Edexcel GCSE Mathematics – Foundation Paper 2 (Calculator) – Nov 2017

What are we being asked to find?

We need to convert a fraction out of one hundred into its decimal form.

Why we do this:

Fractions over 100 represent hundredths. The second decimal place to the right of the decimal point is the hundredths column. Since we are using a calculator paper, we can also simply type \(7 \div 100\).

What this tells us:

The fraction \(\frac{7}{100}\) is equivalent to seven hundredths, which is written as \(0.07\).

What are we being asked to find?

We need to find a number from the 6 times table that falls in the range between 40 and 50.

Why we do this:

By listing out the multiples of 6, we can easily spot which ones fall into our target range.

What this tells us:

There are two correct possible answers. You can write down either one.

Why we do this:

When multiplying algebraic terms, we multiply the numbers (coefficients) together, and write the letters (variables) next to each other to show they are multiplied.

Why we do this:

When any number or variable is multiplied by itself, we use index notation (powers). Specifically, something multiplied by itself once is “squared”.

Why we do this:

We first need to simplify the top of the fraction (the numerator) by collecting “like terms” since both have an ‘n’. Then, we divide the result by the bottom (the denominator).

Why we do this:

To figure out how many oranges Ken bought, we first need to find out how much weight is taken up by the apples, bananas, and peaches.

Why we do this:

The total weight is given in kilograms (1.785 kg), but our other values are in grams. We must convert this to grams (multiply by 1000) so units match. Then, we subtract the known fruit weight to find the total weight of the oranges, and divide by 90g to find how many oranges there are.

Why we do this:

We need to test if 15 tomatoes at 75g each will fit into her requested 1 kg limit. Again, 1 kg is 1000g.

Why we do this:

If Jane’s assumption is wrong, we need to think about how tomatoes vary in the real world to see if getting 15 is possible.

Why we do this:

To find the fraction of students who did not walk, we first need to find out *how many* students did not walk, and place that over the total number of students (60).

Why we do this:

A full circle is \(360^{\circ}\), representing all 60 students. We need to find out how many degrees represent one single student, and multiply that by the number of students for each transport method to draw the sectors accurately.

What this tells us:

Using a protractor, measure and draw each angle starting from the given line. The first section ‘Bus’ takes up exactly a quarter of the circle (\(90^{\circ}\)). Continue round from the new line you draw each time.

Why we do this:

A ratio shows how parts are shared out. To find a fraction, we need to know the number of parts one person gets (the numerator) out of the total number of parts available (the denominator).

Why we do this:

Dan gets \(\frac{1}{4}\) of the sweets, which means out of every 4 sweets, Dan gets 1. The rest must belong to Rosie. We need to find Rosie’s share and then write it as a ratio comparing Rosie to Dan.

What this tells us:

Ratios and fractions represent the same relationship. If one person has 1 part and the other has 3 parts (ratio 3:1), the total is 4 parts, so their fractions are \(\frac{3}{4}\) and \(\frac{1}{4}\).

Why we do this:

To see if Steve is right, we must actually find and count the prime numbers in both ranges. Remember: a prime number is a number greater than 1 that only has two factors: 1 and itself.

What this tells us:

Since 4 is greater than 2, there are more prime numbers between 10 and 20. This means Steve’s statement is incorrect.

Why we do this:

When looking for errors in a graph, always check the axes first. Do the numbers go up evenly? Are the labels correct? Is the data plotted properly?

Why we do this:

A “trend” describes the general direction the data is moving over time. Is it going up, down, or staying exactly the same?

Why we do this:

We are given that the physical distance on the original paper was 5.5 cm[cite: 91]. The scale tells us that every 1 cm on the map equals 0.5 km in real life. We multiply the map distance by the scale factor to find the real distance.

Why we do this:

A bearing is an angle measured in a very specific way: it always starts pointing North (\(000^{\circ}\)) from the starting location, and rotates clockwise until it hits the target line. It must also be written as 3 digits. We need the bearing from Backley, so we place our protractor center on Backley’s cross, line up \(0^{\circ}\) with the North line, and read clockwise to the line going to Cremford.

