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

Linear Programming

Linear Programming is an optimization technique useful for allocating scarce resources among competing demands.

Examples of LP Applications

• Selecting the product mix in a factory to make the best use of machine and labor hours available while maximizing the firm’s profit.
• Scheduling tellers at banks so that needs are met during each hour of the day while minimizing the total cost of labor.
• Determining food ingredient combinations that satisfy stated nutritional requirements at a minimum cost level.

Basic Concepts

• A single objective function states mathematically what is being maximized or minimized.
• Decision variables represent choices that the decision maker can control.
• Constraints are limitations that restrict the decision variables.

Planes in 3D Space

Equations for planes parallel to coordinate planes:

• x = a (parallel to yz-plane)
• y = b (parallel to xz-plane)
• z = c (parallel to xy-plane)

Example: Plane Through a Point

Find the equation of the plane passing through (3, -1, 7) and perpendicular to the vector n = <4, 2, -3>:

4(x - 3) + 2(y + 1) - 3(z - 7) = 0

Example: Plane Through Three Points

Find the equation of the plane through P₁(1, 2, -1), P₂(2, 3, 1), and P₃(3, -1, 2):

P₁P₂ = <1, 1, 2>; P₁P₃ = <2, -3, 3>

n = P₁P₂ × P₁P₃ = <9, 1, -5>

Equation: 9(x - 1) + 1(y - 2) - 5(z + 1) = 0

Example: Line and Plane Parallelism

Determine if the line x = 3 + 8t, y = 4 + 5t, z = -3 - t is parallel to the plane x - 3y + 5z = 12.

The line is parallel to the plane if it is perpendicular... Continue reading "Mastering Planes, Quadratic Surfaces, and 3D Coordinates" »

Golden circle:

This relates to the innovation. If a company is able to attract those who share its beliefs, it is also able to connect easier with innovators.

This is a graph made up of three circles, each inside the other. Each circle has a meaning and a way to apply it to our business efficiently.

We can explain the operation of the Golden Circle in such a way that while most companies ask themselves the answers to the question 'what' as their starting point, they are fully aware of the 'how', but very few companies are clear about the 'why'.

The 'what' responds to what we do and what we want to do. The 'how' responds to the processes through which we carry out our actions, and the 'why' would respond to the reason why we develop our business.... Continue reading "The Golden Circle: Driving Innovation and Quality Management" »

Essential Aspects of Corporate Liquidity Analysis

There are two essential aspects in the analysis of the liquidity of a company. On one hand, there is the quality of the coverage of current debts with the most liquid assets; on the other hand, there is the temporal aspect—that is, the agility of current assets to become cash, as well as the agility of current debts to expire.

Evaluating Debt Coverage and Working Capital

In order to analyze the quality of the coverage of current debts, we start by calculating the working capital and its complementary ratios. The positive value of the working capital serves as an indicator (though never definitive evidence) of a good liquidity situation.

Liquidity Ratios: Acid Test and Cash Ratio

Due to the limitations... Continue reading "Analyzing Corporate Liquidity and Financial Health" »

Market Value Performance Measures

• Market Value Added (MVA): Market Capitalization (Number of shares × Price per share) - Equity.
• Market-to-Book Ratio: Measures value added per dollar invested by shareholders. Calculated as: Market Value of Equity / Book Value of Equity (Total Shareholders' Equity).

Measuring Profitability

• Economic Value Added (EVA): Profit after deducting all costs, including the cost of capital. Formula: [After-tax operating income] - [Cost of Capital × Total Capitalization at start of year (Long-term debt + Total equity)].
• Return on Equity (ROE): Net Income / Equity (start of year).
• Return on Assets (ROA): [(1 - Tax Rate) × Interest Expense + Net Income] / Total Assets (start of year).
• Return on Capital (ROC): [(1 - Tax Rate)

Fundamental Tax Concepts

Taxes: Tax is government revenue created by law and enforceable by taxpayers covered by the same. A tax liability always arises caused by a specific taxable event.

Taxable Event and Residence

Taxable Event: This is the reason or the cause that has been made to pay the tax. It determines what is taxed, whether or not these are linked to an imposition or subject. Exempt tax: No tax is paid because there may be a deficiency.

Taxable Person: This refers to a natural (Physical) or legal person who must pay the tax.

Tax Residence: This is where you reside for Hacienda (taxable purposes). Hacienda is where you live; for legal entities, it is where the act of constitution is performed or where the company is registered. You can... Continue reading "Spanish Tax Systems and VAT Regulations" »

B. Changes in Inventories

The company must make an inventory of the stocks it has, doing a physical count of them and checking the result with the information recorded in the book balance. After verifying this data and correcting any differences, it is necessary to stabilize stock accounts; that is, to correct the accounting balance of these accounts to reflect reality.

C. Verification of Accounting Data

Before determining the result of the accounting fiscal year, the company must verify accounting data by checking, on the one hand, that there are no mistakes in arithmetic or transcription and, on the other, contrasting it with the economic reality. To do this, perform the following inventories and balances:

• Trial Balance: Its purpose is to check

International Accounting Standard No. 10

Events Occurring After the Balance Sheet Date

Objective

The objective of this Standard is to prescribe:

1. when an entity shall adjust its financial statements for events after the balance sheet date and
2. the disclosures that an entity should give about the date on which the financial statements were authorized for issue and about events after the balance sheet date.

The Standard also requires the entity that does not prepare its financial statements under the assumption of going concern, if the events after the balance sheet date indicate that this hypothesis of continuity is not appropriate.

Scope

This Standard is applicable in the accounting and disclosures related to events after the balance sheet date.

The events