maths exam revision

Ratios: Method to simplify $12 : 18$ to its simplest form

Find the highest common factor ($\text{HCF} = 6$) and divide both terms by $6$:

• $12 \div 6 = 2$
• $18 \div 6 = 3$
• Simplest form: $2 : 3$
• Common mistake: Dividing the two sides by different numbers or altering the order of the ratio.

Ratios: Steps to divide $60 in the ratio $2 : 3$

1. Find total parts: $2 + 3 = 5\text{ parts}$
2. Find the value of one part: $\u002460 \div 5 = \u002412$
3. Multiply each ratio part by the single part value:
   • $2 \times \u002412 = \u002424$
   • $3 \times \u002412 = \u002436$

• Check: $\u002424 + \u002436 = \u002460$.

Ratios: Solve for $x$ in the equivalent ratio $4 : 5 = 12 : x$

$$x = 15$$

• Method: Determine the multiplier between the first terms: $12 \div 4 = 3$.
• Multiply the second term by the same factor: $5 \times 3 = 15$.

Ratios: How to simplify a ratio with mixed units such as $20\,\text{mm} : 5\,\text{cm}$

1. Convert both quantities to the same unit first: $5\,\text{cm} = 50\,\text{mm}$.
2. Write the ratio without units: $20 : 50$.
3. Divide both terms by their $\text{HCF}$ ($10$): $2 : 5$.

• Common mistake: Simplifying without converting units ($20 : 5 = 4 : 1$ is incorrect).

Algebra: Simplify by collecting like terms: $4x + 7y - 2x + 3y$

$$2x + 10y$$

• Method: Group terms that have identical variable parts:
  • $(4x - 2x) = 2x$
  • $(7y + 3y) = 10y$
• Common mistake: Adding $2x + 10y$ together to get $12xy$. Unlike terms cannot be combined.

Algebra: Expand using the distributive law: $3(2x - 5)$

$$6x - 15$$

• Method: Multiply the term outside the brackets by every term inside:
  • $3 \times 2x = 6x$
  • $3 \times (-5) = -15$
• Common mistake: Forgetting to multiply the constant term ($3 \times 2x - 5 = 6x - 5$ is incorrect).

Algebra: Factorise completely: $8x + 12$

$$4(2x + 3)$$

• Method:
  1. Find the highest common factor ($\text{HCF}$) of $8x$ and $12$, which is $4$.
  2. Divide each term by $4$: $8x \div 4 = 2x$ and $12 \div 4 = 3$.
  3. Write the $\text{HCF}$ outside the brackets: $4(2x + 3)$.
• Check: $4 \times 2x + 4 \times 3 = 8x + 12$.

Patterns & Linear Relationships: Find the rule and the 10th term for $3, 7, 11, 15, \dots$

• General rule: $\text{Term} = 4n - 1$
• 10th term ($n = 10$): $4(10) - 1 = 39$
• Method: The constant difference is $+4$, giving $4n$. For $n = 1$, $4(1) - 1 = 3$, so the adjustment is $-1$.

Patterns & Linear Relationships: For the rule $y = 3x - 2$, find $y$ when $x = 4$ and find $x$ when $y = 13$

• When $x = 4$: $y = 3(4) - 2 = 12 - 2 = 10$
• When $y = 13$: $13 = 3x - 2 \rightarrow 15 = 3x \rightarrow x = 5$
• Key idea: Substitute the given value into the equation and solve for the unknown.

Patterns & Linear Relationships: Definition and coordinates of $x$- and $y$-intercepts

• $x$-intercept: The point where the graph crosses the $x$-axis, where $y = 0$ ($(x, 0)$).
• $y$-intercept: The point where the graph crosses the $y$-axis, where $x = 0$ ($(0, y)$).
• Common mistake: Swapping the coordinate order when plotting.

Fractions, Decimals & Percentages: Calculate $\frac{2}{3} + \frac{1}{4}$

$$\frac{11}{12}$$

• Method:
  1. Find the lowest common denominator ($\text{LCD} = 12$).
  2. Convert to equivalent fractions: $\frac{2}{3} = \frac{8}{12}$ and $\frac{1}{4} = \frac{3}{12}$.
  3. Add numerators: $\frac{8 + 3}{12} = \frac{11}{12}$.
• Common mistake: Adding straight across ($\frac{2+1}{3+4} = \frac{3}{7}$ is incorrect).

