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

Management Control Instruments

Planning

Establishes lines of action and responsibility within a company's organization to achieve goals. This is executed through programs and embodied in the budget applicable to each program.

Budget

Determines objectives more precisely, specifying amounts and responsible parties. The budget, when applied to the immediate future, is known as operational planning. It is typically performed for a period of days or weeks, using fully quantitative variables and involving the direct participation of each department.

Causes of Deviations

Deviation Definition

A deviation is the difference that arises when comparing expected (planned or budgeted) data with actual results.

Sources of Deviations

• Errors in environmental estimates:

Fundamental Theorems in Probability

The Law of Large Numbers

The Law of Large Numbers is a theorem in probability that describes the average behavior of a succession of random variables as the total number of variables increases. The theorem provides sufficient hypotheses to affirm that the sample average converges to the average of the expectations of the random variables involved.

Chebyshev's Theorem

Chebyshev's Theorem provides an upper bound on the probability that values fall outside a certain range from the average.

Bernoulli's Theorem

Bernoulli's Theorem is a special case of the Law of Large Numbers, which specifies that the approximate frequency of an occurrence converges to the probability p of that occurrence as the experiment is repeated.... Continue reading "Core Principles of Probability and Statistical Inference" »

Financial Mathematics: Outside Discount and Inflation

This set of solved exercises covers key concepts in financial mathematics, focusing on outside discount operations, nominal and real interest rates, and the effects of inflation on purchasing power.

Exercise 1: Outside Discount Calculation

A company has a discount operation with a banking financial institution with a deadline of 23 days. The bank operates with an effective interest rate of 45.76% p.a. discount rate to determine the "outside" discount to be used in the operation.

• Effective Annual Rate (e): 45.76% p.a.
• Term (n): 23 days
• Effective Rate for 23 days (I_f): (1 + 0.4576)^{23 / 360} - 1 = 0.0243648 or 2.43648% per 23 days

To find the equivalent "outside" discount rate (d) for the period:

d... Continue reading "Solving Outside Discount and Inflation Math Problems" »

Case Study 1: Sales Team Head Settlement

Employee Details & Salary Structure

Settlement Calculation Steps

Unpaid Salary

• Period: November 1 - November 3 (3 days)

Monthly Salary = 1,090 + 320 + 90 = 1,500 Euros/month

Unpaid Salary for 3 days = (1,500 / 30) * 3 = 150 Euros

Pro-rata

Tangible Fixed Assets Definition

Tangible fixed assets are a set of rights and resources intended to remain in the business for more than one financial period.

Types of Fixed Assets

• Tangible-Intangible Assets: These assets have a physical entity but are classified as intangible assets or rights subject to economic evaluation.
• Tangible Assets: These assets have a physical entity and are classified as real or personal property, except those that must be classified in other subgroups.

Understanding Depreciation

Depreciation refers to the systematic allocation of the cost of tangible items over their useful life due to three primary reasons:

1. The use made of these elements, which causes deterioration.
2. The passage of time, whether the asset is used or not.

VAT Fees Eligible for Deduction

The following fees can be deducted:

• Submission-entry of goods from another taxable person
• Imports of goods
• Intra-Community acquisitions
• Fees relating to cases of reverse charge

Partial Deduction or Pro Rata

This applies when a company performs two types of activities: operations with deduction and operations without deduction.

The consequence is that only a portion or percentage of the taxes charged can be deducted.

What percentage? To determine the coefficient, calculate the following:

(Sum of output-delivery deduction in the year) / (Sum of all delivery-end of the year)

To calculate the deduction, multiply the percentage by the taxes charged.

Dynamics During the Exercise: Provisionally apply the percentage from the previous... Continue reading "VAT Deduction Rules: Fees, Partial Deduction, and Investment" »

Commercial Sales Documents

Item 5.1: Documents of Sale

• The Order: A document detailing customer demand and the conditions under which items were purchased. It includes the order number, date, location, and identification of both the issuing and receiving companies.
• The Delivery Slip: A document that details and justifies the delivery of goods to the client. It includes vendor ID, buyer identification, shipping characteristics, shipment observations, and the customer signature.
• The Bill (Invoice): A legal document proving the sale of goods or services. It includes the seller's identification, customer ID, invoice number, date, location, description, reference, discounts, tax base (the amount on which the 18% VAT is calculated), applicable surcharges,

Frequency Distribution

1) Loose Data: Data collected that has not been organized numerically. For example, heights of males and females obtained from a survey, arranged alphabetically.

2) Ordination: An arrangement of numerical data in ascending or descending order. The difference between the largest and smallest number is called the data range.

Example:

Data Management Data Range
2.1 1 A: 10 to 1
7.6 2 R: 9
4.9 4
8.10 6 R: range of the data.
7
8
9
10

3) Frequency Distribution: When dealing with large amounts of loose data, it's useful to distribute them into classes or categories and determine the number of individuals belonging to each category, often called a class.

A tabular distribution of data by type or category with its corresponding... Continue reading "Understanding Frequency Distributions: Data Organization" »

Types of Research

Descriptive and Explanatory Research

Descriptive Research: Aims to show a situation or phenomenon.

Explanatory Research: Seeks to uncover the truth of a hypothesis.

Transverse and Longitudinal Research

Transverse Research: Involves studying different variables at a specific point in time.

Longitudinal Research: Involves studying variables over an extended period.

Retrospective and Prospective Research

Retrospective Research: Observation units are classified based on the presence or absence of an effect, and the frequency of causal factors is recorded.

Prospective Research: Individuals are selected based on the presence or absence of exposures (inputs), and the frequency of the effect is recorded.

Case-Control Studies

• Combines advantages

Descriptive Statistics

Classification and description of a dataset.

Statistical Inference

Interpreting results using descriptive techniques for decision-making.

Statistic

A numerical summary or descriptive feature of a total sample of research.

Frequency Polygon

A line graph usually plotted on a frequency histogram.

Experiment

A controlled procedure carried out to determine, demonstrate, or test a hypothesis; a process that yields an outcome.

Median

The midpoint of a data set.

Probability

Measures or quantitatively determines the possibility that an event occurs.

Objective Probability

Probability obtained through numerical calculations or empirical observations.

Classic Approach

Assumes that all events are equally likely to occur.

Event

A possible outcome or... Continue reading "Essential Statistics and Probability Terms Defined" »