Edexcel Foundation June 2017 Paper 3

Question 1 (2 marks)

The table shows the lengths of five rivers.

River	Length (km)
Trent	297
Don	112
Severn	354
Thames	346
Mersey	113

(a) Write down the rivers in order of length. Start with the shortest river. (1 mark)

Ami says,
“The River Thames is more than three times as long as the River Don.”

(b) Show that Ami is correct. (1 mark)

Worked Solution

Part (a): Ordering the rivers

Why we do this:

We need to look at the lengths of the rivers and list them from the smallest number to the largest number.

Working:

Looking at the lengths:

Don = 112 km (Smallest)
Mersey = 113 km
Trent = 297 km
Thames = 346 km
Severn = 354 km (Largest)

✓ (B1) for correctly ordering the rivers (or their lengths).

Part (b): Checking Ami’s statement

Why we do this:

To see if the Thames is “more than three times” the length of the Don, let’s multiply the length of the Don by 3 and compare it to the length of the Thames.

Working:

Calculator Steps:

Length of River Don = $112$
Press: 112 × 3 =
Calculator shows: $336$

Now, we compare this to the River Thames ($346$ km):

\[ 346 > 336 \]

✓ (C1) for showing a correct calculation (e.g., $112 \times 3 = 336$) and comparing it to $346$.

What this tells us:

Because $346$ is a bigger number than $336$, Ami is completely correct. The Thames is indeed more than three times the length of the Don.

Final Answer:

(a) Don, Mersey, Trent, Thames, Severn

(b) $112 \times 3 = 336$. Since $346 > 336$, Ami is correct.

Total: 2 marks

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Question 2 (2 marks)

Cups are sold in packs and in boxes.

There are 12 cups in each pack.
There are 18 cups in each box.

Alison buys $p$ packs of cups and $b$ boxes of cups.

Write down an expression, in terms of $p$ and $b$, for the total number of cups Alison buys.

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Question 3 (2 marks)

Here are four digits.

5 6 1 9

(i) Write down the smallest possible two digit number that can be made with two of the digits. (1 mark)

(ii) Write down the three digit number closest to $200$ that can be made with three of the digits. (1 mark)

Worked Solution

Part (i): Smallest two-digit number

Why we do this:

To make the smallest possible number, we need the smallest digit in the ‘tens’ column, and the next smallest digit in the ‘ones’ column.

Working:

Our digits are: $1, 5, 6, 9$

Smallest digit = $1$ (This goes in the tens place)

Next smallest = $5$ (This goes in the units place)

Result = $15$

✓ (B1) for the correct answer $15$.

Part (ii): Number closest to 200

Why we do this:

We need a 3-digit number. The hundreds digit must be either $1$ (in the 100s) or $5$ (in the 500s) because we only have 1, 5, 6, and 9. A number in the 100s is much closer to 200 than a number in the 500s.

To get as close to 200 as possible from underneath, we need the largest possible number starting with 1.

Working:

Start with $1$ in the hundreds column: $1\_ \_$

We have $5, 6, 9$ left. To make it as large as possible (closest to 200), we put the largest digit in the tens column ($9$) and the next largest in the units column ($6$).

This gives us $196$.

(Checking: $200 – 196 = 4$. The next closest number we could make is $195$, which is further away. $516$ is way too far).

✓ (B1) for the correct answer $196$.

Final Answer:

(i) 15

(ii) 196

Total: 2 marks

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Question 4 (2 marks)

$\frac{4}{5}$ of a number is $32$

Find the number.

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Question 5 (3 marks)

A path is made of white tiles and grey tiles.

$\frac{1}{4}$ of the tiles are white.

(a) Write down the ratio of white tiles to grey tiles. (1 mark)

There is a total of $56$ tiles.

(b) Work out the number of grey tiles. (2 marks)

Worked Solution

Part (a): Converting fraction to ratio

Why we do this:

If $\frac{1}{4}$ of the tiles are white, this means out of every 4 tiles in total, 1 is white. The rest must be grey. A ratio compares the parts to each other (White : Grey).

Working:

Total parts = $4$

White parts = $1$

Grey parts = $4 – 1 = 3$

Ratio of White : Grey = $1 : 3$

✓ (B1) for the correct ratio $1:3$.

Part (b): Calculating the number of grey tiles

Why we do this:

We know the total is 56 tiles. From part (a), the fraction of grey tiles is $\frac{3}{4}$. We need to find $\frac{3}{4}$ of 56.

Alternatively, find the number of white tiles first ($\frac{1}{4}$ of 56) and subtract that from the total.

Working:

Method 1:

Number of white tiles = $56 \div 4 = 14$

Number of grey tiles = Total – White tiles = $56 – 14$

Calculator Steps:

Press: 56 - 14 =
Result: $42$

Method 2:

Grey tiles = $\frac{3}{4}$ of $56$

\[ 56 \div 4 \times 3 = 14 \times 3 = 42 \]

✓ (M1) for a complete correct method (e.g. $56 \div 4 \times 3$ or $56 – 14$).

✓ (A1) for the correct answer $42$.

What this tells us:

Out of the 56 tiles on the path, exactly 42 of them will be grey.

Final Answer:

(a) $1:3$

(b) $42$

Total: 3 marks

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Question 6 (5 marks)

Here is a list of numbers.

12, 15, 14, 17, 22, 19, 13

Bridgit says,
“To work out the median you find the middle number, so the median of these numbers is $17$”

Bridgit’s answer is not correct.

