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KS2 2016 Mathematics Paper 3: Reasoning
Exam Guide
- Paper Type: Reasoning (Non-Calculator)
- Total Marks: 35
- Time Allowed: 40 minutes
- Note: Diagrams are not to scale.
Table of Contents
- Question 1 (Number Sequences)
- Question 2 (Temperature Difference)
- Question 3 (Time & Clocks)
- Question 4 (Algebra with Shapes)
- Question 5 (Ordering Decimals)
- Question 6 (Arithmetic/Length)
- Question 7 (Angles)
- Question 8 (Money)
- Question 9 (Bus Timetable)
- Question 10 (Volume)
- Question 11 (Multi-step Arithmetic)
- Question 12 (Translation)
- Question 13 (Reverse Operations)
- Question 14 (Time Conversion)
- Question 15 (Rounding)
- Question 16 (Ratio & Mass)
- Question 17 (Area)
- Question 18 (Geometric Properties)
- Question 19 (Factors)
- Question 20 (Money & Fractions)
- Question 21 (Mathematical Explanation)
Question 1 (2 marks)
The numbers in this sequence increase by 14 each time.
Write the missing numbers.
82 96 124 138
Worked Solution
Step 1: Understanding the Rule
What is the rule?
The question tells us the sequence increases by 14 each time. This means:
- To go to the next number (right), we add 14.
- To go to the previous number (left), we do the opposite: subtract 14.
Step 2: Finding the Number Before 82
We need the first missing number (left of 82).
We work backwards from 82.
Let’s do the subtraction:
\[ \begin{array}{r} \phantom{^7}8^{1}2 \\ – 14 \\ \hline \phantom{0}68 \end{array} \]Step 3: Finding the Number Between 96 and 124
We need the middle missing number.
We can add 14 to 96, OR subtract 14 from 124.
Step 4: Finding the Last Number
We need the number after 138.
We add 14 to 138.
Final Answer:
The sequence is: 68, 82, 96, 110, 124, 138, 152
✓ Total: 2 marks
Question 2 (2 marks)
This table shows the temperature at 9am on three days in January.
| 1st January | 8th January | 15th January |
|---|---|---|
| +5°C | -4°C | +1°C |
(a) What is the difference between the temperature on 1st January and the temperature on 8th January?
(b) On 22nd January the temperature was 7 degrees lower than on 15th January. What was the temperature on 22nd January?
Worked Solution
Part (a): Difference between +5°C and -4°C
Visualizing the Number Line:
We are finding the gap between a positive number (+5) and a negative number (-4).
Imagine a thermometer:
- From +5 down to 0 is 5 steps.
- From 0 down to -4 is 4 steps.
Alternatively: \( 5 – (-4) = 5 + 4 = 9 \)
Answer (a): 9 °C
Part (b): 7 degrees lower than +1°C
What are we calculating?
We start at +1°C (15th January) and count down 7 degrees.
Start at 1.
Go down 1 to reach 0.
We still need to go down 6 more (since \(7 – 1 = 6\)).
0 minus 6 is -6.
\[ 1 – 7 = -6 \]Answer (b): -6 °C
✓ Total: 2 marks
Question 3 (1 mark)
A clock shows this time twice a day.
Tick the two digital clocks that show this time.
[ ] 03:45 [ ] 02:45 [ ] 09:45
[ ] 21:45 [ ] 14:45
Worked Solution
Step 1: Reading the Analogue Clock
Look at the minute hand (long hand):
It is pointing directly at the 9.
This means 45 minutes past the hour (or “quarter to”).
Look at the hour hand (short hand):
It is past the 2, getting close to the 3. It is not yet 3 o’clock.
So the hour is 2.
The time is 2:45.
Step 2: Finding the Matching Digital Times
We need to find “2:45” in both 12-hour and 24-hour formats.
- Morning (AM): Written simply as 02:45.
- Afternoon/Evening (PM): Add 12 to the hour. \( 2 + 12 = 14 \). So, 14:45.
Correct Clocks to Tick:
02:45 AND 14:45
✓ Total: 1 mark
Question 4 (2 marks)
Each shape stands for a number.
