S2 4th Level Mathematics Revision

Algebraic Expressions and Equations

• Simplification: Combine terms through multiplication and division. For example, expressions like $9a \times 4a$ and the reduction of algebraic fractions such as $\frac{20ab}{2a}$.

• Equations: Solve linear equations with variables on both sides (e.g., $4x + 1 = 2x + 7$) by isolating the variable.

• Inequalities: Solve inequations using similar algebraic principles to determine the range of values for $x$ (e.g., 5x + 1 < 31).

Circle Geometry

• Circumference: Calculate the perimeter using the formula $C = \pi \times d$. Examples include circles with diameters of $3\,cm$, $10\,cm$, and $4.8\,cm$.

• Area: Calculate the interior space using the formula $A = \pi \times r^2$. Relevant inputs include radii of $7\,cm$, $8\,cm$, and $7.5\,mm$, and a diameter of $20\,mm$.

Pythagoras’ Theorem

• Core Calculation: Use $a^2 + b^2 = c^2$ to find the length of the hypotenuse or shorter sides in right-angled triangles.

• Applications:   - Finding diagonal lengths of rectangular objects, such as a Scottish Flag measuring $2.35\,m$ by $1.86\,m$.   - Calculating the vertical height of an isosceles triangle (e.g., a warning sign with a base of $44\,cm$ and side of $48\,cm$).   - Determining horizontal distance ($d$) for projectile paths.

Trigonometry

• Ratios: Utilize SOHCAHTOA ($\sin$, $\cos$, and $\tan$) to find unknown side lengths ($x$) and angle sizes ($x^\circ$).

• Application: Calculating the length of a beam of light based on its height from the floor ($4.62\,m$) and its downward angle of elevation/depression ($70^\circ$).

Percentages and Finance

• Percentage Adjustments: Calculate increases (e.g., $12.5\%$ of 280) and decreases (e.g., $1.28\%$ of 40000).

• Wages and Salaries: Calculate weekly earnings based on hours worked and hourly rates.

• Overtime: Calculate pay using Time and a Half ($\times 1.5$) or Double Time ($\times 2$).

• Take Home Pay: Calculate net income by subtracting deductions (Income Tax, Superannuation, and National Insurance) from gross pay.

Inverse Proportion

• Solve scenarios where the product of two variables remains constant.

• Examples include calculating the time it takes for a different number of workers (6 men vs. 5 men) to complete a task or adjusting travel speed to cover the same distance in less time.

Speed, Distance, and Time (SDT)

• Distance: $D = S \times T$. Note the necessity of converting minutes to fractions of an hour (e.g., $36\,\text{minutes} = 0.6\,\text{hours}$).

• Average Speed: $S = D \div T$. Results are expressed in units such as $\text{mph}$ or $\text{km/hr}$.

• Time: $T = D \div S$. Convert decimal answers back into hours and minutes.