(a) What is wrong with Bridgit’s method? (1 mark)

(b) Work out the range of the numbers in the list. (2 marks)

(c) Work out the mean of the numbers in the list. (2 marks)

Worked Solution

Part (a): Fixing the Median

Why we do this:

The median is indeed the middle number, but there’s a crucial first step that Bridgit forgot. Before finding the middle, the data must always be ordered from smallest to largest.

Working:

Bridgit didn’t put the numbers in size order first.

✓ (C1) for stating a valid reason, e.g., “she must order the numbers first”.

Part (b): Finding the Range

Why we do this:

The range tells us how spread out the numbers are. We find it by subtracting the smallest number in the list from the largest number.

Working:

Highest number = $22$

Lowest number = $12$

\[ \text{Range} = 22 – 12 = 10 \]

✓ (M1) for showing $22 – 12$.

✓ (A1) for the correct answer $10$.

Part (c): Calculating the Mean

Why we do this:

To find the mean (the mathematical average), we need to add up all the values in our list, and then divide that total by how many numbers there are in the list.

Working:

Calculator Steps:

Find the total: 12 + 15 + 14 + 17 + 22 + 19 + 13 = 112
Count the numbers: There are $7$ numbers.
Divide total by count: 112 ÷ 7 = 16

✓ (M1) for adding the numbers and dividing by $7$.

✓ (A1) for the correct answer $16$.

Final Answer:

(a) She must order the numbers first.

(b) 10

(c) 16

Total: 5 marks

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Question 7 (2 marks)

Priti is going to have a meal.

She can choose one starter and one main course from the menu.

Menu
Starter	Main Course
Salad	Pasta
Fish	Rice
Melon	Burger

Write down all the possible combinations Priti can choose.

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Question 8 (2 marks)

Joanne wants to buy a dishwasher.

The dishwasher costs £372.

Joanne will pay a deposit of £36.
She will then pay the rest of the cost in $4$ equal monthly payments.

How much is each monthly payment?

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Question 9 (3 marks)

Davos is a cleaner.

The table shows information about the time it will take him to clean each of four rooms in a house.

Room	Time
Kitchen	2 hours
Sitting room	1 hour 40 minutes
Bedroom	$1\frac{1}{2}$ hours
Bathroom	45 minutes

Davos wants to clean all four rooms in one day.

He will have breaks for a total time of $75$ minutes.
Davos is going to start cleaning at 9 am.

Will he finish cleaning by 4 pm?
You must show all your working.

Worked Solution

Step 1: Finding total time taken

Why we do this:

To easily add these different times together, it’s best to convert everything into the same unit: minutes.

Remember: 1 hour = 60 minutes.

Working:

Kitchen = $2 \times 60 = 120$ mins
Sitting room = $60 + 40 = 100$ mins
Bedroom = $1.5 \times 60 = 90$ mins
Bathroom = $45$ mins
Breaks = $75$ mins

Total time in minutes:

\[ 120 + 100 + 90 + 45 + 75 = 430 \text{ minutes} \]

✓ (P1) for a process to find total time (e.g., adding room times, or converting all to minutes).

Step 2: Compare total time to available time

Why we do this:

Now we need to see if his total time (430 minutes) fits into his workday. He wants to work from 9:00 am to 4:00 pm. Let’s find out how long the workday is, and compare.

Working:

Time available from 9 am to 4 pm = 7 hours.

Convert available time to minutes:

\[ 7 \times 60 = 420 \text{ minutes} \]

Comparing the two:

Total time needed = $430$ mins

Time available = $420$ mins

Since 430 is greater than 420, he does not have enough time.

Alternative approach: Convert total time to hours

Total time needed: $430 \div 60 = 7$ hours and $10$ minutes.

Add to start time: $9\text{am} + 7\text{hrs } 10\text{mins} = 4:10\text{pm}$.

He finishes at 4:10 pm, which is after 4 pm.

✓ (P1) for a complete process to inform final decision (e.g., comparing 420 mins and 430 mins, or finding a finish time of 4:10 pm).

✓ (C1) for stating “No” supported by correct values (e.g., 4:10 pm or 430 mins > 420 mins).

What this tells us:

Davos is 10 minutes short on time. He will need to speed up, skip a break, or finish slightly late!

Final Answer:

No, he will finish at 4:10 pm (or he needs 430 minutes but only has 420 minutes).

Total: 3 marks

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Question 10 (3 marks)

ABC is a straight line.

The length $AB$ is five times the length $BC$.
$AC = 90 \text{ cm}$.

Work out the length $AB$.

Worked Solution

Step 1: Understanding the Ratio

Why we do this:

This is a ratio problem presented as a geometry problem. If $AB$ is 5 times the length of $BC$, we can think of this in “parts”.

Let $BC$ be $1$ part. That means $AB$ is $5$ parts.
The ratio of $AB : BC$ is $5 : 1$.

The whole line segment $AC$ is made up of $AB$ and $BC$ joined together. So, the total number of parts is $5 + 1 = 6$ parts.

Working:

Total parts on the line = $5 \text{ parts} (AB) + 1 \text{ part} (BC) = 6 \text{ parts}$.

Step 2: Finding the length of one part and AB

Why we do this:

We know the total length of $AC$ is $90$ cm. Since $AC$ is $6$ parts, we can divide $90$ by $6$ to find the length of a single part ($BC$). Then, we multiply that by $5$ to find the length of $AB$.

