KS2 Mathematics 2019 Paper 2: Reasoning

Why: We have two numbers and an operation in the first row.

How: \( 4 \times 8 = ? \)

The answer is 32.

Why: Looking at the third column, we have \( 8 \), a missing number, and the answer \( 56 \).

How: \( 8 \times ? = 56 \)

We can work backwards: \( 56 \div 8 = 7 \).

The missing number in the middle of the grid is 7.

Why: Now we know the top number is 4 and the middle number is 3.

How: \( 4 \times 3 = ? \)

The answer is 12.

Check: Does the middle horizontal row work with our found number 7?

\( 3 \times 7 = 21 \). Yes, it matches.

Why: We are asked to find \( 1,000 \) less. This means we need to decrease the digit in the thousands place by 1.

The number is \( \mathbf{9},072 \).

The thousands digit is 9.

How: Subtract 1 from the thousands digit.

\( 9 – 1 = 8 \)

The hundreds, tens, and ones stay the same because we are subtracting zeros from them (\( 072 – 000 = 072 \)).

\( 9,072 – 1,000 = 8,072 \)

All numbers start with \( 1 \) million. We must look at the next digits.

• \( 1,\mathbf{0}09,909 \)
• \( 1,\mathbf{0}23,065 \)
• \( 1,\mathbf{0}09,099 \)
• \( 1,\mathbf{2}30,650 \)

Looking at the digit after the comma:

• \( 1,0… \)
• \( 1,0… \)
• \( 1,0… \)
• \( 1,\mathbf{2}… \) → This has the highest digit (2). So 1,230,650 is 1st (Largest).

Remaining numbers:

• \( 1,0\mathbf{0}9,909 \)
• \( 1,0\mathbf{2}3,065 \)
• \( 1,0\mathbf{0}9,099 \)

\( 1,023,065 \) has a 2 in the ten-thousands place. The others have 0.

So 1,023,065 is 2nd.

Remaining:

• \( 1,009,\mathbf{9}09 \) (hundreds digit is 9)
• \( 1,009,\mathbf{0}99 \) (hundreds digit is 0)

909 is larger than 099.

So 1,009,909 is 3rd and 1,009,099 is 4th (Smallest).

Method: For reflection, every point on the new shape must be the same distance from the mirror line as the original, but on the opposite side.

• Point A (Top Left): 2 squares right of mirror → Reflected to 2 squares left.
• Point B (Top Right): 4 squares right of mirror → Reflected to 4 squares left.
• Point C (Bottom Right): 4 squares right → Reflected to 4 squares left.
• Point D (Bottom Left): 1 square right → Reflected to 1 square left.
• Point E (Inner Point): 3 squares right → Reflected to 3 squares left.

Why: The rule is “increase by 45”. To go backwards (to the left), we must do the opposite: subtract 45.

How:

  155
-  45
─────
  110

So the first number is 110.

Why: To go forwards (to the right), we add 45.

How:

Next number:

  245
+  45
─────
  290

Number after that:

  290
+  45
─────
  335

Why: We are starting with \( 0.3 \) and ending with \( 0.03 \).

The digits are the same (a single 3), but the position has changed. The 3 has moved one place to the right (from the tenths column to the hundredths column).

How: To move digits one place to the right, we divide by 10.

\[ 0.3 \div 10 = 0.03 \]

Why: To read the level, we need to know what each small mark represents.

How: Look at the gap between 3 and 4.

There are 4 spaces between 3 and 4.

\( 1 \text{ litre} \div 4 = 0.25 \text{ litres} \).

So, each small mark counts up by 0.25.

How: Start at 3 and count up the marks.

• 1st mark: \( 3.25 \)
• 2nd mark: \( 3.50 \)
• 3rd mark: \( 3.75 \)

The paint is level with the third mark above 3.

Why: We simply follow the rule: Multiply by 2, then add 3.

How:

1. Multiply 53 by 2:

\[ 53 \times 2 = 106 \]

2. Add 3:

\[ 106 + 3 = 109 \]

The last number is 109.

Why: To go backwards, we must use the inverse (opposite) operations in the reverse order.

Rule: \( \times 2 \rightarrow +3 \)

Inverse Rule: \( -3 \rightarrow \div 2 \)

How:

Start with 25.

1. Subtract 3:

\[ 25 – 3 = 22 \]

2. Divide by 2:

\[ 22 \div 2 = 11 \]

The first number is 11.

