fine for full marks here, as the question didn’t explicitly demand it in its simplest form.

What are we being asked to find?

We need to find the “perimeter” (the total distance around the outside edge) of the real tennis court in metres. We are given the dimensions of the paper drawing in centimetres and a scale of 1:200.

Why we do this:

The scale 1:200 means that 1 unit on the drawing represents 200 units in real life. So, 1 cm on paper is 200 cm in real life. We multiply our paper measurements by 200 to get the real-life centimetres.

Why we do this:

The final answer must be in metres. There are 100 centimetres in 1 metre, so we divide by 100. Finally, a rectangle has 4 sides, so the perimeter is Length + Width + Length + Width.

What this tells us:

By measuring, scaling up, converting units, and applying the perimeter formula, we find the total distance around the real tennis court is 69.2 m. (Note: You could also find the perimeter in cm first, and convert to metres at the very end!)

What are we being asked to find?

We need to look at three mathematical equations for straight lines and figure out which of the 6 pictures correctly shows each line.

Why we do this:

The equation \(y = 2\) means that no matter what the x-value is, the y-value is always exactly 2. This creates a flat, horizontal line that crosses the y-axis at the number 2.

Why we do this:

The equation \(y = x\) means that every y-coordinate is exactly the same as its x-coordinate. It will pass through points like (0,0), (2,2), and (-2,-2). It is a diagonal line sloping upwards from left to right.

Why we do this:

To make this equation easier to graph, we can rearrange it to make \(y\) the subject: subtract \(x\) from both sides to get \(y = -x + 2\). This tells us the line crosses the y-axis at 2 (y-intercept) and goes downwards (negative gradient).

Alternatively, find intercepts: when \(x = 0\), \(y = 2\). When \(y = 0\), \(x = 2\). The line crosses the axes at (0,2) and (2,0).

What are we being asked to do?

We need to organize the raw list of test marks into a stem and leaf diagram. The “stem” will be the tens digits, and the “leaves” will be the units digits. The leaves MUST be in numerical order, and we MUST include a key explaining how to read it.

Why we do this:

To prove Omar is wrong, we need to calculate the actual probability that a randomly chosen student failed. A failure is any mark strictly LESS than the pass mark of 71.

What this tells us:

Omar might have mistakenly thought 5 people failed, or perhaps he just guessed. Using the data directly proves his fraction is incorrect.

What are we being asked to do?

We need to spot the mistake in Jenny’s arithmetic. We have to explain what rule she broke, not just provide the correct answer.

What are we being asked to do?

We need to explain why Rehan’s method for finding the range is incorrect.

What this tells us:

In maths, definitions and rules matter. You can’t just read left-to-right (Jenny) or take the ends of an unsorted list (Rehan) – you must follow the correct procedures!

What are we being asked to find?

We know exactly how much Alan got (£2.50) and what fraction of the total that represents (\(\frac{1}{8}\)). We need to use this to find out how much Bispah got (who received \(\frac{1}{2}\) of the total).

Why we do this:

There are two great ways to do this: we can either figure out the total amount of money first, or we can compare their fractions directly. Since \(\frac{1}{2}\) is exactly \(4 \times \frac{1}{8}\), comparing fractions is very quick!

Why we do this:

To find the ratio of Alan to Chan, we need to know how much Chan gets. We can do this either by working out Chan’s fraction of the whole, or by working out his actual money value.

What this tells us:

For every £1 Alan receives, Chan receives £3. Working entirely in fractions is often faster than working out the exact monetary values, but both methods work perfectly!

What are we being asked to find?

We need to solve for the unknown variable \(x\). We are given a formula hint: we know the lengths of the base and height of the right-angled triangle in terms of \(x\), and we know the total numerical area. We can use the area of a triangle formula to build an equation.

Why we do this:

The area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\). Here, the base is \(3x\) and the height is \(3x\). We set this equal to 162 and use algebra to work backwards to find \(x\).

What this tells us:

If \(x=6\), then the sides are \(3 \times 6 = 18\) cm. Let’s check: Area = \(0.5 \times 18 \times 18 = 162\). It matches perfectly!