Working:

Calculator Steps:

Value of 1 part: 90 ÷ 6 = 15

So, $BC = 15 \text{ cm}$.

✓ (P1) for $90 \div 6 = 15$ (or connecting AB and BC by ratio).

Now, find $AB$ (which is 5 parts):

Calculate AB: 15 × 5 = 75

✓ (P1) for a complete method to find length AB, e.g., $90 \div 6 \times 5$.

✓ (A1) for the correct answer $75$.

What this tells us:

We can check: If $AB = 75$ and $BC = 15$, then $75 + 15 = 90$ (which matches $AC$). And $75$ is exactly $5 \times 15$. Perfect!

Final Answer:

75 cm

Total: 3 marks

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Question 11 (4 marks)

\[ T = 4v + 3 \]

(a) Work out the value of $T$ when $v = 2$. (2 marks)

(b) Make $v$ the subject of the formula $T = 4v + 3$. (2 marks)

Worked Solution

Part (a): Substituting a value into a formula

Why we do this:

We need to find the numerical value of $T$. The formula tells us that to get $T$, we multiply $v$ by $4$ (because $4v$ means $4 \times v$) and then add $3$. We are given that $v = 2$, so we substitute (swap) the letter $v$ for the number $2$.

Working:

Start with the formula:

\[ T = 4v + 3 \]

Substitute $v = 2$:

\[ T = 4(2) + 3 \] \[ T = 8 + 3 \]

✓ (M1) for substituting $v = 2$ correctly (e.g., seeing $4 \times 2 + 3$ or $8 + 3$).

\[ T = 11 \]

✓ (A1) for the correct answer $11$.

Part (b): Rearranging the formula

Why we do this:

“Make $v$ the subject” means we need to rearrange the equation so it reads “$v = \text{something}$”. Right now, $v$ is multiplied by 4 and then 3 is added to it. We need to do the reverse operations in reverse order: subtract 3, then divide by 4.

Working:

Start with the original equation:

\[ T = 4v + 3 \]

Step 1: Subtract 3 from both sides to isolate the term with $v$.

\[ T – 3 = 4v \]

✓ (M1) for a correct first step to rearrange by isolating $4v$ (e.g., $T – 3 = 4v$) OR dividing each term by 4.

Step 2: Divide both sides by 4 to get $v$ by itself.

\[ \frac{T – 3}{4} = v \]

We usually write the subject on the left side:

\[ v = \frac{T – 3}{4} \]

✓ (A1) for the fully correct answer.

What this tells us:

We can test our new formula using part (a). If $T = 11$, then $v = \frac{11 – 3}{4} = \frac{8}{4} = 2$. It matches exactly, proving our rearrangement is correct!

Final Answer:

(a) $11$

(b) $v = \frac{T – 3}{4}$

Total: 4 marks

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Question 12 (5 marks)

The diagram shows a cube of side length 2 cm.

Vera says,
“The volume of any solid made with 6 of these cubes is $48\text{ cm}^3$”

(a) Is Vera correct? You must show your working. (2 marks)

(b) (i) Draw a cuboid that can be made with 6 of these cubes. Write the dimensions of the cuboid on your diagram. (1 mark)

(ii) Work out the surface area of your cuboid. (2 marks)

Worked Solution

Step 1: Finding the volume (Part a)

Why we do this:

To check Vera’s statement, we need to find the total volume of 6 cubes. The total volume is just the volume of one cube multiplied by 6.

Working:

Volume of one cube = $\text{length} \times \text{width} \times \text{height}$

\[ \text{Volume of 1 cube} = 2 \times 2 \times 2 = 8 \text{ cm}^3 \]

✓ (M1) for a method to find the volume of one cube.

Volume of 6 cubes = $6 \times \text{Volume of 1 cube}$

\[ \text{Total Volume} = 6 \times 8 = 48 \text{ cm}^3 \]

Since the total volume is $48\text{ cm}^3$, Vera is correct.

✓ (C1) for stating “Yes” with supporting evidence.

Step 2: Drawing the cuboid (Part bi)

Why we do this:

We need to arrange 6 cubes into a solid rectangular shape (cuboid). There are two ways to do this:

Place them in a straight line: $1 \times 6$ cubes. Dimensions would be $2\text{ cm} \times 12\text{ cm} \times 2\text{ cm}$.
Place them in a $2 \times 3$ grid: Dimensions would be $4\text{ cm} \times 6\text{ cm} \times 2\text{ cm}$.

We’ll show the $4\text{ cm} \times 6\text{ cm} \times 2\text{ cm}$ option below.

Working:

✓ (B1) for drawing a correct cuboid with dimensions labeled (e.g., $4\text{ cm} \times 6\text{ cm} \times 2\text{ cm}$ or $2\text{ cm} \times 12\text{ cm} \times 2\text{ cm}$).

Step 3: Calculating Surface Area (Part bii)

Why we do this:

Surface area is the total area of the outside of the shape. A cuboid has 6 rectangular faces (front/back, top/bottom, left/right). We find the area of each pair and add them together.

Working:

Using our dimensions of $4\text{ cm}$, $6\text{ cm}$, and $2\text{ cm}$:

Front face area = $4 \times 6 = 24\text{ cm}^2$ (and the back is also 24)
Top face area = $6 \times 2 = 12\text{ cm}^2$ (and the bottom is also 12)
Side face area = $4 \times 2 = 8\text{ cm}^2$ (and the other side is also 8)

Total Surface Area:

\[ 2 \times (24 + 12 + 8) \] \[ 2 \times (44) = 88 \text{ cm}^2 \]

✓ (M1) for finding the area of 3 or more faces of their cuboid and adding.

