Notes, summaries, assignments, exams, and problems for Mathematics

Geometry Fundamentals

Understanding Angles

An angle is the portion of a plane formed by two rays (semi-straight lines) sharing a common endpoint. The rays are called the sides of the angle, and the common endpoint is called the vertex.

Angle Designation

• By three letters, with the vertex letter always in the middle (e.g., ∠ABC).
• By the letter of the vertex (e.g., ∠B).
• By a number or a Greek letter, often placed near the vertex (e.g., ∠α).

Types of Angles

• Adjacent angles: Angles that share a common vertex and a common side.
• Linear Pair: Two adjacent angles whose non-common sides form a straight line (sum is 180°).
• Right angle: An angle that measures exactly 90°.
• Straight angle: An angle that measures exactly 180°.
• Complementary angles: Two angles

This analysis delves into the financial performance of the company for the years 2007 and 2008, focusing on key liquidity and operational efficiency metrics such as working capital and various maturity periods.

Working Capital Calculation (2007-2008)

The working capital is calculated as the difference between current assets and current liabilities. Here are the results for 2007 and 2008:

• 2008: Current Assets (2,788,286) - Current Liabilities (3,020,313) = -232,027
• 2007: Current Assets (2,741,769) - Current Liabilities (2,953,480) = -211,711

Both periods show negative working capital. This indicates that the company's current liabilities exceed its current assets. The negative working capital remained almost constant across both years.

Analysis of

Linear Equations

First-degree algebraic expressions with several unknowns.

Linear equations with 2 unknowns correspond to the general equation: ax + by = c

Solution of a Linear Equation

A solution is any pair of values for the unknowns that verifies the equation.

If x1, y1 are real numbers, the pair (x1, y1) is a solution of the linear equation in two unknowns if: ax1 + by1 = c.

Linear equations with 2 unknowns have infinite solutions.

Graphical Representation

The equation ax + by = c is a straight line. Each point on this line is a solution of the equation.

Systems of Linear Equations

A linear system of two linear equations with two unknowns is an algebraic expression of the form:

ax + by = c
a'x + b'y = c'

Solution of a System

A solution is any pair of... Continue reading "Solving Linear Equations and Systems" »

Understanding Accounting and Financial Statements

Accounting, within economics, is the study of business assets, the rules, and the scientific basis for the registration of economic information within an enterprise.

Key Components of Annual Accounts

Annual accounts, comprising the balance sheet, profit and loss account, the statement of changes in equity, and the memory, form a unified set. These accounts are financial statements that report on the company's results and its financial position. They include the following:

• The Balance Sheet

  Reports on the financial situation of the company at the close of a given fiscal year.

• The Profit and Loss Account

  Reports on the performance of the company for a year from its business operations.

• The Statement of

Measures of Central Tendency:

Arithmetic Mean

Used for:

Intervals and pooled data (xi = class mark)

Data are not grouped (tables).

Data are grouped (tables) but no intervals.

Median

The middle value in a sorted dataset.

For an odd number of observations, it's the middle number.

Sort data from lowest to highest before finding the median.

Mode

The value that appears most frequently in a dataset.

Mid-Range

RM = (Maximum Value + Minimum Value) / 2

Geometric Mean

G = ⁿ√(x₁ * x₂ * ... * xn)

Harmonic Mean

H = n / ( (1/x₁) + (1/x₂) + ... + (1/xn) )

Quadratic Mean (Root Mean Square)

Q = √[ (x₁² + x₂² + ... + xn²) / n ]

Percentile

A measure indicating the value below which a given percentage of observations in a group falls.

Kth Percentile

Measures of... Continue reading "Statistical Measures: Central Tendency, Dispersion, and Form" »

Practice 4: Salaries, Overtime, SSO, PHL

Basic Salary

• = (Additional Table at $[$D$24] * Time Worked)

Total Overtime

• Fx = IF(logical test = E10 [Overtime] > 40)
• True-Value: E10 - 40
• False-value: 0

Triple Overtime

• Fx = IF Function
• Logic Test: Total Overtime [G10 > 8]
• True-Value: G10 - 8
• False-value: 0

Payment of Extra Time Triple

• Triple Overtime (H10) * Pay Per Hour (D24) * 3

Double Overtime

• = G10 (Total Overtime) - H10 (Triple Overtime)

Payment of Extra Time Double

• = J10 (Double Extra Time) * D24 (Pay per hour) * 2

Compulsory Social Security (SSO)

• = (F10 [Basic Salary] + I10 [Payment of Triple Overtime] + K10 [Payment of Double Overtime]) * 5%

Housing Policy Act (HPL)

• = (F10 [Basic Salary] + I10 [Payment of Triple Overtime] + K10 [Payment of Double Overtime]

Least Squares Adjustment: Fundamental Models

1. Conditional Equations Model

This section outlines the formulation and solution for conditional equations in least squares adjustment.

• B: Jacobian matrix of the conditional equations with respect to observations.
• A: Vector representing the sum or constant terms of the conditional equations.
• P: Weight matrix of observations, calculated as 1 / σ₀².
• Q: Covariance matrix of observations, Q = P&supminus;¹.
• Qe: Covariance matrix of the conditional equations, Qe = B * Q * B&supT;.
• K: Vector of Lagrange multipliers, K = -Qe&supminus;¹ * A.
• V: Vector of residuals, V = Q * B&supT; * K.
• L̂: Adjusted observations, L̂ = V + Lobserved.

Linearized Conditional Observation Equation

A common form of... Continue reading "Least Squares Adjustment Models and Error Propagation" »

Data Visualization: Types and Applications

Graphics are a visual representation of data, prioritizing simplicity, easy interpretation, and adherence to standards.

Classification of Graphics

• Structural graphics: Representing a single set of data.
• Relational graphics: Connecting two sets of data.
• Special graphics.

Types of Graphics

• Organizational Charts: Represent the structure of an organization.
• Classification Plans: Represent elements of a total set, subdivided into smaller subsets. These can be square or pyramidal.
• Schedules: Use current statistics, ordering data over time to identify peaks and trends.
• Histograms: Represent class intervals or monthly values, highlighting extreme values.
• Function Tables: Establish mutual relationships between two sets

The definition of a weighted average:

What is a Weighted Average?

A weighted average is the result of multiplying each number in a set by a value assigned to it (its weight), and then calculating the arithmetic mean of the resulting products. It's used when the components contributing to the average are not equally important.

For example, if a teacher states that an examination is worth 40% of the final mark, another is worth 35%, and a third is worth 25%, the weighted average would be calculated as follows:

mediaPond = (ex1 * 40 + ex2 * 35 + ex3 * 25) / 100

Basically, it's an average of a dataset that allows you to define the degree of importance for each data point's contribution to the average.

If the data are 2, 3, 5, 7, 9, 6, 8, the average... Continue reading "Understanding Weighted Averages: Definition and Applications" »