Edexcel GCSE Mathematics
Foundation Tier – Paper 1 (Non-Calculator)
June 2018

Working:

The number is \(6324\).

• The thousands digit is \(6\).
• The next digit to the right (hundreds) is \(3\).

Because \(3\) is less than \(5\), we round down. This means the \(6\) stays the same, and the remaining digits become zeros.

Working:

Looking at the negatives: \(-6\) is further left on the number line than \(-5\), so \(-6\) is the smallest. Next is \(-5\).

Then comes \(0\).

Finally, we order the positive numbers: \(6\) is smaller than \(12\).

Ordering them gives: \(-6, -5, 0, 6, 12\).

✓ (B1) Correct ordering. [cite: 103]

Working:

Let’s pad all the numbers to 3 decimal places:

• \(0.078 \rightarrow 0.078\)
• \(0.78 \rightarrow 0.780\)
• \(0.87 \rightarrow 0.870\)
• \(0.708 \rightarrow 0.708\)

Now, ignoring the “0.”, we can compare the parts like whole numbers: \(078, 780, 870, 708\).

Order from smallest to largest:

\(078 \rightarrow 0.078\)

\(708 \rightarrow 0.708\)

\(780 \rightarrow 0.78\)

\(870 \rightarrow 0.87\)

✓ (B1) Correct ordering. [cite: 103]

Working:

We can simplify this fraction by dividing both the top (numerator) and bottom (denominator) by \(20\):

✓ (B1) Correct fraction. (Note: \( \frac{20}{100} \) or equivalent like \( \frac{2}{10} \) is acceptable for the mark). [cite: 103]

Working:

• \(\frac{4}{16}\) : Both divide by \(4\) \(\rightarrow\) \(\frac{1}{4}\)
• \(\frac{2}{8}\) : Both divide by \(2\) \(\rightarrow\) \(\frac{1}{4}\)
• \(\frac{15}{60}\) : Both divide by \(15\) \(\rightarrow\) \(\frac{1}{4}\)
• \(\frac{3}{9}\) : Both divide by \(3\) \(\rightarrow\) \(\frac{1}{3}\)

Working:

Let’s list the first few multiples of \(7\):

\(7 \times 1 = 7\) (Odd)

\(7 \times 2 = 14\) (Even)

Working:

✓ (B1) Correct simplification.

Working:

First part: \(8a – 3a = 5a\)

Second part: \(5a + 2a = 7a\)

✓ (B1) Correct simplification.

Working:

(a) The letter at \(1\) on the scale is D.

(b) The letter situated between \(0\) and \(\frac{1}{2}\) is B.

✓ (B1) for D, (B1) for B.

Working:

Total counters = \(12\)

Counters we know = \(3\) (red) \(+ 1\) (blue) \(+ 2\) (yellow) = \(6\)

Green counters = \(12 – 6 = 6\)

✓ (M1) for finding the number of green counters.

Working:

Number of yellow or green counters = \(2\) (yellow) \(+ 6\) (green) = \(8\)

Probability = \(\frac{8}{12}\)

Simplify the fraction by dividing top and bottom by \(4\):

✓ (M1) for adding the probabilities or totals correctly (\(\frac{8}{12}\) seen).

✓ (C1) for a correct conclusion supported by accurate figures.

Working:

Cost of \(1 \text{ kg} = 54 \div 3\)

Using the bus stop method:

   1 8
  ----
3| 524

\(3\) goes into \(5\) once, remainder \(2\).

\(3\) goes into \(24\) eight times.

So, \(1 \text{ kg}\) costs \(£18\).

✓ (M1) method to find the cost of \(1 \text{ kg}\) (\(54 \div 3\)).

Working:

✓ (A1) Correct final answer.

Working:

(a) The diagram clearly shows a line extending from the center cross to the circumference. This is the radius.

(b) The question asks for the name of a straight line “touching” the circle. This fits the definition of a tangent.

✓ (B1) for “Radius”.

✓ (B1) for “Tangent”.

Working:

Total Cost = Plane Tickets + Villa + Car Hire

Total Cost = \(£1280 + £640 + £220\)

   1 2 8 0
     6 4 0
+    2 2 0
----------
   2 1 4 0
   1 1
----------

The total cost for the group is \(£2140\).

✓ (P1) for a start to the process (adding the values).

Working:

Tim’s share = \(2140 \div 4\)

Using the bus stop method:

     5 3 5
    -------
 4| 2211420

• \(4\) goes into \(2\) zero times, carry the \(2\).
• \(4\) goes into \(21\) five times, remainder \(1\).
• \(4\) goes into \(14\) three times, remainder \(2\).
• \(4\) goes into \(20\) five times exactly.

✓ (P1) for a full process to find the cost per person.

