Why we do this: To find the closest number, we calculate the difference (distance) between each number and 500. The smallest difference is the closest.

Why we do this: We look for the largest difference calculated above.

Why we do this: To order them, we look at the number of digits and the value of the highest place value columns first.

Why we do this: Start from the rightmost column. We need a number that adds to 4 to make 5.

Why we do this: The result ends in 1. Since $5 + \text{something}$ cannot equal 1, it must equal 11. This means we carry 1.

Why we do this: Add the digits we have plus the carried 1.

Why we do this: “Total” means addition. We need to add the population of Formby (24,478) and Telford (166,640).

Why we do this: “Difference” means subtraction. We subtract Dover (34,087) from Bedford (82,448).

Why we do this: We check the properties of each number to see which sets (Prime, Square, Even) they belong to.

Why we do this: For a reflection, every point on the new shape must be the same distance from the mirror line as the original point, but on the opposite side.

Why we do this: The middle fraction is $\frac{8}{12}$. We can simplify it or compare it to others.

Why we do this: Now compare $\frac{8}{12}$ to $\frac{4}{\square}$.

Why we do this: It helps to write all numbers with the same number of decimal places to avoid confusion.

0.25 is the target.

Why we do this: We know the cost of 6 pencils (£1.68). 3 pencils is half of 6 pencils.

Why we do this: The second statement says “3 pencils and 1 rubber cost £1.09”. We now know the 3 pencils part is £0.84.

Why we do this: $\frac{3}{4}$ means for every 4 equal parts, we shade 3. Or if there are 8 parts, we shade 6 ($\frac{6}{8} = \frac{3}{4}$).

• Square: Divided into 4 triangles. Shade 3 triangles.
• Grid Shape: Divided into squares with diagonals. Total 6 triangles? Or 8? Let’s check: 3 squares x 2 triangles = 6 triangles. $\frac{3}{4}$ of 6 is 4.5? That doesn’t work.
• Let’s re-examine Shape 2 (L-shape). It’s 4 squares? Top row 2, Middle row 1? No.
• It looks like 4 squares in an upside down L. 3 across top, 1 down?
• Let’s count sections. If we shade 3/4, the number of sections must be a multiple of 4. Likely 4, 8, 12.
• Counting from screenshot p12: Top row has 3 squares. Bottom left has 1 square. Total 4 squares. Each square cut in half? No, only some?
• Actually, the grid lines show 4 squares total. 3 in top row, 1 in bottom left. Each has a diagonal. Total 8 triangles.
• Target: $\frac{3}{4}$ of 8 = 6 triangles.
• Circle: Divided into 8 sectors. $\frac{3}{4}$ of 8 = 6 sectors.

Why we do this: We have kg and g. We must work in the same unit. Let’s convert kg to g.

1 kg = 1000 g.

Why we do this: To find the number of days, divide the total amount by the amount used per day.

Why we do this: We need to isolate $q$. We do the inverse operations in reverse order.

Why we do this: We know 20 boxes = 140 cm. Division gives the height of a single box.

Why we do this: If 3 boxes are removed, there are 17 boxes left. Or we can subtract the height of 3 boxes.

Why we do this: A common multiple is a number that is in both times tables.

Why we do this: Look at the gap between 20°C and 30°C. 25°C is exactly halfway.

Why we do this: Since 25 is halfway between 20 and 30, the answer will be halfway between 68 and 86.

Why we do this: The angle marked $160^\circ$ and angle $a$ are vertically opposite angles. Vertically opposite angles are equal.

Why we do this: The shape formed is a quadrilateral (4-sided shape). The sum of angles in a quadrilateral is $360^\circ$.

The angles are:

• Two right angles ($90^\circ$ each)
• Angle $a$ ($160^\circ$)
• Angle $b$

Why we do this: $70 \div \text{something} = 3.5$ is the same as $70 \div 3.5 = \text{something}$.

Note: 10 jars cost £6.90. She makes 20 jars, so she needs 2 sets of jars.

Why we do this: The question gives the coordinates of all three vertices of A in the diagram: $(-10, 40)$, $(20, 30)$, and $(10, 10)$.

The question says B has new vertices $(10, -10)$ and $(20, -30)$.

These correspond to reflecting $(10, 10)$ and $(20, 30)$.

We need to reflect the third vertex: $(-10, 40)$.

Rule: To reflect in the x-axis, the x-coordinate stays the same, and the y-coordinate changes sign (positive becomes negative).

$(x, y) \rightarrow (x, -y)$