KS2 Mathematics Paper 3: Reasoning (2017)

What are we asked? We need to find a number that we divide 75 by to get 7.5.

Why do we do this? This is a place value question. Comparing the starting number (75) with the answer (7.5), we can see the digits are the same, but the decimal point has moved.

How do we find the missing number? When a number becomes smaller by one decimal place (moves one column to the right on a place value chart), it has been divided by 10.

What this tells us: The missing number is 10.

What are we asking? We know the total amount (£80) and the amount per person (£16). We need to find the number of people.

Why do we do this? This is a division problem: \( \text{Total} \div \text{Amount per person} = \text{Number of people} \). Alternatively, we can think of it as “How many 16s go into 80?”.

Method: We can count up in 16s until we reach 80.

Check: Does \( 16 \times 5 = 80 \)? Yes, \( 10 \times 5 = 50 \) and \( 6 \times 5 = 30 \), and \( 50 + 30 = 80 \).

What is the condition? The answer must be a multiple of 10. This means the answer must end in a 0.

How do we get a 0 at the end? When multiplying integers, we get a 0 at the end if we multiply a number ending in 5 by an even number (like 2, 4, 6, 8).

Strategy: We have the digits 5, 6, and 9. We need to use them all: two for the 2-digit number and one for the 1-digit number. One of the numbers being multiplied MUST end in 5, and the other MUST be even (end in 6), to produce a 0.

Step 1: Read the graph.

Find 3 am on the bottom axis (x-axis) and go down to the line. The value on the side (y-axis) is −5°C.

Find 3 pm on the bottom axis and go up to the line. The value on the side is 2°C.

Step 2: Calculate the difference.

We need to find how much warmer 2°C is than −5°C. This is the difference between the numbers.

Step 1: Start with the 3 pm temperature.

From part (a), we know the temperature at 3 pm is 2°C.

Step 2: Apply the change.

The question says it was “4 degrees lower”. This means we subtract 4.

What do we need? We need the difference between the target (£360) and what they have (£57.73). This is a subtraction.

Important: Line up the decimal points. £360 can be written as 360.00 to match the two decimal places in £57.73.

Check: Does \( 302.27 + 57.73 = 360.00 \)?

\( 0.27 + 0.73 = 1.00 \)

\( 302 + 57 + 1 = 360 \). Correct.

What is 5:30 pm? In the 24-hour clock used in the table, we add 12 to the hour.

\( 5 + 12 = 17 \), so 5:30 pm is 17:30.

Method: Look at the “Arrives Paris” column. We need a time that is 17:30 or earlier.

What is the question asking? The latest time he can leave.

The arrival time 17:26 corresponds to the departure time 14:01.

The next train (leaving 14:31) arrives at 17:53, which is too late.

What are the corners? Let’s look at the three corners of the original triangle:

• Top corner: \( (-5, -1) \)
• Bottom-left corner: \( (-5, -4) \)
• Bottom-right corner: \( (-2, -4) \)

Rule: “7 right, 5 up”.

Mathematically: Add 7 to the x-coordinate, Add 5 to the y-coordinate.

Check: The new shape should look exactly the same size and orientation, just in a different place.

What are factors? Numbers that divide into 30 exactly.

Now list numbers that divide into 15 exactly.

Goal: Find numbers that are in the first list but NOT in the second list.

Method: Scan the timetable for “English” and note the day and time.

Method: Calculate the time difference for each slot.

Method: Add up all the times.

Problem: One value is in millilitres (568 ml) and the other is in litres (“half a litre”). We need them in the same unit.

Conversion: 1 litre = 1000 millilitres.

Half a litre = \( 1000 \div 2 = 500 \) ml.

Method: Subtract the amount poured out from the total.

Definition: The diameter is the distance across the full circle passing through the centre. The radius is the distance from the centre to the edge.

Relationship: The radius is exactly half of the diameter.

Adam buys 6 bags. Each bag has 24 balloons.

Chen buys 3 bags. Each bag has 12 balloons.

Adam claims he has four times as many. We need to check if Chen’s total multiplied by 4 equals Adam’s total.

A pentagon must have 5 straight sides.