✓ (A1) Correct final answer.

Working:

Let’s pick an even number, like \(12\).

The factors of \(12\) are: \(1, 2, 3, 4, 6, 12\).

Notice that \(3\) is a factor of \(12\), and \(3\) is an odd number. Also, \(1\) is a factor of every number, and \(1\) is odd.

✓ (C1) for a correct example (e.g., “\(3\) is a factor of \(12\)” or “\(1\) is a factor of \(6\)”).

Working:

Let’s make a two-digit odd number: \(23\).

\(23\) is an odd number because it ends in \(3\).

However, the first digit is \(2\), which is an even digit. Therefore, not all digits in \(23\) are odd.

✓ (C1) for a correct example (e.g., \(23\) or \(45\)).

Working:

Looking at the 2015 column, the grey bar (desktops) goes from \(0\) up to exactly the line for \(100\).

✓ (B1) Correct answer: \(100\).

Working:

Let’s find the number of laptops sold each year:

• 2015: The laptop bar starts at \(100\) and ends at \(260\).
  Laptops sold \(= 260 – 100 = 160\)
• 2016: The laptop bar starts at \(120\) and ends at \(340\).
  Laptops sold \(= 340 – 120 = 220\)
• 2017: The laptop bar starts at \(160\) and ends at \(440\).
  Laptops sold \(= 440 – 160 = 280\)

Total laptops \(= 160 + 220 + 280\)

   1 6 0
   2 2 0
+  2 8 0
--------
   6 6 0
   1
--------

✓ (M1) for reading at least 3 required figures from the graph correctly.

✓ (M1) for adding the three differences together.

✓ (A1) Correct final sum: \(660\).

Working:

Let’s look at the growth of each item from 2015 to 2017:

• Desktops: \(100\) (in 2015) grew to \(160\) (in 2017). Increase \(= 60\).
• Laptops: \(160\) (in 2015) grew to \(280\) (in 2017). Increase \(= 120\).
• Tablets: \(320 – 260 = 60\) (in 2015) grew to \(780 – 440 = 340\) (in 2017). Increase \(= 340 – 60 = 280\).

Tablets clearly had the greatest increase.

✓ (B1) for identifying Tablets.

✓ (C1) for valid statement showing evidence (e.g., increased by \(280\) from \(60\) to \(340\)).

Working:

The graph only shows us the number of items sold (volume). It does not provide any information about the cost to make them or the price they are sold for.

For example, if the profit on one desktop is £500, but the profit on a tablet is only £10, selling more tablets wouldn’t necessarily mean more total profit.

Therefore, Alex is incorrect because we lack financial data.

✓ (C1) for “No” supported by a valid reason (e.g., “we do not know costs or profit margins”).

Working:

Length of the two initial cuts = \(2 \times 45 = 90 \text{ cm}\)

Remaining length = \(240 – 90\)

   1
   2 4 0
-    9 0
--------
   1 5 0
--------

There is \(150 \text{ cm}\) of wire remaining.

✓ (P1) for a start to the process (e.g. \(240 – 90 = 150\)).

Working:

Let’s count up in \(40\)s:

• 1 piece = \(40 \text{ cm}\)
• 2 pieces = \(80 \text{ cm}\)
• 3 pieces = \(120 \text{ cm}\)
• 4 pieces = \(160 \text{ cm}\) (Too much! We only have \(150 \text{ cm}\))

So, Peter can cut a maximum of \(3\) full \(40 \text{ cm}\) lengths. (He will have \(30 \text{ cm}\) left over as waste).

✓ (P1) for a complete process to find the number of pieces.

✓ (A1) Correct final answer.

Working:

1. Gavin:

Gavin saves \(28\%\). (This is already in our desired format).

2. Harry:

Harry spends \(\frac{3}{4}\). The whole salary is represented by \(1\) (or \(\frac{4}{4}\)).

Harry saves \(= 1 – \frac{3}{4} = \frac{1}{4}\).

To convert \(\frac{1}{4}\) to a percentage, we multiply by \(100\):

✓ (P1) for a process to work with \(\frac{3}{4}\) (e.g. finding \(25\%\)).

3. Isabel:

Isabel’s saves : spends ratio is \(3 : 7\).

Total parts in the ratio \(= 3 + 7 = 10\) parts.

This means she saves \(3\) parts out of the total \(10\) parts, which is the fraction \(\frac{3}{10}\).

To convert \(\frac{3}{10}\) to a percentage, we multiply top and bottom by \(10\) to get it out of \(100\):

✓ (P1) for a process to work with the ratio \(3:7\) (e.g. finding \(\frac{3}{10}\) or \(30\%\)).

