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

Differences Between Delivery Notes and Invoices

Delivery Note: A provisional document justifying the dispatch of goods. It does not include VAT.

Invoice: A definitive document providing legal accreditation. It is valid for any claim and includes VAT.

Another important difference is that invoices are legally required to be kept for 6 years, while retaining delivery notes is not mandatory for the same period.

Sales Transaction Documentation

Common documents involved in sales transactions include:

• The order sheet
• The delivery note
• The invoice
• The expenses sheet
• Remittance advice
• Receipt
• Voucher or promissory note
• Check
• Bill of exchange

What Are Quantity Discounts (Rappels)?

These are discounts granted by the seller to the buyer for purchasing goods exceeding... Continue reading "Key Commercial Documents Explained" »

Understanding Sequences, Progressions, and Functions

Sequences

Sequences are unlimited strings of real numbers. Each of the numbers that form a sequence is a term and is designated with a letter and an index that indicates its position in the sequence. The general term is the algebraic expression used to calculate any term, depending on the index.

Recurrent Sequences

Recurrent sequences are those in which terms are defined based on one given earlier, according to a known algebraic expression.

Arithmetic Progressions

A sequence of rational numbers is an arithmetic progression if each term is obtained from the previous one by adding a fixed number, or difference, usually represented by *d*. The general term is: W = A1 + (n-1) * d.

Geometric Progressions

A... Continue reading "Understanding Sequences, Progressions, and Functions in Math" »

Finding the Vertex Coordinates of a Parallelogram

The points A (-2, 3, 1), B (2, -1, 3), and C (0, 1, -2) are consecutive vertices of the parallelogram ABCD.

(a) Find the Vertex Coordinates of D

If ABCD are the vertices of a parallelogram, free vectors AB and DC are equal:

• AB = (4, -4, 2)
• DC = (-x, 1 - y, -2 - z)

Equating coordinates, we have x = -4, y = 5, and z = -4. The missing point is D (-4, 5, -4).

(b) Equation of the Line Through B and Parallel to Diagonal AC

The line passes through point B (2, -1, 3) and has a direction vector AC = (2, -2, -3). Its continuous equation is:

(x - 2) / 2 = (y + 1) / -2 = (z - 3) / -3

(c) Equation of the Plane Containing the Parallelogram

We can use point B (2, -1, 3) and the vectors BA = (-4, 4, -2) and BC = (-2,... Continue reading "Solving Problems with Parallelograms, Lines, and Planes in 3D Space" »

Working Capital

Working Capital measures the capacity for payment in the ordinary course of business activity. It's calculated as: Current Assets (CA) - Current Liabilities (CL)

• CA > CL: Positive Working Capital. The business has the potential for investment. Working Capital should never exceed 10% of CA, as these are idle funds.
• CA < CL: Negative Working Capital. This may indicate a suspension of payments or insolvency. It usually signifies mismanagement in the negotiation of ordinary business activity, but it doesn't always mean a bad situation.

Acid Test

The Acid Test measures a company's capacity to meet all of its short-term debts. It's calculated as: (Current Liabilities - Treasury) / Available. This indicates immediate liquidity;... Continue reading "Understanding Key Financial Ratios for Businesses" »

Valuation Criteria

Historical Cost

For Assets: The purchase price or production cost. This includes the amount of cash paid or payable, plus the fair value of any other consideration given for the acquisition. All costs directly related to the acquisition and necessary to bring the asset to operating condition are included.

For Liabilities: The value corresponding to the consideration received in exchange for incurring the debt. In some cases, it is the amount of cash expected to be paid to settle the liability in the ordinary course of business.

Fair Value

The price that would be received to sell an asset or paid to transfer a liability in an orderly transaction between knowledgeable, willing market participants at the measurement date. It is determined... Continue reading "Understanding Asset and Liability Valuation Concepts" »

Sampling Stage

Sample Size:

• Depends on population size
• The type of sample
• Feature (parameter) of the population that is analyzed
• The maximum permissible error in the estimation of the parameters

The Form: The information collected can be classified as:

• Acts or behaviors that can be compared
• Information: analyzes the degree of respondents' knowledge on specific topics
• Opinions or Judgments: such as perceived service quality
• Attitudes or predispositions of mind: We are looking for what is thought in relation to something
• Motives or explanations for specific behaviors: the question is, why certain views or acts?
• Possible future behavior: it may ask whether or not to consume a prepared product.

Concept and Structure of Questionnaires

The questionnaire is a way... Continue reading "Effective Questionnaire Design for Market Research" »

Food and Nutrition

Food is the act of providing sustenance to the human body, while nutrition encompasses the physiological processes by which the body receives, transforms, and utilizes the chemical components in food.

Inherited Diseases

• Hemophilia
• Huntington's disease
• Cystic fibrosis
• Color blindness
• Phenylketonuria

Measures of Dispersion

Measures of dispersion indicate how close the data are to the average.

Vaccines

Vaccines are preparations containing killed or attenuated microorganisms. They are introduced into our bodies to produce antibodies that kill the organism, providing immunity.

Common Pathogens

Pathogens such as bacteria and fungi are the most common causes of infectious diseases.

Health and Disease

Health is a state of complete physical, mental,... Continue reading "Key Concepts in Health, Nutrition, and Statistics" »

IT Project Management and Departmental Structure

Resource Management in IT Projects

Resource management projects often involve:

• IT professionals (informaticians).
• SAP systems, often developed in Germany.
• Project integrity, focusing on achieving target costs and timelines.

Evolving Role of the IT Department

The aim is for the Information Technology (IT) department to become more self-sufficient, reducing its reliance on other functions like accounting or human resources.

Within the IT department, there are project managers, analysts, and programmers. Each functional area typically has a manager, and similarly, the IT department should also have its own manager.

IT Department as a Cost Center

While most functions within an organization generate revenue,... Continue reading "Optimizing IT Operations: Systems, Costs, and Users" »

Exercise 01: Call Center Call Time

Suppose that the time needed for customer service calls in a telephone call center follows a Normal distribution with a mean of 8 minutes and a standard deviation of 2 minutes.

1. What is the probability that a call lasts less than five minutes?
2. What is the probability that a call lasts more than 9.5 minutes?
3. What is the probability that a call lasts between 7 and 10 minutes?
4. 75% of calls require at least how long for service?

Exercise 02: Rabbit Weight Classification

The distribution of the weights of rabbits raised on a farm may well be represented by a Normal distribution with a mean of 5 kg and a standard deviation of 0.9 kg. An abattoir buys 5000 rabbits and intends to classify them according to weight as follows:... Continue reading "Statistics Exercises: Normal Distribution" »

Working Papers

These are the documents in which the auditor records the data and information obtained during their review, as well as the results of the tests performed.

Summary Warrant

A sample of major accounts that are an item.

Worksheet

This is the card that shows the groups or items that comprise the financial statements.

Decree of Detail

The items that comprise an account balance of a major account or any other account.

Certificates of Verification

The work performed to verify the correctness of an item or transaction.

Situations for Audit Work Plan

• Ensure that the service is clearly controlled.
• Be fully aware of what is being offered.

Conditions to Perform the Audit

• Maintain independence of mind regarding the specific client.
• Possess the ability to