Fractions, Decimals & Percentages: Calculate $\frac{3}{5} \div \frac{2}{7}$

$$\frac{21}{10} = 2\frac{1}{10}$$

• Method: Use Keep, Change, Flip (KCF):
  1. Keep $\frac{3}{5}$.
  2. Change $\div$ to $\times$.
  3. Flip $\frac{2}{7}$ to $\frac{7}{2}$.
  4. Multiply numerators and denominators: $\frac{3 \times 7}{5 \times 2} = \frac{21}{10}$.

Fractions, Decimals & Percentages: Convert $0.35$ into a fraction in simplest form and a percentage

• As a percentage: $0.35 \times 100\% = 35\%$
• As a fraction: $\frac{35}{100} = \frac{7}{20}$ (dividing numerator and denominator by $5$).
• Method: Two decimal places represent hundredths.

Fractions, Decimals & Percentages: Calculate $15\%$ of $\u002480$

$$\u002412$$

• Mental strategy:
  • $10\%\text{ of }\u002480 = \u00248$
  • $5\%\text{ of }\u002480 = \u00244$
  • $15\% = \u00248 + \u00244 = \u002412$
• Formula method: $\frac{15}{100} \times 80 = 0.15 \times 80 = 12$.

Angles: Key angle sum rules for complementary, supplementary, and revolution

• Complementary angles: Sum to $90^\circ$
• Supplementary angles: Sum to $180^\circ$
• Angles at a point (revolution): Sum to $360^\circ$
• Vertically opposite angles: Exactly equal to each other.

Angles: The three angle pairs formed by parallel lines and a transversal

1. Corresponding angles (F-shape): Equal ($\angle 1 = \angle 2$)
2. Alternate angles (Z-shape): Equal ($\angle 1 = \angle 2$)
3. Co-interior angles (C- or U-shape): Supplementary ($\angle 1 + \angle 2 = 180^\circ$)

• Common mistake: Assuming co-interior angles are equal instead of supplementary.

Angles: Calculate $x$ if two co-interior angles on parallel lines are $2x$ and $80^\circ$

$$x = 50^\circ$$

• Method: Co-interior angles sum to $180^\circ$:
  • $2x + 80^\circ = 180^\circ$
  • $2x = 100^\circ$
  • $x = 50^\circ$
• Reason: Co-interior angles on parallel lines are supplementary.

Triangles & Quadrilaterals: Interior angle sums and exterior angle property

• Triangle angle sum: Interior angles add to $180^\circ$.
• Quadrilateral angle sum: Interior angles add to $360^\circ$.
• Exterior angle of a triangle: Equal to the sum of the two opposite interior angles ($d = a + b$).

Triangles & Quadrilaterals: In an isosceles triangle, the vertex angle is $40^\circ$. Find each base angle.

Each base angle is $70^\circ$.

• Method:
  1. Base angles in an isosceles triangle are equal.
  2. Subtract the vertex angle from the total sum: $180^\circ - 40^\circ = 140^\circ$.
  3. Divide by $2$: $140^\circ \div 2 = 70^\circ$.

Triangles & Quadrilaterals: Distinguishing properties of a Rhombus compared to a Parallelogram

• A rhombus has all $4$ sides equal in length (a general parallelogram only has opposite sides equal).
• The diagonals of a rhombus bisect each other at right angles ($90^\circ$).
• The diagonals of a rhombus bisect the vertex angles.

Equations: Solve the two-step equation $3x + 5 = 26$

$$x = 7$$

• Steps:
  1. Subtract $5$ from both sides: $3x = 21$
  2. Divide both sides by $3$: $x = 7$
• Check: $3(7) + 5 = 21 + 5 = 26$.

Equations: Solve the two-step equation $\frac{x}{4} - 3 = 5$

$$x = 32$$

• Steps:
  1. Add $3$ to both sides: $\frac{x}{4} = 8$
  2. Multiply both sides by $4$: $x = 8 \times 4 = 32$
• Common mistake: Multiplying by $4$ first without multiplying the $-3$ term.

Equations: Solve the equation with grouping symbols: $2(x - 4) = 14$

$$x = 11$$

• Method 1 (Expand first): $2x - 8 = 14 \rightarrow 2x = 22 \rightarrow x = 11$
• Method 2 (Divide first): $x - 4 = 14 \div 2 \rightarrow x - 4 = 7 \rightarrow x = 11$.

