maths term 3

Financial Maths

Financial mathematics involves understanding how money grows or depreciates over time, particularly through interest. We primarily use two types of interest calculations: simple interest and compound interest.

1. Simple Interest: This type of interest is calculated only on the initial principal amount. It's often used for short-term loans or simple savings accounts where interest is not reinvested.

   • The formula is given by: $A = P(1 + i \cdot n)$

   • A represents the final amount accumulated after n periods.

   • P represents the principal amount (the initial sum of money).

   • i represents the interest rate per period (expressed as a decimal, e.g., 5% is 0.05).

   • n represents the number of periods (e.g., years, months).

   • Rearrangements allow us to solve for different variables:

     • To find the Principal (P): $P = \frac{A}{1 + i \cdot n}$ (Useful for determining how much to invest now to reach a future goal).

     • To find the Rate (i): $i = \frac{A - P}{P \cdot n}$ (Useful for calculating the effective interest rate paid or earned).

     • To find the Number of periods (n): $n = \frac{A - P}{P \cdot i}$ (Useful for determining how long it takes to reach a financial goal).

2. Compound Interest: This type of interest is calculated on the initial principal and also on the accumulated interest from previous periods. It leads to faster growth and is commonly used for long-term investments, savings, and most loans.

   • The formula is: $A = P(1 + i)^n$

   • The variables A, P, i, n hold the same meaning as in simple interest.

   • The key difference is that the base $(1 + i)$ is raised to the power of $n$, reflecting interest earning interest.

   • Rearrangements include:

     • To find the Principal (P): $P = \frac{A}{(1 + i)^n}$ (To find the present value of a future amount).

     • To find the Rate (i): $i = \sqrt[n]{\frac{A}{P}} - 1$ (To determine the annual growth rate of an investment).

     • To find the Number of periods (n): $n = \frac{\log(A/P)}{\log(1 + i)}$ (Requires logarithms to solve for the exponent).

3. Foreign Exchange (FX) Rates: These rates determine the value of one currency in terms of another and have significant economic impacts.

   • Impacts: FX rates affect the cost of imported goods (a weaker local currency makes imports more expensive), the competitiveness of exports (a weaker local currency makes exports cheaper for foreign buyers), and the cost of international travel.

   • Conversion Example: If $1\,\text{USD} = 18.50\,\text{R}$:

     • Converting USD to R (local currency): Multiply the USD amount by the rate. E.g., $200\,\text{USD} = 200 \times 18.50 = 3700\,\text{R}$.

     • Converting R to USD (foreign currency): Divide the R amount by the rate. E.g., $9250\,\text{R} = \frac{9250}{18.50} = 500\,\text{USD}$.

4. Hire Purchase (HP): This is a credit agreement where a consumer agrees to pay for an item in installments over a period, ultimately owning it once all payments are made. It typically involves a deposit and then fixed monthly payments, often incurring significant interest.

   • Total Cost of the item under HP: $\text{Total Cost} = \text{Deposit} + (\text{Monthly Payment} \times \text{Months})$

   • Example: A fridge costs $7000 but by HP, it requires a deposit of$2000 and 20 monthly payments of $300.</p><ul><li><p>Total paid:$2000 + (300 \times 20) = 2000 + 6000 = 8000\, \text{R}$.</p></li><li><p>Total interest paid:$8000 - 7000 = 1000\, \text{R}$. This$1000 represents the cost of borrowing.

Algebraic Fractions

Algebraic fractions are fractions that contain variables in the numerator, denominator, or both. Mastering them requires strong factoring skills and an understanding of exponent laws.

1. Factorising: This is the process of breaking down an expression into a product of simpler ones. It's crucial for simplifying fractions.

   • Common Factor: E.g., $3x + 6 = 3(x + 2)$.

   • Difference of Squares: E.g., $x^2 - 9 = (x - 3)(x + 3)$.

   • Trinomials: E.g., $2x^2 + 5x + 3 = (2x + 3)(x + 1)$. (Look for two numbers that multiply to $a \cdot c$ and add to $b$).

   • Grouping: E.g., $ax + ay + bx + by = a(x + y) + b(x + y) = (a + b)(x + y)$.

2. Simplifying Algebraic Fractions: This involves factorising both the numerator and the denominator and then canceling out any common factors. It's important to state any restrictions on the variable(s) that would make the original denominator zero.

   • Example: $\frac{x^2 - 9}{x^2 - x - 6} = \frac{(x - 3)(x + 3)}{(x - 3)(x + 2)} = \frac{x + 3}{x + 2}$

   • Domain Restriction: Here, the original denominator is zero if $x = 3$ or $x = -2$. So, the simplified fraction is valid for $x \neq 3, -2$.