✓ (A1) for the correct surface area based on their cuboid (either $88$ or $104$).

What this tells us:

Even though the volume is always the same ($48\text{ cm}^3$) no matter how you arrange the 6 cubes, the surface area changes depending on the arrangement. A long thin stick of cubes has more surface area ($104\text{ cm}^2$) than a blocky grid ($88\text{ cm}^2$).

Final Answer:

(a) Yes, $2 \times 2 \times 2 = 8$ and $8 \times 6 = 48$

(b)(i) Cuboid drawn with dimensions e.g. $4\text{ cm by } 6\text{ cm by } 2\text{ cm}$

(b)(ii) 88 (or 104 if arranged 1×6)

Total: 5 marks

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Question 13 (5 marks)

The size of the largest angle in a triangle is 4 times the size of the smallest angle.

The other angle is $27^\circ$ less than the largest angle.

Work out, in degrees, the size of each angle in the triangle.
You must show your working.

Worked Solution

Step 1: Forming Algebraic Expressions

Why we do this:

We are given relationships between the three angles of a triangle. The easiest way to solve this is to let the smallest angle be a letter (like $x$) and write the other angles in terms of $x$.

Working:

Let the smallest angle = $x$
The largest angle is 4 times the smallest = $4x$
The other angle is $27^\circ$ less than the largest = $4x – 27$

✓ (P1) for finding two of $x$, $4x$, and $4x – 27$.

Step 2: Solving the Equation

Why we do this:

We know a very important fact about triangles: the three interior angles always add up to $180^\circ$. We can use this to create an equation and find the value of $x$.

Working:

Set up the equation:

\[ x + 4x + (4x – 27) = 180 \]

✓ (P1) for an equation summing their three angles to $180$.

Simplify by collecting the $x$ terms:

\[ 9x – 27 = 180 \]

✓ (P1) for correctly simplifying the algebraic expression.

Solve for $x$:

Add 27 to both sides:

\[ 9x = 180 + 27 \] \[ 9x = 207 \]

✓ (P1) for a correct process to solve their equation.

Divide by 9:

Calculator Steps:

Press: 207 ÷ 9 =
Result: $23$

\[ x = 23 \]

Step 3: Calculating all three angles

Why we do this:

The question asks for the size of each angle, not just $x$. We must substitute $x = 23$ back into our expressions from Step 1.

Working:

Smallest angle: $x = 23^\circ$
Largest angle: $4x = 4 \times 23 = 92^\circ$
Other angle: $4x – 27 = 92 – 27 = 65^\circ$

✓ (A1) for three correct angles (order irrelevant).

What this tells us:

Check the answers: do they add up to 180? $23 + 92 + 65 = 180$. Is the largest (92) four times the smallest (23)? Yes ($23\times 4=92$). Is the middle one (65) exactly 27 less than the largest? Yes ($92-27=65$). The math is perfect.

Final Answer:

$23^\circ, 65^\circ, 92^\circ$

Total: 5 marks

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Question 14 (6 marks)

Andy went on holiday to Canada.
His flights cost a total of £1500

Andy stayed for 14 nights.
His hotel room cost $196 per night.

Andy used wifi for 12 days.
Wifi cost $5 per day.

The exchange rate was $1.90 to £1

(a) Work out the total cost of the flights, the hotel room and wifi. Give your answer in pounds. (5 marks)

(b) If there were fewer dollars to £1, what effect would this have on the total cost, in pounds, of Andy’s holiday? (1 mark)

Worked Solution

Step 1: Find the total cost in Dollars (Part a)

Why we do this:

We have costs in two different currencies (pounds for flights, dollars for hotel and wifi). Before we convert currencies, let’s group all the dollar costs together so we only have to do the conversion once.

Working:

Hotel total cost: $196 per night for 14 nights

\[ 196 \times 14 = \$2744 \]

✓ (P1) for a process to find total room cost.

Wifi total cost: $5 per day for 12 days

\[ 5 \times 12 = \$60 \]

✓ (P1) for a process to find total wifi cost.

Total dollar cost = $2744 + $60 = $2804

Step 2: Convert to Pounds and find final total

Why we do this:

The exchange rate is £1 = $1.90. To change from dollars to pounds, we must divide by 1.90. Once everything is in pounds, we can add it to the flight cost to get the grand total.

Working:

Convert $2804 into pounds:

Calculator Steps:

Press: 2804 ÷ 1.90 =
Result: $1475.78947…$

✓ (P1) for using exchange rate appropriately (e.g., $2804 \div 1.90$).

Now, add the cost of the flights (£1500) which was already in pounds:

\[ \text{Total Cost} = 1500 + 1475.78947… = 2975.78947… \]

✓ (P1) for a process to find the total cost in £.

We always round money to 2 decimal places (pence):

\[ \text{£}2975.79 \]

✓ (A1) for a correct final answer between 2975 to 2976.

Step 3: Reasoning about exchange rates (Part b)

Why we do this:

We are asked what happens if there are “fewer dollars to £1” (e.g., maybe £1 only gets you $1.50). The hotel still costs $2804. If £1 gets fewer dollars, your pounds are weaker. You will need to spend more pounds to get the same $2804 to pay for the hotel.