What are we being asked to find?

We are dividing two numbers written in standard form. We can either type this directly into a scientific calculator using the standard form button, or work it out by separating the decimals and the powers of 10.

Why we do this:

When dividing in standard form, you divide the front numbers and you subtract the powers (because \(10^a \div 10^b = 10^{a-b}\)). Finally, ensure the answer is correctly formatted in standard form (\(A \times 10^n\), where \(1 \le A < 10\)).

What this tells us:

Because the number 2.3 is already between 1 and 10, no further adjustments are needed for it to be in proper standard form.

What are we being asked to find?

We need to state the statistical term for the pattern the crosses make on the graph.

Why we do this:

Looking at the graph, as the age (x-axis) increases, the time (y-axis) goes down. A pattern sloping downwards from left to right is called a “negative” correlation.

What are we being asked to do?

We are told Kristina (11 years old, 12 seconds) is an “outlier”. We need to explain why this specific point is given that label.

Why we do this:

If you imagine plotting Kristina’s point on the graph at (11, 12), you would see it sits very far below all the other crosses for 11-year-olds (who are running around 14.5 to 16 seconds). An outlier is a point that doesn’t fit the main pattern.

What are we being asked to do?

We need to state whether Debbie’s prediction is reliable based on the graph, and explain why.

Why we do this:

Debbie is 15. The oldest girls plotted on the graph are under 14 years old. Making predictions outside the range of your actual data is called “extrapolation,” and it is unreliable because we don’t know if the downward trend continues (maybe runners stop getting faster after age 14).

What this tells us:

You can only trust a line of best fit within the limits of the data you actually collected. Anything outside that is a guess!

What are we being asked to do?

“Expand” means multiply out the brackets. “Simplify” means collect all the like terms (put all the \(p\)’s together and all the normal numbers together) once the brackets are gone.

Why we do this:

We multiply everything inside the first bracket by 5, and everything inside the second bracket by \(-2\). Be very careful with the negative signs!

Why we do this:

Now we write out the full expanded expression and combine the matching terms to tidy it up.

What this tells us:

The complex algebraic statement we started with is mathematically identical to the much simpler statement \(9p + 13\).

What are we being asked to do?

Before we can draw a triangle with an “equal area”, we first have to figure out exactly what the area of the given trapezium is. We do this by counting the squares along its parallel sides and its height.

Why we do this:

Now we need a triangle that also has an area of 18 cm². The formula for a triangle’s area is \( \frac{1}{2} \times \text{base} \times \text{height} \).

So, we need \( \frac{1}{2} \times \text{base} \times \text{height} = 18 \). This means the base multiplied by the height must equal 36.

What this tells us:

Different shapes can hold the exact same amount of space (area). As long as your triangle’s base \(\times\) height equals 36, it perfectly matches the trapezium’s area.

What are we being asked to do?

We need to spot two separate mistakes Amir made when filling out his diagram. We do this by checking the two main rules of probability trees:

1. The probabilities on branches coming from a single point must always add up to exactly 1.
2. If the event is independent (like rolling the same dice again), the probability of success doesn’t randomly change from the first throw to the second.

Why we do this:

Let’s check the first set of branches, and then check the second set of branches.

What this tells us:

Always double-check that your branch pairs sum to 1, and ensure you place the correct probability next to its corresponding label.

What are we being asked to do?

We need to find the size of the angle at vertex B. We know two sides of a right-angled triangle, so we must use trigonometry.

Why we do this:

First, label the sides relative to the angle we want (ABC):

• The side opposite the right angle is the Hypotenuse (\(AB = 11\)).
• The side next to our angle is the Adjacent (\(BC = 7\)).

Since we have Adjacent and Hypotenuse, we use CAH: \( \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \).

Why we do this:

We are asked what happens specifically to the value of \(\cos ABC\) (which is the fraction \(\frac{\text{Adj}}{\text{Hyp}}\)), NOT the angle itself.