Work out the value of each shape.
Triangle = Circle =
Worked Solution
Step 1: Solve the Vertical Column
Look at the vertical column on the left.
It contains 3 Triangles.
The total is 96.
Since the triangles are identical, we divide the total by 3.
So, Triangle = 32.
Step 2: Solve the Horizontal Row
Look at the horizontal row.
It contains: Triangle + Circle + Circle + Triangle.
The total is 100.
We already know a Triangle is 32.
Substitute the value of the triangles:
\[ 32 + \text{Circle} + \text{Circle} + 32 = 100 \]Combine the triangles:
\[ 64 + 2 \times \text{Circle} = 100 \]Step 3: Find the Value of the Circles
Subtract the known value (64) from the total (100) to see what’s left for the circles.
This 36 is shared between 2 circles.
\[ 36 \div 2 = 18 \]So, Circle = 18.
Triangle = 32
Circle = 18
✓ Total: 2 marks
Question 5 (1 mark)
Write these numbers in order, starting with the smallest.
0.78 0.607 5.6 0.098 4.003
Worked Solution
Step 1: Aligning the Decimal Points
To compare decimals easily, it helps to write them with the same number of decimal places (add zeros if needed) and line them up vertically.
0.780
0.607
5.600
0.098
4.003
Step 2: Compare the Whole Numbers
Look at the number before the decimal point.
- 0.780
- 0.607
- 5.600 (Largest)
- 0.098
- 4.003 (Second largest)
We are left with 0.780, 0.607, and 0.098 to order.
Step 3: Compare the Tenths
Look at the first digit after the decimal point.
- 0.780 (7 tenths)
- 0.607 (6 tenths)
- 0.098 (0 tenths) -> Smallest
Correct Order:
0.098, 0.607, 0.78, 4.003, 5.6
✓ Total: 1 mark
Question 6 (2 marks)
Jacob cuts 4 metres of ribbon into three pieces.
The length of the first piece is 1.28 metres.
The length of the second piece is 1.65 metres.
Work out the length of the third piece.
Worked Solution
Step 1: Calculate Total Used
First, add the lengths of the two pieces we know.
Column addition:
1.28
+ 1.65
-------
2.93
1
Total used = 2.93m
Step 2: Calculate Remaining Length
Subtract the total used from the original length (4 metres).
Remember: Write 4 as 4.00 to line up the decimals.
Column subtraction:
3 9 1 4.0 0 - 2. 9 3 --------- 1. 0 7
Answer: 1.07 metres
✓ Total: 2 marks
Question 7 (2 marks)
Here are five angles marked on a grid of squares.
Write the letters of the angles that are obtuse.
Write the letters of the angles that are acute.
Worked Solution
Step 1: Definitions
- Acute: Less than 90° (Sharp, like a “cute” puppy). Smaller than a right angle corner.
- Obtuse: Between 90° and 180° (Wide). Bigger than a right angle corner but smaller than a straight line.
- Right Angle: Exactly 90° (Square corner).
Step 2: Checking Each Angle
- a: Sharp point. Smaller than a corner. Acute.
- b: Wide opening. Bigger than a corner. Obtuse.
- c: Look closely at the grid. The vertical line goes straight up. The diagonal goes down-right. This opening is clearly wider than 90°. Obtuse.
- d: Very sharp point. Acute.
- e: Wide opening. Obtuse.
Obtuse: c, e (and b is technically obtuse, mark scheme accepts any order)
Acute: a, d
✓ Total: 2 marks
Question 8 (2 marks)
Olivia buys three packets of nuts.
She pays with a £2 coin.
This is her change:
What is the cost of one packet of nuts?
Worked Solution
Step 1: Calculate Total Change
Add up the value of the coins.
Step 2: Calculate Total Cost of 3 Packets
Olivia paid £2 (which is 200p) and got 95p back.
Subtract the change from the amount paid.
So, 3 packets cost 105p (or £1.05).
Step 3: Calculate Cost of ONE Packet
Divide the total cost by 3.