Working:

Mathematically, if we divide $2804 by a smaller number (like 1.50 instead of 1.90), the resulting pound value will be larger.

Therefore, the total cost of the holiday will go up.

✓ (C1) for stating the total price would rise (or cost more).

Final Answer:

(a) £2975.79

(b) The total cost would rise/increase.

Total: 6 marks

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Question 15 (6 marks)

$\mathcal{E} = \{\text{odd numbers less than 30}\}$
$A = \{3, 9, 15, 21, 27\}$
$B = \{5, 15, 25\}$

(a) Complete the Venn diagram to represent this information. (4 marks)

A number is chosen at random from the universal set, $\mathcal{E}$.

(b) What is the probability that the number is in the set $A \cup B$? (2 marks)

Worked Solution

Step 1: Completing the Venn diagram

Why we do this:

To fill a Venn diagram correctly, it’s best to work from the “inside out”.

Write down all numbers in the universal set $\mathcal{E}$.
Find any numbers that are in BOTH set A and set B. These go in the middle overlap (intersection).
Put the rest of A’s numbers in the “A only” section.
Put the rest of B’s numbers in the “B only” section.
Put any numbers from $\mathcal{E}$ that haven’t been used yet on the outside of the circles.

Working:

$\mathcal{E}$ = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29} (These are all 15 odd numbers under 30)

Intersection (in both): The number 15 is in both $A$ and $B$.

Only in $A$: 3, 9, 21, 27

Only in $B$: 5, 25

Outside (neither A nor B): 1, 7, 11, 13, 17, 19, 23, 29

✓ (B1) for labels A and B on diagram.

✓ (M1) for just 15 in the intersection.

✓ (M1) for correctly placing just 5, 25 in B OR just 3,9,21,27 in A OR the 8 outside numbers.

✓ (C1) for all numbers correctly placed.

Step 2: Finding the probability

Why we do this:

$A \cup B$ means the “Union” of A and B. This refers to anything that is inside Circle A, OR inside Circle B, OR in both. We just count how many numbers are inside the circles, and put it over the total number of numbers available.

Working:

Total numbers in Universal Set = $15$

Numbers inside the circles ($A \cup B$) = {3, 9, 21, 27, 15, 5, 25}

Count of numbers in $A \cup B$ = $7$

\[ \text{Probability} = \frac{\text{Successful outcomes}}{\text{Total outcomes}} = \frac{7}{15} \]

✓ (P1) for a numerator of 7 (or a denominator of 15).

✓ (A1) for the correct fraction $\frac{7}{15}$.

Final Answer:

(a) Diagram fully populated as above.

(b) $\frac{7}{15}$

Total: 6 marks

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Question 16 (3 marks)

Solve the simultaneous equations

$3x + y = -4$
$3x – 4y = 6$

Worked Solution

Step 1: Eliminating one variable

Why we do this:

We have two variables ($x$ and $y$). We cannot solve for both at once. However, we notice that both equations have exactly a “$3x$” in them. If we subtract the second equation from the first, the $3x$ terms will cancel each other out (eliminate), leaving us with only $y$.

Working:

Let’s label our equations:

(1) $3x + y = -4$

(2) $3x – 4y = 6$

Subtract equation (2) from equation (1):

\[ (3x – 3x) + (y – -4y) = -4 – 6 \]

Be very careful with negative numbers! $y – (-4y)$ becomes $y + 4y$.

\[ 0 + 5y = -10 \]

✓ (M1) for a correct method to eliminate one variable.

Divide by 5:

\[ y = -2 \]

Step 2: Finding the second variable

Why we do this:

Now that we know $y = -2$, we can substitute this value back into either of the original equations to figure out the value of $x$.

Working:

Substitute $y = -2$ into Equation (1):

\[ 3x + (-2) = -4 \] \[ 3x – 2 = -4 \]

Add 2 to both sides:

\[ 3x = -4 + 2 \] \[ 3x = -2 \]

✓ (M1) for substituting the found value back into an equation.

Divide by 3:

\[ x = -\frac{2}{3} \]

✓ (A1) for both $x = -\frac{2}{3}$ and $y = -2$ fully correct.

What this tells us:

We can check our answers by substituting both into equation (2):
$3(-\frac{2}{3}) – 4(-2) = -2 + 8 = 6$. Since it equals 6, we know our answers are 100% correct!

Final Answer:

$x = -\frac{2}{3}$

$y = -2$

Total: 3 marks

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Question 17 (2 marks)

The table shows some information about the dress sizes of 25 women.

Dress size	Number of women
8	2
10	9
12	8
14	6

(a) Find the median dress size. (1 mark)

3 of the 25 women have a shoe size of 7.

Zoe says that if you choose at random one of the 25 women, the probability that she has either a shoe size of 7 or a dress size of 14 is

$\frac{3}{25} + \frac{6}{25} = \frac{9}{25}$

(b) Is Zoe correct? You must give a reason for your answer. (1 mark)

Worked Solution

Part (a): Finding the median from a frequency table

Why we do this:

There are 25 women in total. The median is the value in the exact middle when they are ordered by size. The position of the median is $\frac{n+1}{2}$. We then count cumulatively through the table until we reach that person.

Working:

Median position = $\frac{25 + 1}{2} = \frac{26}{2} = 13$

We need to find the dress size of the 13th woman.

The first 2 women wear size 8. (cumulative: 2)
The next 9 women wear size 10. (cumulative: $2 + 9 = 11$)
The next 8 women wear size 12. (cumulative: $11 + 8 = 19$)

Since the 13th woman falls in the group that takes us from 12 to 19, she must wear a dress size 12.