Perimeter & Circumference: Formulas for the circumference of a circle

• Using radius: $C = 2\pi r$
• Using diameter: $C = \pi d$
• Common mistake: Confusing radius and diameter; always check if the question provides $r$ or $d$ ($d = 2r$).

Perimeter & Circumference: Find the perimeter of a semicircle with diameter $14\,\text{cm}$ (use $\pi \approx \frac{22}{7}$)

$$36\,\text{cm}$$

• Steps:
  1. Curved boundary: $\frac{1}{2} \times \pi d = \frac{1}{2} \times \frac{22}{7} \times 14 = 22\,\text{cm}$
  2. Add the straight base diameter: $22 + 14 = 36\,\text{cm}$
• Common mistake: Forgetting to add the straight edge to complete the closed perimeter.

Perimeter & Circumference: Find the perimeter of a regular hexagon with side length $4.5\,\text{cm}$

$$27\,\text{cm}$$

• Method: A regular hexagon has $6$ equal sides.
• $P = 6 \times s = 6 \times 4.5\,\text{cm} = 27\,\text{cm}$.

Area: Formulas for the area of a Triangle, Parallelogram, and Trapezium

• Triangle: $A = \frac{1}{2}bh$
• Parallelogram: $A = bh$
• Trapezium: $A = \frac{1}{2}(a + b)h$, where $a$ and $b$ are the parallel sides.
• Crucial rule: Always use the perpendicular height ($h$), never the slant height.

Area: Calculate the area of a circle with radius $r = 7\,\text{m}$ (use $\pi \approx \frac{22}{7}$)

$$154\,\text{m}^2$$

• Formula: $A = \pi r^2$
• Calculation: $A = \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 22 \times 7 = 154\,\text{m}^2$
• Common mistake: Multiplying radius by $2$ instead of squaring ($r^2 = r \times r$).

Area: Step-by-step strategy for calculating composite area

1. Split the shape into non-overlapping standard geometric shapes (e.g., rectangles, triangles).
2. Calculate missing side lengths using opposite parallel sides.
3. Compute the area of each separate component ($A<em>1, A</em>2, \dots$).
4. Add (or subtract for hollow regions) the individual areas to find $A_{\text{total}}$.

Area: Calculate the area of a trapezium with parallel sides $6\,\text{cm}$ and $10\,\text{cm}$, and perpendicular height $5\,\text{cm}$

$$40\,\text{cm}^2$$

• Calculation:
  • $A = \frac{1}{2}(a + b)h$
  • $A = \frac{1}{2}(6 + 10) \times 5$
  • $A = \frac{1}{2} \times 16 \times 5 = 8 \times 5 = 40\,\text{cm}^2$.

Volume & Capacity: General volume formulas for any right prism and cylinder

• Any right prism: $V = Ah$, where $A$ is the uniform cross-sectional area and $h$ is height/length.
• Cylinder (circular cross-section): $V = \pi r^2 h$
• Units: Volume is measured in cubic units ($\text{cm}^3$, $\text{m}^3$).

Volume & Capacity: Key conversion rules connecting volume to capacity

• $1\,\text{cm}^3 = 1\,\text{mL}$
• $1000\,\text{cm}^3 = 1\,\text{L}$
• $1\,\text{m}^3 = 1000\,\text{L} = 1\,\text{kL}$
• Common mistake: Assuming $1\,\text{m}^3 = 1000\,\text{mL}$ instead of $1000\,\text{L}$.

Volume & Capacity: Calculate the volume of a triangular prism with triangle base $6\,\text{cm}$, triangle height $4\,\text{cm}$, and prism length $10\,\text{cm}$

$$120\,\text{cm}^3$$

• Steps:
  1. Find the area of the triangular cross-section: $A = \frac{1}{2} \times 6 \times 4 = 12\,\text{cm}^2$
  2. Multiply by the length of the prism: $V = A \times h = 12 \times 10 = 120\,\text{cm}^3$.

Volume & Capacity: Calculate the exact volume of a cylinder with radius $3\,\text{cm}$ and height $10\,\text{cm}$ in terms of $\pi$

$$90\pi\,\text{cm}^3$$

• Method: $V = \pi r^2 h$
• Substitute values: $V = \pi \times 3^2 \times 10 = \pi \times 9 \times 10 = 90\pi\,\text{cm}^3$
• (Approximate decimal value: $\approx 282.74\,\text{cm}^3$).