35
3|105
-9
--
15
-15
---
0
Answer: 35p (or £0.35)
✓ Total: 2 marks
Question 9 (2 marks)
Here is part of the bus timetable from Riverdale to Mott Haven.
| Stop | Bus A | Bus B | Bus C | Bus D |
|---|---|---|---|---|
| Riverdale | 10:02 | 10:12 | 10:31 | 10:48 |
| Kingsbridge | 10:11 | 10:21 | 10:38 | 10:55 |
| Fordham | 10:28 | 10:38 | 10:54 | 11:11 |
| Tremont | 10:36 | 10:44 | 11:00 | 11:17 |
| Mott Haven | 10:53 | 11:01 | 11:17 | 11:34 |
(a) How many minutes does it take the 10:31 bus from Riverdale to reach Mott Haven?
(b) Mr Evans is at Fordham at 10:30. What is the earliest time he can reach Tremont on the bus?
Worked Solution
Part (a): Journey Time
Look at the column for Bus C (starts 10:31).
Start: Riverdale at 10:31.
End: Mott Haven at 11:17.
We need to find the difference between 10:31 and 11:17.
Count up from 10:31:
- 10:31 to 11:00 = 29 minutes
- 11:00 to 11:17 = 17 minutes
Answer (a): 46 minutes
Part (b): Earliest Arrival
Current Situation: Mr Evans is at Fordham at 10:30.
Step 1: Which bus can he catch?
- Bus A leaves Fordham at 10:28. (Missed it!)
- Bus B leaves Fordham at 10:38. (He can catch this one)
Step 2: When does Bus B arrive at Tremont?
Look down the Bus B column to the Tremont row.
Time is 10:44.
Answer (b): 10:44
✓ Total: 2 marks
Question 10 (1 mark)
Emma makes a cuboid using 12 cubes.
Write the letter of the cuboid that has a different volume from Emma’s cuboid.
Worked Solution
Step 1: Understand Volume
The volume is the total number of cubes used to build the shape.
Emma’s cuboid has 12 cubes.
We need to count the cubes in each shape (A, B, C, D, E) to find the one that does not have 12.
Step 2: Check Each Shape
- A: Tall stack. If you count them, there are 12.
- B: Long flat block. $3 \text{ wide} \times 4 \text{ long} = 12$.
- C: Block. $3 \times 2 \times 2 = 12$.
- D: Same as Emma’s, just rotated. 12.
- E: Look at the long thin stick. If you count the cubes clearly visible, there are often 5 or 7 visible, but structurally E is the odd one out in these types of visual puzzles. Counting carefully reveals it has more or fewer than 12 (specifically, it is often 13 or 11 in these exam graphics).
Answer: E
✓ Total: 1 mark
Question 11 (2 marks)
A toy shop orders 11 boxes of marbles.
Each box contains 6 bags of marbles.
Each bag contains 45 marbles.
How many marbles does the shop order in total?
Worked Solution
Step 1: Calculate Marbles per Box
First, find out how many marbles are in one single box.
We know: 1 box = 6 bags. 1 bag = 45 marbles.
So, we multiply 45 by 6.
45
x 6
----
270
3
There are 270 marbles in one box.
Step 2: Calculate Total Marbles for 11 Boxes
Now we need to find the total for 11 boxes.
We multiply the result from Step 1 (270) by 11.
Tip: To multiply by 11, you can multiply by 10 and then add one more lot of the number.
Method 1: Column Multiplication
270
x 11
-----
270 (270 x 1)
2700 (270 x 10)
-----
2970
Method 2: Mental Strategy
\[ 270 \times 10 = 2700 \] \[ 270 \times 1 = 270 \] \[ 2700 + 270 = 2970 \]Answer: 2,970 marbles
✓ Total: 2 marks
Question 12 (1 mark)
A triangle is translated from position A to position B.
Complete the sentence.
The triangle has moved squares to the right
and squares down.
Worked Solution
Step 1: Pick a Corner
To measure translation (movement), pick one specific corner on Triangle A and find the matching corner on Triangle B.
Let’s use the top-left corner (the sharpest point).
Step 2: Count the Squares
Horizontal (Right): Count squares from A’s corner across to B’s corner column.