What this tells us:

In any fraction, if you make the bottom number (denominator) smaller while keeping the top number the same, the overall value of the fraction gets bigger.

What are we being asked to do?

Before we can find the exact number of red counters, we need to know the probability of getting a red counter.

Why we do this:

The sum of all probabilities in a complete set is always 1. We can subtract the known probabilities to find what’s left for red and white combined. Then, we use the clue (“red is twice white”) to split that remainder.

Why we do this:

We know that a probability of 0.45 corresponds to exactly 18 physical counters (the blue ones). We can use this to find the total number of counters in the bag, and then take our Red probability (0.2) of that total to find the number of red counters.

Why we do this:

A probability of 0.5 is exactly the same as the fraction \(\frac{1}{2}\). We need to explain why taking half of the total items must mean the total is even.

What are we being asked to do?

We need to “solve” the equation, which means using algebra to find the exact value of \(x\) that makes both sides equal.

Why we do this:

Fractions make algebra tricky. The whole left side is divided by 2. To get rid of this divide-by-two, we must do the opposite: multiply EVERYTHING on both sides by 2.

Why we do this:

We want all the \(x\)’s on one side of the equals sign, and all the plain numbers on the other side. Let’s move the \(x\)’s to the right (to keep them positive) and the numbers to the left.

What this tells us:

The only number in the universe that makes \(\frac{5 – x}{2}\) exactly equal to \(2x – 7\) is \(3.8\) (or \(\frac{19}{5}\)).

What are we being asked to do?

To find the missing angles inside a polygon, we first need to know what all the angles should add up to. A pentagon has 5 sides.

Why we do this:

The formula for the sum of interior angles of any polygon is \((n – 2) \times 180^\circ\), where \(n\) is the number of sides. This works because you can split a 5-sided shape into 3 triangles.

Why we do this:

We know three angles: \(90^\circ\) (the square symbol at D), \(115^\circ\), and \(125^\circ\). We don’t know angles ABC and BCD, but we know BCD is twice as big as ABC. Using algebra makes this easy to solve: let angle ABC be \(x\), so angle BCD is \(2x\).

Why we do this:

The question specifically asks for angle BCD, not ABC. Since BCD is \(2x\), we must multiply our answer by 2.

What this tells us:

Let’s do a final check: \( 90 + 115 + 125 + 70 + 140 = 540 \). The maths works out perfectly!

What are we being asked to do?

Because the triangles are “similar”, it means one is a perfectly magnified version of the other. We need to find the “Scale Factor” (the multiplier) that links them, and use it to find the missing lengths.

Why we do this:

The arcs on the angles show us which corners match. Angle A matches D (1 arc). Angle B matches E (2 arcs). This means the side connecting A and B (8.4 cm) matches the side connecting D and E (12.6 cm).

Why we do this:

We want to find DF (on the large triangle). Its matching side on the small triangle is AC (the side between the 1-arc and 3-arc angles), which is 6.4 cm. To go from small to large, we multiply by the scale factor.

Why we do this:

We want to find CB (on the small triangle). Its matching side on the large triangle is FE (the side between the 3-arc and 2-arc angles), which is 15 cm. To go from large to small, we do the opposite of multiplying: we divide by the scale factor.

What this tells us:

Similar triangles always keep their exact proportions. If one side is 50% longer, all sides are 50% longer.

What are we being asked to do?

“Make \(g\) the subject” means we need to rearrange the formula so that it looks like \( g = \text{everything else} \). We have to peel away the operations surrounding \(g\) one by one, using inverse (opposite) operations.

Why we do this:

The giant square root covers the entire right-hand side. It is the “outermost” layer trapping \(g\). The opposite of square rooting is squaring (power of 2), so we must square both sides of the equation.

Why we do this:

Next, the expression \((g + 6)\) is being divided by 2. The opposite of dividing by 2 is multiplying by 2. After that, we just subtract 6 to leave \(g\) completely alone.

What this tells us:

If we know the value of \(T\), we can now plug it directly into this new formula to instantly find \(g\). We’ve successfully built a “calculator” for \(g\).