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

Fundamentals of Ratios and Proportions

Defining Ratios and Proportions

• Ratio: A ratio of two numbers is their quotient.
• Proportion: A proportion exists when we have two ratios whose quotients are equal.

Key Properties of Proportions

The terms in a proportion are often referred to as means and extremes (or ends).

Example: $2/4 = 3/6$

The Fundamental Property of Proportions states that the product of the extremes is equal to the product of the means. This property is essential for finding an unknown term.

Constant of Proportionality

The constant of proportionality is the ratio (quotient) of any of the corresponding terms in the proportion.

Understanding Proportionality

Magnitudes and Measurement

A magnitude is a measurable characteristic, such as:

• Length
• Volume
• Weight
• Mass
• Temperature

Direct

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" »

1.1 Theory of Sets

Theory of sets. Why is it said to be a mathematical system and language? It is said to be a mathematical system because it contains a set of operations, theorems, functions and relations, and it underpins areas such as algebra, geometry, calculus and more.

Set theory is an appropriate tool for structured thinking and for developing the capacity to analyze and design solutions for particular problems. It allows focusing on a problem as a whole by removing what is irrelevant and highlighting the essentials.

Set theory facilitates the visualization of relationships between all component parts of a problem as well as each part individually. It lets us combine elements within its own methodology and use deductive reasoning together... Continue reading "Set Theory Fundamentals: Definitions, Notation and Examples" »

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" »

The Architecture of Basilica San Lorenzo

The Basilica San Lorenzo was built during the Quattrocento by the architect Filippo Brunelleschi, with construction conducted between 1422 and 1470. It is inspired by the Church of the Holy Cross (Gothic). The structure is divided into three naves and lateral chapels. In the middle of the transept is a cupola, which is surrounded by chapels. Opposite the middle of the transept is the main chapel.

Geometric and Mathematical Design

Its plan is a Latin cross with three naves and side chapels, forming a basilica that is almost extremely longitudinal. In the transept, which has little development, he set a dome with a lantern. The space was mathematically and geometrically modulated by the circle inscribed in

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)

1. Thales' Theorem and Similar Triangles

Thales' Theorem is fundamental in geometry. It states that if a straight line is drawn parallel to one side of a triangle, it divides the other two sides proportionally, creating a smaller triangle that is similar to the original one.

The theorem is valid only for similar triangles. For example, given a triangle ABC, if we trace a segment MN parallel to one side, we obtain a smaller triangle AMN similar to ABC.

Criteria for Triangle Similarity

Two triangles are considered similar if they satisfy any of the following premises:

• Their corresponding sides are proportional.
• They have two corresponding angles equal (which implies all three angles are equal).
• They have one equal angle, and the sides forming that angle