✓ (B1) for the correct answer $12$.

Part (b): Testing Probability Logic

Why we do this:

Zoe is using the “OR rule” by adding the two probabilities together. However, you can only directly add probabilities if the two events are “mutually exclusive” (meaning they cannot happen at the same time).

Working:

Could a woman have BOTH a dress size 14 AND a shoe size 7? Yes, absolutely.

Because some of those 6 women with a dress size 14 might also be in the group of 3 women with a shoe size 7, Zoe is potentially double-counting them.

✓ (C1) for stating “No” and giving a valid reason, e.g., “The events are not mutually exclusive” or “A woman could have both a dress size 14 and a shoe size 7”.

Final Answer:

(a) 12

(b) No, because the events are not mutually exclusive (a person could be in both categories).

Total: 2 marks

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Question 18 (5 marks)

Daniel bakes 420 cakes.

He bakes only vanilla cakes, banana cakes, lemon cakes and chocolate cakes.

$\frac{2}{7}$ of the cakes are vanilla cakes.
$35\%$ of the cakes are banana cakes.

The ratio of the number of lemon cakes to the number of chocolate cakes is $4:5$

Work out the number of lemon cakes Daniel bakes.

Worked Solution

Step 1: Finding Vanilla and Banana cakes

Why we do this:

To find out how many lemon cakes there are, we first need to find out how many cakes are left over after taking away the vanilla and banana cakes. We’ll calculate the exact numbers of vanilla and banana cakes first.

Working:

Vanilla cakes: $\frac{2}{7}$ of 420

Calculator: 420 ÷ 7 × 2 = $120$

✓ (P1) for finding the number of vanilla cakes.

Banana cakes: $35\%$ of 420

Calculator: 420 × 0.35 = $147$

✓ (P1) for finding the number of banana cakes.

Step 2: Finding the remaining cakes

Why we do this:

By subtracting the vanilla and banana cakes from the total, we will know exactly how many cakes are meant to be shared between lemon and chocolate.

Working:

Total of vanilla and banana = $120 + 147 = 267$

Remaining cakes (Lemon + Chocolate) = $420 – 267 = 153$

✓ (P1) for finding the number of remaining cakes.

Step 3: Sharing using ratio

Why we do this:

The remaining 153 cakes are split in the ratio $4:5$ (Lemon : Chocolate). The total number of ratio “parts” is $4 + 5 = 9$. We divide 153 by 9 to find the value of one part, and then multiply by 4 to find the lemon cakes.

Working:

Total ratio parts = $4 + 5 = 9$

Value of 1 part = $153 \div 9 = 17$

Number of Lemon cakes (4 parts) = $17 \times 4$

Calculator: 17 × 4 = $68$

✓ (P1) for a full process to find the number of lemon cakes.

✓ (A1) for the correct final answer of 68.

What this tells us:

Daniel bakes exactly 68 Lemon cakes. If we wanted, we could find Chocolate ($17 \times 5 = 85$) and check: $120 + 147 + 68 + 85 = 420$. It balances perfectly.

Final Answer:

68

Total: 5 marks

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Question 19 (4 marks)

In the diagram, $AB$, $BC$ and $CD$ are three sides of a regular polygon $P$.

Show that polygon $P$ is a hexagon.
You must show your working.

Worked Solution

Step 1: Analyzing the angles at point B (or C)

Why we do this:

Look at point B on the diagram. Three different shapes meet around this point: a square, the regular 12-sided polygon, and polygon $P$.

Angles around a single point always add up to $360^\circ$. If we can find the angle of the square and the angle of the 12-sided polygon, we can subtract them from $360^\circ$ to find the interior angle of polygon $P$.

Working:

Interior angle of a square = $90^\circ$

Now, let’s find the interior angle of the regular 12-sided polygon (dodecagon):

Formula for exterior angle: $360 \div \text{number of sides}$

\[ \text{Exterior angle} = 360 \div 12 = 30^\circ \]

Formula for interior angle: $180 – \text{exterior angle}$

\[ \text{Interior angle} = 180 – 30 = 150^\circ \]

✓ (M1) for a complete method to find the interior or exterior angle of the dodecagon.

Step 2: Finding the interior angle of Polygon P

Why we do this:

We know two of the three angles around point B. We subtract them from $360^\circ$.

Working:

\[ \text{Interior angle of P} = 360 – 90 – 150 \] \[ \text{Interior angle of P} = 120^\circ \]

✓ (M1) for a complete method to find the interior angle of polygon P.

Step 3: Proving it’s a hexagon

Why we do this:

A hexagon is a 6-sided shape. We must prove that a regular polygon with an interior angle of $120^\circ$ has exactly 6 sides. We do this by reversing our earlier exterior angle formula.

Working:

If interior angle = $120^\circ$

Then exterior angle = $180 – 120 = 60^\circ$

Number of sides = $360 \div \text{Exterior Angle}$

\[ \text{Number of sides} = \frac{360}{60} = 6 \]

✓ (A1) for 30 and 120 (or 150 and 120) identified in calculations.

✓ (C1) for a complete solution, fully supported by accurate figures.

What this tells us:

Because the polygon has 6 sides, we have conclusively proven that it is a hexagon. You cannot get this mark if your working is missing or incorrect.

