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" »

Study Focus: Techniques for systematizing repair processes and improving duration.

Study Methods: Analyzes processes by studying the sequence of movements and operations executed by employees. The study considers all factors impacting the result: job, equipment and tools, facilities, and workers.

Process to Follow for Improvement

• Selecting work to perform
• Record facts
• Examine actions
• Develop a new improved method
• Implement the new method

Record of Activities

This involves recording, through direct observation, all operations and tasks that are part of the job being studied.

Dingbats Actions

• Operation (Round): Performed when changing the properties of a piece during assembly or disassembly.
• Shipping (Arrow): When a piece is displaced.
• Inspection (Square)

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" »

Soil Volume Calculation for Excavation

This section details the calculation of soil volume extracted between two distinct profiles, a common task in civil engineering and surveying projects.

Profile Dimensions and Separation

• First Profile Surface Area (St): 32 m²
• Second Profile Cross-Section: A trapezoid with a height of 3m, a lower base of 6m, and an upper base of 17m.
• Distance Separating Profiles (d): 54m

Calculating the Second Profile's Surface Area (Sd)

The area of the trapezoidal second profile is calculated as:

Sd = [(Lower Base + Upper Base) / 2] × Height

Sd = [(6 + 17) / 2] × 3 = 34.5 m²

Calculating Partial Volumes (Vt and Vd)

Using a specific volume computation method for irregular shapes:

Vt = 0.5 × (St)² / (St + Sd) × d

Vt = 0.5 × (32)

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" »

Survival Analysis Fundamentals

Understanding Survival Analysis

The main objective of survival analysis techniques is to determine differences between two or more treatments applied to a set of individuals. Each individual receives a particular treatment, and the effect (response) is measured by the occurrence of a specific event of interest (e.g., default) and the time elapsed from the start of observation until the aforementioned event occurs. Survival analysis techniques apply to data with the following characteristics:

• The dependent variable (or response variable) is the time that elapses until the individual experiences a specific event of interest, often termed death. Therefore, while the individual does not experience the event of interest,

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" »

La gestion des stocks adéquate est celle qui minimise les niveaux de stocks.

Une rupture de stock survient lorsque le niveau de stocks d'un article particulier est insuffisant pour les besoins de l'entreprise, l'empêchant de respecter ses engagements envers les clients ou l'obligeant à interrompre le processus de production. Pour éviter cette situation, il convient de disposer d'un stock de sécurité.

Le niveau du stock de sécurité est le niveau le plus bas de stocks que l'entreprise devrait avoir pour que la production ne soit pas interrompue.

Coûts de la gestion des stocks

La détention de stocks entraîne divers coûts :

• Coût d'approvisionnement : C'est le coût d'achat du produit. Cp = pQ
• Coût de passation de commande : C'est le coût