   • Addition/Subtraction: To add or subtract algebraic fractions, you must find a common denominator. For example, $\frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy}$.

3. Exponent Laws in Fractions: These laws apply directly to variables with exponents in algebraic fractions.

   • Product Rule: $x^a \cdot x^b = x^{a+b}$

   • Quotient Rule: $\frac{x^a}{x^b} = x^{a-b}$ (Example: $\frac{x^5}{x^2} = x^{5-2} = x^3$).

   • Power Rule: $(x^a)^b = x^{ab}$

   • Zero Exponent: $x^0 = 1$ (for $x \neq 0$).

   • Negative Exponent: $x^{-a} = \frac{1}{x^a}$.

Graphs

Graphs provide a visual representation of relationships between variables, typically $x$ and $y$, on a Cartesian plane.

1. Substitution and Tables: To plot a graph, you substitute various x-values into an equation (or function) to find corresponding y-values, generating ordered pairs $(x, y)$. These pairs are then plotted.

   • Systematic Approach: Choose a range of x-values (e.g., -2, -1, 0, 1, 2) and calculate $y$ for each.

   • Example: For $y = 2x + 1$:

     • If $x = -1$, $y = 2(-1) + 1 = -1$ (pair: $(-1, -1)$).

     • If $x = 0$, $y = 2(0) + 1 = 1$ (pair: $(0, 1)$).

     • If $x = 1$, $y = 2(1) + 1 = 3$ (pair: $(1, 3)$).

2. Domain and Range: These define the extent of the graph.

   • Domain: The set of all possible input values (x-values) for which the relation or function is defined. For linear graphs, the domain is typically all real numbers ($x \in \mathbb{R}$).

   • Range: The set of all possible output values (y-values) that the relation or function can produce. For linear graphs, the range is also typically all real numbers ($y \in \mathbb{R}$).

3. Forms and Intercepts of Linear Equations:

   • Standard Form (Slope-Intercept Form): $y = mx + c$

     • m is the gradient (slope) of the line, representing the rate of change of $y$ with respect to $x$ ($\text{rise/run}$). A positive $m$ means an upward slope, negative $m$ means a downward slope.

     • c is the y-intercept, the point where the line crosses the y-axis (i.e., when $x = 0$).

   • Intercepts:

     • y-intercept: Set $x = 0$ and solve for $y$. For $y = 2x + 3$, the y-intercept is $(0, 3)$.

     • x-intercept: Set $y = 0$ and solve for $x$. For $y = 2x + 3$, $0 = 2x + 3 \implies 2x = -3 \implies x = -\frac{3}{2}$. The x-intercept is $(-\frac{3}{2}, 0)$.

4. Drawing Linear Graphs:

   • Table Method: Plot several points from a table of values and connect them with a straight line.

   • Gradient-Intercept Method: Plot the y-intercept $(0, c)$. Then use the gradient $m = \frac{\text{rise}}{\text{run}}$ to find a second point (e.g., if $m = 2 = \frac{2}{1}$, move 1 unit right and 2 units up from the y-intercept).

   • Double Intercept Method: Find and plot both the x-intercept and the y-intercept, then draw a straight line through these two points.

5. Equations of Given Linear Graphs:

   • Step 1: Find the gradient (m). Choose any two distinct points $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ on the line. Calculate $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.

   • Step 2: Find the y-intercept (c). If the y-intercept is visible, read it directly. Otherwise, substitute the calculated $m$ and the coordinates of one point $(x, y)$ into $y = mx + c$ and solve for $c$.

   • Step 3: Write the equation. Substitute $m$ and $c$ into $y = mx + c$.

6. Vertical and Horizontal Lines:

   • Vertical Lines: Have an undefined gradient. Their equation is always of the form $x = a$, where $a$ is the x-intercept. E.g., the line passing through $(3, 0)$ and $(3, 5)$ is $x = 3$.

   • Horizontal Lines: Have a gradient of zero. Their equation is always of the form $y = b$, where $b$ is the y-intercept. E.g., the line passing through $(0, 4)$ and $(5, 4)$ is $y = 4$.

7. Parallel and Perpendicular Lines:

   • Parallel Lines: Have the same gradient. If $L<em>1$ is parallel to $L</em>2$, then $m<em>1 = m</em>2$.

   • Perpendicular Lines: Their gradients multiply to $-1$. If $L<em>1$ is perpendicular to $L</em>2$, then $m<em>1 \cdot m</em>2 = -1$. This means one gradient is the negative reciprocal of the other (e.g., if $m<em>1 = 2$, then $m</em>2 = -\frac{1}{2}$).

Geometry (Part 2)

This section delves into the properties of quadrilaterals, the fundamental Pythagoras Theorem, and its application in solving geometric problems.