Check: Does 11 work? \( 11 \times 2 = 22 \). \( 22 + 3 = 25 \). Yes.

Why: We can think of this as a missing number problem.

\[ ? \times 7 + 85 = 953 \]

To find the missing number, we work backwards from the answer using inverse operations.

How: The last thing Jack did was add 85. So, we subtract 85 from 953.

  953
-  85
─────
  868

How: The first thing Jack did was multiply by 7. So, we divide our current number by 7.

\[ 868 \div 7 \]

    124
  ┌────
7 │ 868

Why: We need to translate the word problem into a mathematical expression.

• “Each ticket costs £24” means if you buy \( n \) tickets, you pay \( n \times 24 \).
• “There is a £3 charge” means a single fee added on top, regardless of how many tickets.

How:

Cost = (Cost of tickets) + (Booking fee)

Cost = (number of tickets \( \times \) 24) + 3

Why: We need to convert “one-quarter” into a decimal.

How:

\[ \frac{1}{4} = 1 \div 4 \]

We can also think of it as half of a half. Half of 1 is 0.5. Half of 0.5 is 0.25.

Answer: 0.25

Why: Change is the difference between the money given and the cost.

How: £2.00 – £1.35

  1 9 1
  2.0 0
- 1.3 5
───────
  0.6 5

Answer: £0.65 or 65p

Compare: \( \frac{7}{10} \) vs \( 0.07 \)

Convert fraction to decimal: \( \frac{7}{10} = 0.7 \)

Now compare 0.7 and 0.07.

0.7 is much bigger (it has 7 tenths, the other has 0 tenths).

So, \( \frac{7}{10} \mathbf{>} 0.07 \)

Compare: \( \frac{23}{1000} \) vs \( 0.23 \)

Convert fraction to decimal: \( \frac{23}{1000} = 0.023 \) (23 thousandths)

Convert 0.23 to thousandths to compare easier: \( 0.230 \)

0.023 is smaller than 0.230.

So, \( \frac{23}{1000} \mathbf{<} 0.23 \)

What we know:

• The base line is 8 cm.
• The angle at the left end is 35°.
• The angle at the right end is a right angle (90°).

How to draw:

1. Start with the 8 cm line provided.
2. Place your protractor on the left end of the line. Mark 35°. Draw a long light line through this mark.
3. Go to the right end of the line. Draw a vertical line straight up (90°) using a set square or protractor.
4. The point where your 35° line and your vertical line cross is the third corner of the triangle.

Number: 39,476

Target: Ten-thousands digit is 3 (representing 30,000).

Check: Look at the next digit (thousands) -> 9.

Since 9 is 5 or more, we round UP.

30,000 becomes 40,000.

Answer: 40,000

Number: 39,476

Target: Thousands digit is 9.

Check: Look at the next digit (hundreds) -> 4.

Since 4 is less than 5, we round DOWN (stay the same).

The 39,000 stays as it is.

Answer: 39,000

Number: 39,476

Target: Hundreds digit is 4.

Check: Look at the next digit (tens) -> 7.

Since 7 is 5 or more, we round UP.

400 becomes 500.

Answer: 39,500

Why: We need to find the portion of children who chose Orange out of the total.

Orange = 15

Total = 60

Fraction = \( \frac{15}{60} \)

How: Both numbers can be divided by 15.

15 goes into 60 exactly 4 times.

\[ \frac{15}{60} = \frac{1}{4} \]

How: We know that \( \frac{1}{4} \) is a quarter.

A quarter of 100% is 25%.

Answer: 25%

Why: Multiplication must be done before Addition or Subtraction.

How: Calculate \( 2 \times 2 \) first.

\[ 2 \times 2 = 4 \]

How: Put the 4 back into the equation.

\[ 6 + 4 – \Box = 6 \]

Add the first part: \( 6 + 4 = 10 \).

Now we have: \( 10 – \Box = 6 \)

How: \( 10 – ? = 6 \)

\( 10 – 6 = 4 \)

The missing number is 4.

Why: A regular hexagon has 6 equal sides.

How:

\[ \text{Perimeter} = 6 \times 8 \text{ cm} \]

\[ \text{Perimeter} = 48 \text{ cm} \]

Why: The square has the same perimeter as the hexagon (48 cm). A square has 4 equal sides.