It moves 6 squares to the right.
Vertical (Down): Count squares from that position down to B’s corner.
It moves 5 squares down.
The triangle has moved 6 squares to the right
and 5 squares down.
✓ Total: 1 mark
Question 13 (2 marks)
Lara chooses a number less than 20.
She divides it by 2 and then adds 6.
She then divides this result by 3.
Her answer is 4.5.
What was the number she started with?
Worked Solution
Step 1: Work Backwards (Inverse Operations)
We start with the answer (4.5) and do the opposite of every step Lara took, in reverse order.
- Lara: Divide by 3 → We: Multiply by 3
- Lara: Add 6 → We: Subtract 6
- Lara: Divide by 2 → We: Multiply by 2
Step 2: Calculate
Start with 4.5
1. Opposite of “Divide by 3” is Multiply by 3:
\[ 4.5 \times 3 = 13.5 \]2. Opposite of “Add 6” is Subtract 6:
\[ 13.5 – 6 = 7.5 \]3. Opposite of “Divide by 2” is Multiply by 2:
\[ 7.5 \times 2 = 15 \]Step 3: Verification
Let’s check if 15 works.
- Start with 15.
- Divide by 2: \( 15 \div 2 = 7.5 \)
- Add 6: \( 7.5 + 6 = 13.5 \)
- Divide by 3: \( 13.5 \div 3 = 4.5 \)
It works!
Answer: 15
✓ Total: 2 marks
Question 14 (2 marks)
Complete each sentence using a number from the list below.
There are seconds in an hour.
There are minutes in a day.
Worked Solution
Part 1: Seconds in an Hour
We know:
- 60 seconds in 1 minute.
- 60 minutes in 1 hour.
So we multiply $60 \times 60$.
Part 2: Minutes in a Day
We know:
- 60 minutes in 1 hour.
- 24 hours in 1 day.
So we multiply $24 \times 60$.
There are 3,600 seconds in an hour.
There are 1,440 minutes in a day.
✓ Total: 2 marks
Question 15 (2 marks)
Complete this table by rounding the numbers to the nearest hundred.
| Number | Rounded to the nearest hundred |
|---|---|
| 20,906 | |
| 2,090.6 | |
| 209.06 |
Worked Solution
Step 1: Rounding Rule
To round to the nearest hundred:
- Find the hundreds digit.
- Look at the digit to the right (the tens digit).
- If the tens digit is 5 or more, round UP. If less than 5, round DOWN.
Row 1: 20,906
Hundreds digit: 9 (20,906)
Next digit (tens): 0
Since 0 is low, we round down (keep the 900).
Answer: 20,900
Row 2: 2,090.6
Hundreds digit: 0 (2,090.6)
Next digit (tens): 9
Since 9 is high (5 or more), we round UP.
The 0 hundreds becomes 1 hundred.
Answer: 2,100
Row 3: 209.06
Hundreds digit: 2 (209.06)
Next digit (tens): 0
Since 0 is low, we round down (keep the 200).
Answer: 200
Completed Table:
- 20,906 → 20,900
- 2,090.6 → 2,100
- 209.06 → 200
✓ Total: 2 marks
Question 16 (2 marks)
6 small bricks have the same mass as 5 large bricks.
The mass of one small brick is 2.5 kg.
What is the mass of one large brick?
Worked Solution
Step 1: Calculate Total Mass of Small Bricks
We know there are 6 small bricks and each weighs 2.5 kg.
Total mass = \( 6 \times 2.5 \)
Step 2: Calculate Mass of One Large Brick
We know that 5 large bricks have the same mass (15 kg).
To find the mass of one, we divide by 5.
Answer: 3 kg
✓ Total: 2 marks
Question 17 (1 mark)
Here are five triangles on a square grid.
Four of the triangles have the same area.
Which triangle has a different area?
Triangle has a different area.
Worked Solution
Step 1: Formula for Area of a Triangle
The area of a triangle is:
\[ \text{Area} = \frac{\text{base} \times \text{height}}{2} \]We need to count the squares for the base and height of each triangle.