Working:

• Gavin = \(28\%\)
• Harry = \(25\%\)
• Isabel = \(30\%\)

Comparing these, \(30\%\) is the largest number, so Isabel saves the most.

✓ (A1) for all three correct comparative values (e.g. \(28\%\), \(25\%\), \(30\%\)).

✓ (C1) for clearly stating Isabel with supported evidence.

Working:

First, find \(10\%\) of \(160\):

Next, find \(5\%\) of \(160\) by halving the \(10\%\) amount:

✓ (M1) for a clear method to find \(15\%\) (e.g. showing \(10\% = 16\) and \(5\% = 8\)).

Working:

✓ (A1) Correct final answer.

Working:

Substitute \(x = 5\) and \(y = -2\) into the formula:

Now, do the multiplications first (following order of operations):

Put it back together:

✓ (M1) for showing the substitution \(4 \times 5\) and \(3 \times -2\). [cite: 108]

✓ (A1) Correct final answer. [cite: 108]

Working:

Multiply \(4e\) by \(e\):

Multiply \(4e\) by \(2\):

Combine them:

✓ (B2) Fully correct expansion. (Award B1 if only \(4e^2\) or \(8e\) is correct). [cite: 108]

Working (Method 1: Divide first):

Divide both sides by \(3\):

Add \(4\) to both sides to isolate \(m\):

✓ (M1) for a correct first step (e.g., \(m – 4 = 7\) or expanding to \(3m – 12 = 21\)). [cite: 108]

✓ (A1) Correct final answer of \(11\). [cite: 108]

Working:

Chocolates with nuts = \(\frac{1}{4}\)

Chocolates without nuts = \(1 – \frac{1}{4} = \frac{3}{4}\)

Working:

Ratio of (with nuts) : (no nuts) = \(\frac{1}{4} : \frac{3}{4}\)

Multiply both sides by \(4\) to remove the denominators:

The question asks for the form \(1 : n\), and our answer is exactly in that form where \(n = 3\).

✓ (M1) for \(\frac{1}{4} : \frac{3}{4}\) or an equivalent unsimplified ratio like \(25:75\). [cite: 108]

✓ (A1) Correct final answer of \(1 : 3\). [cite: 108]

Working:

Multiples of \(5\) between \(14\) and \(26\):
\(A = \{15, 20, 25\}\)

Odd numbers between \(14\) and \(26\):
\(B = \{15, 17, 19, 21, 23, 25\}\)

✓ (M1) for correctly listing either set with no incorrect numbers. [cite: 109]

Working:

Combining \(A\) and \(B\):
\(A \cup B = \{15, 17, 19, 20, 21, 23, 25\}\)

✓ (A1) Correct list of numbers with no repeats. [cite: 109]

Working:

Looking at our two lists:
\(A = \{ \mathbf{15}, 20, \mathbf{25} \}\)
\(B = \{ \mathbf{15}, 17, 19, 21, 23, \mathbf{25} \}\)

The numbers that appear in both are \(15\) and \(25\). These are the odd multiples of \(5\) in that range.

✓ (C1) for either listing \(15, 25\) or describing them as “odd multiples of \(5\)”. [cite: 109]

Working:

First, add the whole numbers:

Now, make the fractions have a common denominator of \(28\):

Add the fractions:

Combine the whole number and fraction:

✓ (M1) for method to find common denominator with at least one fraction correct. [cite: 109]

✓ (A1) Correct final mixed number or improper fraction (\(\frac{95}{28}\)). [cite: 109]

Working:

Convert \(1\frac{1}{5}\) to an improper fraction:

The problem is now:

Keep, Change, Flip (Multiply by the reciprocal):

✓ (M1) for converting to improper and multiplying by reciprocal. [cite: 109]

Now, simplify by dividing top and bottom by their greatest common factor, which is \(3\):

The question requires a mixed number answer. How many \(5\)s fit into \(8\)?

One \(5\) fits in, with \(3\) left over, so it’s \(1\frac{3}{5}\).

✓ (A1) Correct mixed number in simplest form. [cite: 109]

Working:

Ratio of Flats : Bungalows = \(8 : 5\)

We know actual Bungalows = \(50\)

To get from the ratio part (\(5\)) to the actual amount (\(50\)), we multiply by \(10\).

Since we multiplied the bungalows by \(10\), we must multiply the flats ratio by \(10\) to maintain the proportion:

✓ (P1) process started, finding \(80\) flats. [cite: 112]

Working:

Ratio of Houses : Flats = \(7 : 4\)

We know actual Flats = \(80\)

To get from the ratio part (\(4\)) to the actual amount (\(80\)), we must multiply by \(20\).