How:

\[ \text{Side of square} = 48 \div 4 \]

\[ \text{Side} = 12 \text{ cm} \]

Why: Area of a square = side \(\times\) side.

How:

\[ \text{Area} = 12 \times 12 \]

\[ \text{Area} = 144 \text{ cm}^2 \]

Analysis: It ends in a 5.

Any number ending in 5 (except 5 itself) can be divided by 5.

\( 95 \div 5 = 19 \). So, 95 is not prime.

Analysis: It ends in 7, so it’s odd. Let’s check if it divides by 3.

Rule: Add the digits: \( 8 + 7 = 15 \).

Since 15 is in the 3 times table, 87 can be divided by 3.

\( 87 \div 3 = 29 \). So, 87 is not prime.

Analysis: It’s not even. It doesn’t end in 5. Digits add to 17 (not div by 3). Not in 7 times table (\( 7 \times 12 = 84 \)).

89 is the prime number.

Why: We know the rate is every 4 seconds. We need to know how many “4 seconds” are in a minute.

1 minute = 60 seconds.

How:

\[ 60 \div 4 = 15 \]

The machine pours 15 times in one minute.

Why: Each pour is 250 ml.

How:

\[ 15 \times 250 \]

Calculation Trick: \( 10 \times 250 = 2500 \). \( 5 \times 250 = 1250 \).

\( 2500 + 1250 = 3750 \text{ ml} \)

Why: The question asks for the answer in litres.

1 Litre = 1000 millilitres.

How:

\[ 3750 \div 1000 = 3.75 \]

Why: Percent means “out of 100”.

\[ 20\% = \frac{20}{100} \]

We can simplify this fraction: Divide top and bottom by 20.

\[ \frac{20 \div 20}{100 \div 20} = \frac{1}{5} \]

So we are looking for fractions equivalent to \( \frac{1}{5} \).

• \( \frac{1}{20} \): Not equal to \( \frac{1}{5} \). (False)
• \( \frac{20}{40} \): Simplifies to \( \frac{1}{2} \) (50%). (False)
• \( \frac{1}{5} \): This matches exactly. (True)
• \( \frac{3}{15} \): Divide top and bottom by 3 &rightarrow; \( \frac{1}{5} \). (True)
• \( \frac{2}{100} \): Simplifies to \( \frac{1}{50} \) (2%). (False)

Dimensions: The grid is 6 squares wide and 5 squares high.

Goal: Make two squares of different sizes and one rectangle.

Strategy: Find the largest square possible first.

The grid is 5 squares high, so we can make a 5 by 5 square.

Cut 1: Cut vertically after the 5th column. This leaves a 5×5 square on the left and a 1×5 rectangle strip on the right.

Now we need another square from the remaining 1×5 strip.

Cut 2: Cut horizontally after the 1st row in the strip. This creates a 1 by 1 square at the top.

Result:

• Shape 1: 5×5 Square (Large)
• Shape 2: 1×1 Square (Small) – Different size ✅
• Shape 3: 1×4 Rectangle (Remainder) ✅

Why: We look for values smaller than 10.

Data:

• Mon: 8.1 (Below)
• Tue: 9.3 (Below)
• Wed: 11.9 (Above)
• Thu: 11.8 (Above)
• Fri: 12.4 (Above)

Result: 2 days out of 5.

Answer: \( \frac{2}{5} \)

Why: Mean = Sum of values ÷ Number of values.

Sum:

   8.1
   9.3
  11.9
  11.8
+ 12.4
──────
  53.5

Divide: \( 53.5 \div 5 \)

    10.7
  ┌─────
5 │ 53.5

Answer: 10.7 °C

Why: Volume = Length \(\times\) Width \(\times\) Height.

Dimensions: 6 cm, 3 cm, 4 cm.

Calculation:

\[ 6 \times 3 \times 4 = 18 \times 4 = 72 \text{ cubes} \]

Rule: Add 5 cm to each dimension.

• Length: \( 6 + 5 = 11 \) cm
• Width: \( 3 + 5 = 8 \) cm
• Height: \( 4 + 5 = 9 \) cm

Calculation:

\[ 11 \times 8 \times 9 \]

\( 11 \times 8 = 88 \)

\( 88 \times 9 \)

   88
 ×  9
─────
  792

Why: “Difference” means subtract.

How:

\[ 792 – 72 = 720 \]