• Top Left: 5 sides (Pentagon).
• Top Right: 5 sides (Pentagon).
• Middle Left (Arrowhead): 4 sides drawn? Let’s trace it: Up, Down-Left, Up-Left, Down. Wait, looking at the arrowhead shape \( \text{M} \). Vertices are Bottom-Left, Top-Left, Middle-V, Top-Right, Bottom-Right. It has 5 vertices and 5 sides. It is a pentagon.
• Middle Right (Bowtie): 6 sides. (Hexagon).
• Bottom: 6 sides (Hexagon).

We are left with the first three shapes.

Acute angle: Less than 90° (sharp corner).

• Top Left (Convex Pentagon): All angles look obtuse (wider than 90°).
• Top Right (House): Bottom two are right angles (90°). Top one is acute. Side ones are obtuse. Only 1 acute angle.
• Middle Left (Arrowhead):
  • Bottom left point: Acute.
  • Top left point: Acute.
  • Top right point: Acute.
  • Bottom right point: Acute.
  • Middle ‘V’: This is a reflex angle (greater than 180°) on the inside.
  This shape has exactly 4 acute angles.

We know 1 mango = £1.35. We need the cost of 2 mangoes.

We know 3 pineapples cost the same amount (£2.70).

To find the cost of one, we divide by 3.

• Parallel: Lines that never meet (like train tracks).
• Perpendicular: Lines that meet at a right angle (90° or square corner).

• A: No parallel lines (sides slope). No perpendicular lines.
• C: Curved.
• E:
  • The three horizontal bars are parallel to each other.
  • The horizontal bars meet the vertical back bar at 90° (perpendicular).
  • Result: Has BOTH.
• L: Has perpendicular lines (corner), but no parallel lines.
• Z: Top and bottom lines are parallel. Diagonal line is not perpendicular (angles are acute/obtuse).

William and Ally take 450 each. We need to find the total they took.

Subtract the amount taken from the total in the box.

Adam and Chen share the remaining 1500 equally (divide by 2).

To compare numbers easily, convert the mixed numbers to decimals.

Square Numbers: Numbers made by multiplying a whole number by itself (\(1 \times 1\), \(2 \times 2\), etc.).

We only need squares smaller than 22.

Subtract each square number from 22. Check if the result is a prime number (has exactly two factors: 1 and itself).

Problem: A number rounded to the nearest 10 is 50. What could that number be?

Range: Numbers from 45 up to (but not including) 55 round to 50.

Problem: Dev started with a whole number. This means the result after multiplying by 4 must be a multiple of 4.

We need to check which numbers in our list (45-54) are in the 4 times table.

Length: 3 cm longer than 20 cm.

Width: 2 cm narrower than 20 cm.

Look at the numbers we have next to each other: \( 1 \) and \( 1\frac{5}{8} \).

The difference is \( \frac{5}{8} \).

Let’s check if this works for the next number: \( 1\frac{5}{8} + \frac{5}{8} = 1\frac{10}{8} \).

\( \frac{10}{8} = 1\frac{2}{8} = 1\frac{1}{4} \). So \( 1 + 1\frac{1}{4} = 2\frac{1}{4} \).

This matches the next number in the sequence. The rule is + \(\frac{5}{8}\).

To find the number before 1, we subtract the step.

To find the number after \( 2\frac{1}{4} \), we add the step.

The top width and bottom width of the diagram are the same.

Top Width: \( w + 34 \)

Bottom Width: \( 18 + w + 9 + w \)

Set the two expressions equal to each other.

Subtract \(w\) from both sides, then subtract 27 from both sides.

Look at the \(b\) values: 9, 19, 29, 39.

How much do they go up by each time? The step is 10.

This means the first part of the rule involves multiplying \(a\) by 10.

Test \( 10 \times a \) with the first row.

Formula: Volume of a cube = \( \text{side} \times \text{side} \times \text{side} \).

Problem: Volume of B is also 216. We know two dimensions: 6 cm and 4 cm. We need the length (\(L\)).

Volume of Cuboid = \( \text{Length} \times \text{Width} \times \text{Height} \).

Divide the volume by 24.