Why we do this:

Shape P is a triangle. The formula for the area of a triangle is \(\frac{1}{2} \times \text{base} \times \text{height}\). Because it’s drawn on a centimetre grid, we can just count the squares to find the lengths of the base and the straight vertical height.

What this tells us:

Quadrilateral Q has two pairs of equal-length adjacent (next to each other) sides. The top two slanted sides are the same length, and the bottom two slanted sides are the same length. It has one line of symmetry down the middle. This specific shape is called a kite.

Why we do this:

Before we can write a ratio, we need the total numerical value for each part. The fete is given, but we need to calculate the totals for the quiz and the membership fees.

Why we do this:

The question asks for the ratio in a specific order: Fete : Quiz : Membership. Once written, we must simplify it by finding the largest number that divides evenly into all three parts.

Why we do this:

A frequency tree splits a total amount into different categories. Each time a branch splits, the numbers in the bubbles at the end of the branches must add up to the number in the bubble before it. We use the information given to fill in the missing numbers.

Why we do this:

Probability is the number of successful outcomes divided by the total number of possible outcomes. The question says a person is chosen AT RANDOM from the “do not have a garden” group. This means our denominator is the total number of people without a garden. The numerator is the number of females in that specific group.

Why we do this:

To compare their rates, we need to know how much money each person makes in exactly one hour. For Ellie, we multiply the number of hats she makes in an hour by how much she gets paid per hat.

Why we do this:

Reaze is paid a total amount for multiple hours. To find his hourly rate, we must divide his total pay by the total hours worked.

What this tells us:

By comparing the two values we just calculated (£7.82 vs £7.60), we can clearly see and prove that Ellie’s hourly rate is higher.

Why we do this:

The question asks for a specific numerical example. We just need to pick any two odd numbers, substitute them into the expression \(2(a + b)\), calculate the result, and check if it is in the 4 times table.

Why we do this:

To prove something is “always” true, we can’t just use numbers (like in part a). We have to use logic. What do we know about adding two odd numbers together? An odd number plus an odd number always makes an even number.

What this tells us:

Another way to write the proof is algebraically. Let \(a = 2m + 1\) and \(b = 2n + 1\).
\(2(a+b) = 2(2m + 1 + 2n + 1) = 2(2m + 2n + 2) = 4m + 4n + 4 = 4(m+n+1)\). Since 4 is factored out, it is a multiple of 4.

Why we do this:

To know how much oil Mr Page bought in February to “fill the tank completely”, we first need to know how big the tank is. We can figure this out by calculating how many litres he bought in November and adding it to the 1000 litres already in the tank.

Why we do this:

In February, the tank has 600 litres in it. To fill it back up to its 2500 litre capacity, we subtract what’s already there from the total capacity.

Why we do this:

The price of oil went up by 4%. We must calculate the new price per litre, and then multiply it by the 1900 litres he bought to find the final bill.

What this tells us:

Mr Page had to buy more oil in February than in November (1900L vs 1500L), and the price had gone up. Therefore, it makes sense that his total bill (£988) is considerably higher than his November bill (£750).

Why we do this:

To solve for \(x\), we need to get all the \(x\)’s on one side of the equals sign. Right now, there is an \(x\) trapped inside a bracket on the right side. We must multiply everything inside that bracket by the 3 on the outside to release it.

Why we do this:

Now we collect all the \(x\) terms on the left side, and all the regular numbers on the right side. We do this by doing the opposite operation: subtract \(3x\) from both sides, and add \(6\) to both sides.

Why we do this:

We know what \(2x\) is, but we want to find out what just one \(x\) is. We divide both sides by 2.

What this tells us:

We can check our answer by substituting \(1.5\) back into the original equation: Left side = \(5(1.5) – 6 = 7.5 – 6 = 1.5\). Right side = \(3(1.5 – 1) = 3(0.5) = 1.5\). Both sides are equal, so our answer is perfectly correct.