1. Definitions of Quadrilaterals (Four-sided Polygons):

   • Square: All four sides are equal; all four interior angles are $90^\circ$; diagonals are equal in length, bisect each other at $90^\circ$, and bisect the angles.

   • Rectangle: Opposite sides are equal and parallel; all four interior angles are $90^\circ$; diagonals are equal in length and bisect each other.

   • Rhombus: All four sides are equal; opposite angles are equal; diagonals bisect each other at $90^\circ$ and also bisect the angles of the rhombus.

   • Parallelogram: Opposite sides are equal and parallel; opposite angles are equal; consecutive angles are supplementary ($180^\circ$); diagonals bisect each other.

   • Trapezium (or Trapezoid): Has exactly one pair of parallel sides. The non-parallel sides can be of unequal length.

   • Kite: Has two distinct pairs of equal-length adjacent sides. One diagonal is the perpendicular bisector of the other diagonal, and it also bisects the angles at the vertices it connects.

2. Pythagoras Theorem: This theorem applies exclusively to right-angled triangles (triangles with one $90^\circ$ angle).

   • Theorem: The square of the length of the hypotenuse (the side opposite the right angle, denoted as $c$) is equal to the sum of the squares of the lengths of the other two sides (legs, denoted as $a$ and $b$).

   • Formula: $c^2 = a^2 + b^2$

   • Example 1 (Finding hypotenuse): If $a = 3$ and $b = 4$, then $c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.

   • Example 2 (Finding a leg): If $c = 10$ and $a = 6$, then $b^2 = c^2 - a^2 = 10^2 - 6^2 = 100 - 36 = 64$, so $b = \sqrt{64} = 8$.

3. Solving Geometric Problems: This typically involves applying definitions, theorems, and formulas to find unknown angles, lengths, areas, or perimeters within geometric figures.

   • Triangle Properties: Sum of angles in a triangle is $180^\circ$; properties of isosceles, equilateral, and right-angled triangles.

   • Quadrilateral Properties: Using the properties defined above, e.g., opposite angles of a parallelogram are equal.

   • Pythagorean Theorem: Crucial for problems involving right triangles or creating right triangles from other shapes (e.g., finding the height of an isosceles triangle).

   • Parallel Lines: When a transversal line intersects parallel lines, alternate interior angles are equal, corresponding angles are equal, and consecutive interior angles are supplementary.

Transformations

Transformations involve changing the position, orientation, or size of a geometric figure. The original figure is called the pre-image, and the transformed figure is the image.

1. Types of Transformations:

   • Translation (Slide): Moving every point of a figure by the same distance in the same direction. It preserves size and shape.

   • Reflection (Flip): Flipping a figure across a line (the line of reflection). It preserves size and shape but reverses orientation.

   • Rotation (Turn): Turning a figure about a fixed point (the center of rotation) by a specific angle. It preserves size and shape.

   • Dilation (Resizing): Changing the size of a figure by a scale factor from a fixed point (the center of dilation). It preserves shape but changes size.

2. Reflection Examples (Coordinate Rules):

   • In the x-axis: $(x, y) \to (x, -y)$ (Negate the y-coordinate).

   • In the y-axis: $(x, y) \to (-x, y)$ (Negate the x-coordinate).

   • In the line $y = x$: $(x, y) \to (y, x)$ (Swap the x and y coordinates).

   • In the line $y = -x$: $(x, y) \to (-y, -x)$ (Swap and negate both coordinates).

   • In the origin: $(x, y) \to (-x, -y)$ (Negate both x and y coordinates).

3. Translation Example: Moving a point by a vector $(a, b)$ means adding $a$ to the x-coordinate and $b$ to the y-coordinate.

   • Rule: $(x, y) \to (x + a, y + b)$.

   • Example: Move point A$(2, 3)$ by vector $(3, -2)$. The image A' will be $(2 + 3, 3 + (-2)) = (5, 1)$.

4. Identifying Transformations from Coordinates:

   • Translation: All coordinates shift by the same constant amounts $(x<em>2 = x</em>1 + a, y<em>2 = y</em>1 + b)$.

   • Reflection: The coordinates follow one of the reflection rules (e.g., $(x, y) \to (x, -y)$ for x-axis reflection, or $(x, y) \to (-x, y)$ for y-axis reflection).

   • Rotation (e.g., 90 degrees counter-clockwise about origin): $(x, y) \to (-y, x)$. For 180 degrees: $(x, y) \to (-x, -y)$. For 270 degrees counter-clockwise (or 90 degrees clockwise): $(x, y) \to (y, -x)$.

   • Dilation (centered at origin with scale factor $k$): $(x, y) \to (kx, ky)$. The distances from the origin are scaled by $k$.