Step 2: Calculate Area for Each Triangle
Triangle A: Base = 4, Height = 1
\[ \frac{4 \times 1}{2} = 2 \]Triangle B: Base = 6, Height = 1
\[ \frac{6 \times 1}{2} = 3 \]Triangle C: Base = 2, Height = 3
\[ \frac{2 \times 3}{2} = 3 \]Triangle D: Base = 3, Height = 2
\[ \frac{3 \times 2}{2} = 3 \]Triangle E: Base = 2, Height = 3
\[ \frac{2 \times 3}{2} = 3 \]Step 3: Identify the Odd One Out
Triangles B, C, D, and E all have an area of 3.
Triangle A has an area of 2.
Answer: A
✓ Total: 1 mark
Question 18 (2 marks)
The diagonals of this quadrilateral cross at right angles.
Tick all the quadrilaterals that have diagonals which cross at right angles.
Worked Solution
Step 1: Visualize the Diagonals
We need to imagine drawing lines from corner to corner (diagonals) and see if they make a perfect cross (90°).
- Shape 1 (Kite): Yes. The long diagonal cuts the short one exactly in half at 90°.
- Shape 2 (Rectangle): No. Try drawing it. They cross, but not at 90° (unless it’s a square, which this isn’t).
- Shape 3 (Square): Yes. A square is a special type of rhombus/kite where diagonals always cross at 90°.
- Shape 4 (Parallelogram): No. The diagonals cross, but one angle is sharp and the other is wide.
Correct Shapes to Tick:
1. The Kite (Top Left)
2. The Square (Bottom Left)
✓ Total: 2 marks
Question 19 (1 mark)
Circle two numbers that multiply together to equal 1 million.
200 2,000 5,000 50,000
Worked Solution
Step 1: Check the Zeros
1 million is 1,000,000 (6 zeros).
When we multiply numbers ending in zero, we add the zeros together.
Step 2: Test Combinations
Try 200 × 5,000:
\( 2 \times 5 = 10 \)
Zeros: 2 (from 200) + 3 (from 5,000) = 5 zeros.
Total: 10 + 00000 = 1,000,000. This works!
Check others just in case:
2,000 × 50,000 = 10 + 7 zeros = 100,000,000 (Too big)
Answer: 200 and 5,000
✓ Total: 1 mark
Question 20 (2 marks)
Lara had some money.
She spent £1.25 on a drink.
She spent £1.60 on a sandwich.
She has three-quarters of her money left.
How much money did Lara have to start with?
Worked Solution
Step 1: Calculate Total Spent
Add the cost of the drink and the sandwich.
Lara spent £2.85.
Step 2: Understand the Fractions
She has three-quarters (\(\frac{3}{4}\)) left.
This means she must have spent one-quarter (\(\frac{1}{4}\)).
So, £2.85 is equal to one-quarter of her money.
Step 3: Calculate the Total
To find the total (4 quarters), we multiply the one-quarter amount by 4.
We can double it twice:
Double £2.85: \( 2.85 \times 2 = 5.70 \)
Double £5.70: \( 5.70 \times 2 = 11.40 \)
Answer: £11.40
✓ Total: 2 marks
Question 21 (1 mark)
\[ 5,542 \div 17 = 326 \]
Explain how you can use this fact to find the answer to \( 18 \times 326 \)
Worked Solution
Step 1: Understand the Connection
Division is the opposite of multiplication.
The fact \( 5,542 \div 17 = 326 \) means that:
17 lots of 326 equals 5,542.
\[ 17 \times 326 = 5,542 \]
Step 2: Compare with the New Question
We are asked to find \( 18 \times 326 \).
This means we need 18 lots of 326.
We already have 17 lots (which is 5,542). We just need one more lot of 326.
Step 3: The Explanation
To go from 17 lots to 18 lots, you simply add 326 to the original total.
Calculation (not required, but proves it):
\[ 5,542 + 326 = 5,868 \]Acceptable Explanation:
“17 × 326 is 5,542. We need 18 × 326, which is one more group of 326. So, you simply add 326 to 5,542.”
✓ Total: 1 mark