Final Answer:

Interior angle of dodecagon = $150^\circ$.
Interior angle of P = $360 – 90 – 150 = 120^\circ$.
Exterior angle of P = $180 – 120 = 60^\circ$.
Sides = $360 \div 60 = 6$, so it is a hexagon.

Total: 4 marks

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Question 20 (4 marks)

The density of apple juice is $1.05 \text{ g/cm}^3$.

The density of fruit syrup is $1.4 \text{ g/cm}^3$.

The density of carbonated water is $0.99 \text{ g/cm}^3$.

$25 \text{ cm}^3$ of apple juice are mixed with $15 \text{ cm}^3$ of fruit syrup and $280 \text{ cm}^3$ of carbonated water to make a drink with a volume of $320 \text{ cm}^3$.

Work out the density of the drink.
Give your answer correct to 2 decimal places.

Worked Solution

Step 1: Calculating the Mass of each ingredient

Why we do this:

The formula for density is: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$.

To find the density of the final mixture, we need its Total Mass and its Total Volume. The question gives us the Total Volume ($320 \text{ cm}^3$). But we have to calculate the total mass by finding the mass of each individual liquid and adding them together.

Rearranging the formula gives: $\text{Mass} = \text{Density} \times \text{Volume}$.

Working:

Apple Juice Mass: $1.05 \times 25 = 26.25 \text{ g}$
Fruit Syrup Mass: $1.4 \times 15 = 21 \text{ g}$
Carbonated Water Mass: $0.99 \times 280 = 277.2 \text{ g}$

✓ (P1) for correctly calculating at least one of the component masses.

Step 2: Finding Total Mass and Final Density

Why we do this:

We add the three masses together to find the Total Mass. Then we divide this Total Mass by the Total Volume to find the overall Density of the new drink.

Working:

Total Mass = $26.25 + 21 + 277.2$

Calculator: 26.25 + 21 + 277.2 = $324.45 \text{ g}$

✓ (P1) for a complete process to find the total mass.

Now, calculate Density:

\[ \text{Total Density} = \frac{\text{Total Mass}}{\text{Total Volume}} \] \[ \text{Total Density} = \frac{324.45}{320} \]

Calculator: 324.45 ÷ 320 = $1.01390625$

✓ (P1) for complete process to find the density.

The question asks for 2 decimal places. We look at the third decimal place (3), which is less than 5, so we keep it as it is:

\[ 1.01 \]

✓ (A1) for the correct answer rounded to $1.01$.

What this tells us:

Since the vast majority of the drink is carbonated water (density 0.99), we’d expect our final density to be very close to 0.99, pulled up slightly by the heavy syrup and apple juice. 1.01 makes perfect logical sense.

Final Answer:

$1.01 \text{ g/cm}^3$

Total: 4 marks

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Question 21 (2 marks)

Show that these two triangles are mathematically similar.

Worked Solution

Step 1: Checking for a common scale factor

Why we do this:

For two shapes to be “mathematically similar”, one must be an exact enlargement of the other. This means every side on the smaller triangle must be multiplied by the exact same number (the scale factor) to get the corresponding side on the larger triangle. We need to pair up the sides (shortest with shortest, longest with longest) and check if the multiplier is identical for all three.

Working:

Pair up the corresponding sides:

Shortest sides: $7.5$ and $3$
Middle sides: $10$ and $4$
Hypotenuses (longest sides): $12.5$ and $5$

Divide the large sides by the small sides to find the scale factor:

Calculator Steps:

7.5 ÷ 3 = $2.5$
10 ÷ 4 = $2.5$
12.5 ÷ 5 = $2.5$

✓ (M1) for a method to divide a pair of corresponding sides (e.g., $7.5 \div 3$) or stating the scale factor is $2.5$.

What this tells us:

Because every single pair gives exactly the same answer (2.5), the large triangle is a perfect $2.5\times$ magnification of the small one. They are definitively mathematically similar.

✓ (C1) for showing that all sides are enlarged by the same factor (or showing all corresponding angles are equal).

Final Answer:

$\frac{7.5}{3} = 2.5$, $\frac{10}{4} = 2.5$, $\frac{12.5}{5} = 2.5$.
Since all pairs of corresponding sides share the exact same scale factor (2.5), the triangles are mathematically similar.

Total: 2 marks

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Question 22 (4 marks)

(a) Complete the table of values for $y = \frac{6}{x}$ (2 marks)

x	0.5	1	1.5	2	3	4	5	6
y		6		3		1.5

(b) On the grid below, draw the graph of $y = \frac{6}{x}$ for values of $x$ from $0.5$ to $6$. (2 marks)

Worked Solution

Step 1: Completing the Table (Part a)

Why we do this:

To plot a graph, we need coordinates. We find the missing $y$ values by substituting the given $x$ values into the equation $y = \frac{6}{x}$ (which means 6 divided by x).

Working:

Calculator Steps:

For $x = 0.5$: 6 ÷ 0.5 = $12$
For $x = 1.5$: 6 ÷ 1.5 = $4$
For $x = 3$: 6 ÷ 3 = $2$
For $x = 5$: 6 ÷ 5 = $1.2$
For $x = 6$: 6 ÷ 6 = $1$

x	0.5	1	1.5	2	3	4	5	6
y	12	6	4	3	2	1.5	1.2	1

✓ (B2) for a fully correct table. (B1 is awarded for 3 or 4 correct values).