Multiply the houses ratio by \(20\) to find the actual number of houses:

✓ (P1) full process to solve problem. [cite: 112]

✓ (A1) Correct final answer of \(140\). [cite: 112]

Working:

Convert kg to g (\(1 \text{ kg} = 1000 \text{ g}\)):

Divide the total weight by the weight of one bag:

Simplify by dividing both by \(10\) (crossing off a zero):

✓ (P1) for full process to find the number of bags (\(20\)).

Working:

Revenue = \(20 \text{ bags} \times 65\text{p}\)

Convert pence back to pounds:

✓ (P1) for process to find total revenue (\(20 \times 0.65 = 13\)).

Working:

Actual Profit = Revenue \(- \text{Cost}\)
Profit \(= £13 – £10 = £3\)

Percentage Profit = \( \left( \frac{\text{Profit}}{\text{Original Cost}} \right) \times 100 \)

✓ (P1) for process to find percentage profit (\( (13 – 10) \div 10 \times 100 \)).

✓ (A1) Correct final answer of \(30\).

Working:

• Total Distance: \(3069.25 \approx 3000 \text{ miles}\)
• Average Speed: \(15.12 \approx 15 \text{ mph}\)
• Hours per day: \(8 \text{ hours}\)

✓ (P1) for using rounded values in a correct process.

Working:

Estimated distance per day = Speed \(\times\) Time

Now find the total days:

Simplify by dividing both sides by \(10\):

Let’s halve both numbers: \(150 \div 6\)

Halve again: \(75 \div 3 = 25\)

So, Juan will take approximately \(25\) days.

✓ (P1) for a full process to find the number of days (e.g., \(3000 \div 15 \div 8\)).

✓ (A1) Correct estimated answer of \(25\).

Working:

Because his average speed has increased, he is covering more distance each day. Therefore, it will take him fewer days to complete the race overall.

(Alternatively, you could note that for an estimate, you might still round \(16.27\) down to \(15\), meaning the estimated answer wouldn’t change, but stating it takes “fewer days” is the most direct logical conclusion.)

✓ (C1) for a valid statement referring to the time taken.

Working (Accurate Diagram):

✓ (M1) for drawing an isosceles triangle with base \(6\) and height \(4\).

✓ (A1) for a fully correct, accurate diagram.

Working:

Area of the square base:

The base is a square with sides of \(6 \text{ cm}\).

Area of one triangular face (e.g., face VBC):

The base of the triangle is \(6 \text{ cm}\). The slanted height of the triangle down its face (line \(VM\)) is \(5 \text{ cm}\). Be careful not to use the vertical \(4 \text{ cm}\) height of the pyramid inside.

✓ (M1) method to find the area of a triangular face.

Total Surface Area:

Add the base and the four triangular faces together:

✓ (M1) method to find the total surface area (\(4 \times 15 + 6 \times 6\)).

✓ (A1) Correct numerical answer.

✓ (B1) Correct units (\(\text{cm}^2\)).

Working:

The \(x\)-coordinate of \(A\) is \(6\). The \(x\)-coordinate of \(B\) is \(38\).

Total horizontal width of the diagram = \(38 – 6 = 32\)

✓ (P1) process to find width or height (\(38 – 6 = 32\)).

Horizontally, the diagram is made up of exactly \(4\) squares placed side-by-side without overlapping left-to-right.

Side length of one square = \(32 \div 4 = 8\)

✓ (P1) process to find side of square (\(32 \div 4 = 8\)).

Working:

\(x\)-coordinate of \(C\) = (start \(x\)) + (\(2\) square widths)

✓ (P1) process to find x-coordinate (\(6 + 16\)).

Working:

\(y\)-coordinate of \(C\) = (top \(y\)) – (\(2\) square heights)

✓ (P1) process to find y-coordinate (\(36 – 16\)).

Working:

• If \(x = -3\), \(y = 1 – 4(-3) = 1 + 12 = 13\) \(\rightarrow (-3, 13)\)
• If \(x = -1\), \(y = 1 – 4(-1) = 1 + 4 = 5\) \(\rightarrow (-1, 5)\)
• If \(x = 0\), \(y = 1 – 4(0) = 1 – 0 = 1\) \(\rightarrow (0, 1)\)
• If \(x = 2\), \(y = 1 – 4(2) = 1 – 8 = -7\) \(\rightarrow (2, -7)\)
• If \(x = 3\), \(y = 1 – 4(3) = 1 – 12 = -11\) \(\rightarrow (3, -11)\)

✓ (B1) for at least 2 correct points calculated.

Working:

✓ (B2) for a correct line segment through at least 3 points.

✓ (B3) for a fully correct line drawn continuously between \(x = -3\) and \(x = 3\).

Working:

✓ (M1) for finding \(2a\).

Working:

✓ (A1) Correct final column vector.