Why we do this:

To find a profit, we need to know the total amount of money she brings in (revenue). We do this by multiplying the number of bottles by the selling price of each bottle.

Why we do this:

Profit is the money she gets to keep after paying her original costs. Profit = Total Revenue – Total Cost.

Why we do this:

To find the percentage profit, we use the standard formula: \(\frac{\text{Profit}}{\text{Original Cost}} \times 100\). It is crucial to divide by the original cost (£5.64), not the selling price.

What this tells us:

Emily made a 6.4% return on the money she initially spent buying the water.

Why we do this:

To find the distance between two points, we first need the entire perimeter of the circle (the circumference). The formula for the circumference is \(\pi \times d\), where \(d\) is the diameter.

Why we do this:

The 8 points divide the circle’s edge into 8 equal gaps (arcs). We divide the total circumference by 8 to find the length of one single gap.

Why we do this:

The formula for the mean distance is: \(\text{Mean} = \frac{\text{Total Distance Walked}}{\text{Number of points}}\). We need to see if either of those two numbers has changed.

Why we do this:

To find a probability, we need to know the total number of parts in the bag. We can pick a small, easy number for the smallest group (yellow) and build the rest from there to find the full ratio.

Why we do this:

The probability of picking a yellow cube is the number of yellow parts divided by the total number of all parts combined.

Why we do this:

Rotating an object \(180^{\circ}\) about the origin \((0,0)\) flips both of its coordinates. For every point \((x, y)\), the new point is \((-x, -y)\). We apply this rule to the corners of shape T to find shape A.

Why we do this:

A translation moves a shape without rotating or flipping it. The vector \(\binom{-1}{-3}\) tells us exactly how to move it: the top number is the \(x\) movement (left/right) and the bottom number is the \(y\) movement (up/down). So, we must move every point 1 unit left and 3 units down.

What this tells us:

Here is how the final grid should look with all three shapes correctly plotted.

Why we do this:

When multiplying terms with the same base (here, the base is \(p\)), we add their powers. So, \(p^A \times p^B = p^{A+B}\).

Why we do this:

When you have a power raised to another power (indicated by brackets), you multiply the powers together. So, \((A^B)^C = A^{B \times C}\).

Why we do this:

The expression mixes bases of 100, 1000, and 10. To combine them using index laws, they all need to share the same base (base 10). We must rewrite 100 and 1000 as powers of 10 first.

Why we do this:

To find angle CDA, we need the lengths of the right-angled triangle on the right side. We know its height is 6 cm, but we don’t know its base yet. First, we use Pythagoras’ theorem on the left right-angled triangle to find its base.

Why we do this:

The total length of the bottom base AD is 24 cm. This is made up of three parts: the left triangle’s base (4.5 cm), the middle rectangle’s base (10 cm), and the right triangle’s base. We subtract to find the right triangle’s base.

Why we do this:

We now have a right-angled triangle on the right with an Opposite side of 6 cm and an Adjacent side of 9.5 cm. To find the angle, we use SOH CAH TOA. Since we have Opposite and Adjacent, we use Tan (\(\tan\)).

Why we do this:

This tests your ability to type complex formulas accurately into your calculator. It is essential that your calculator is in DEGREES mode (usually shown by a small ‘D’ at the top of the screen). You must also ensure you close the brackets around the angles.

Why we do this:

We need to round to 2 decimal places. We look at the 3rd decimal place to decide whether to round the 2nd decimal place up or leave it the same.

Why we do this:

We want to find \(x\). First, we isolate \(x^2\) by dividing by 2. Then, we take the square root to find \(x\). Remember that taking a square root of a number gives both a positive AND a negative answer, because e.g., \(-6 \times -6 = 36\).

Why we do this:

To expand two brackets, every term in the first bracket must be multiplied by every term in the second bracket. We can use the FOIL method (First, Outside, Inside, Last). Then we collect the like terms in the middle to simplify.

Why we do this:

Factorising \(x^2 + 6x + 9\) means putting it back into double brackets \((x + ?)(x + ?)\). We need to find two numbers that multiply to make the end number (+9) and add to make the middle number (+6).