Step 2: Plotting the Graph (Part b)

Why we do this:

Now we map our table pairs onto the graph as coordinates: $(0.5, 12)$, $(1, 6)$, $(1.5, 4)$, etc. Because $y = \frac{6}{x}$ is a reciprocal curve, the points should form a smooth curve that swoops downwards, getting closer and closer to the axes but never touching them.

Working:

✓ (M1) for correctly plotting at least 6 points from their table.

✓ (A1) for drawing a fully correct, smooth curve passing through all points.

Final Answer:

(a) Missing values: 12, 4, 2, 1.2, 1

(b) Smooth reciprocal curve plotted accurately on the grid.

Total: 4 marks

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Question 23 (4 marks)

Harley’s house has a value of £160 000 correct to 2 significant figures.

(a) (i) Write down the least possible value of the house. (1 mark)

(ii) Write down the greatest possible value of the house. (1 mark)

The value of Rita’s house increased by $5\%$.
Her house then had a value of £210 000.

(b) Work out the value of Rita’s house before the increase. (2 marks)

Worked Solution

Step 1: Finding Error Intervals (Part a)

Why we do this:

When a number is rounded, it represents a range of possible real values. £160,000 is rounded to 2 significant figures. The 1 and the 6 are the significant figures, meaning it was rounded to the nearest 10,000.

To find the bounds, we halve the rounding degree (half of 10,000 is 5,000). We subtract 5,000 for the lower bound and add 5,000 for the upper bound.

Working:

Least possible value (Lower Bound): $160000 – 5000 = 155000$

✓ (B1) for correct least value $155000$.

Greatest possible value (Upper Bound): $160000 + 5000 = 165000$

Note: By convention in math, the upper bound is written as the exact halfway point, even though practically it would round up. So 165000 (or 164999.99) is accepted.

✓ (B1) for correct greatest value $165000$.

Step 2: Reverse Percentages (Part b)

Why we do this:

This is a reverse percentage problem. Do NOT just find 5% of £210,000 and subtract it. That is the most common mistake! The 5% increase was calculated on the original, unknown price, not the new price.

An increase of $5\%$ means the new value is $100\% + 5\% = 105\%$ of the original. To go backwards to the original $100\%$, we must divide by the multiplier $1.05$.

Working:

Let Original Price = $x$

\[ x \times 1.05 = 210000 \]

✓ (M1) for recognising that $210000 = 105\%$ or dividing by $1.05$.

To find $x$, we divide:

Calculator Steps:

Press: 210000 ÷ 1.05 =
Result: $200000$

✓ (A1) for the correct answer $200000$.

What this tells us:

We can check this going forwards: 5% of £200,000 is £10,000. £200,000 + £10,000 = £210,000. It works perfectly.

Final Answer:

(a)(i) £155 000

(a)(ii) £165 000

(b) £200 000

Total: 4 marks

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Edexcel GCSE Mathematics – June 2017Foundation Tier – Paper 3 (Calculator)

Mark Scheme Legend

Table of Contents

Question 1 (2 marks)

Worked Solution

Part (a): Ordering the rivers

Part (b): Checking Ami’s statement

Question 2 (2 marks)

Worked Solution

Step 1: Translating words into algebra

Question 3 (2 marks)

Worked Solution

Part (i): Smallest two-digit number

Part (ii): Number closest to 200

Question 4 (2 marks)

Worked Solution

Step 1: Finding what one fifth represents

Step 2: Finding the whole number

Question 5 (3 marks)

Worked Solution

Part (a): Converting fraction to ratio

Part (b): Calculating the number of grey tiles

Question 6 (5 marks)

Worked Solution

Part (a): Fixing the Median

Part (b): Finding the Range

Part (c): Calculating the Mean

Question 7 (2 marks)

Worked Solution

Step 1: Systematic Listing

Question 8 (2 marks)

Worked Solution

Step 1: Finding the remaining amount to pay

Step 2: Calculating the monthly payments

Question 9 (3 marks)

Worked Solution

Step 1: Finding total time taken

Step 2: Compare total time to available time

Question 10 (3 marks)

Worked Solution

Step 1: Understanding the Ratio

Step 2: Finding the length of one part and AB

Question 11 (4 marks)

Worked Solution

Part (a): Substituting a value into a formula

Part (b): Rearranging the formula

Question 12 (5 marks)

Worked Solution

Step 1: Finding the volume (Part a)

Step 2: Drawing the cuboid (Part bi)

Step 3: Calculating Surface Area (Part bii)

Question 13 (5 marks)

Worked Solution

Step 1: Forming Algebraic Expressions

Step 2: Solving the Equation

Step 3: Calculating all three angles

Question 14 (6 marks)

Worked Solution

Step 1: Find the total cost in Dollars (Part a)

Step 2: Convert to Pounds and find final total

Step 3: Reasoning about exchange rates (Part b)

Question 15 (6 marks)

Worked Solution

Step 1: Completing the Venn diagram

Step 2: Finding the probability

Question 16 (3 marks)

Worked Solution

Step 1: Eliminating one variable

Step 2: Finding the second variable

Question 17 (2 marks)

Worked Solution

Part (a): Finding the median from a frequency table

Part (b): Testing Probability Logic

Question 18 (5 marks)

Worked Solution

Step 1: Finding Vanilla and Banana cakes

Step 2: Finding the remaining cakes

Step 3: Sharing using ratio

Question 19 (4 marks)

Worked Solution

Edexcel GCSE Mathematics – June 2017
Foundation Tier – Paper 3 (Calculator)