Definitive Study Guide for the PHUMA University Entrance Examination

REAL NUMBER SYSTEMS AND ARITHMETIC PROPERTIES

The set of real numbers is classified into four fundamental sub-sets based on their origin and numerical characteristics. Natural numbers, denoted by the symbol $\mathbb{N}$, are the oldest and simplest numbers used for counting and quantifying objects; they are always positive and include the set $\mathbb{N} = \{0, 1, 2, 3, \dots, +\infty\}$. Integers, represented by the symbol $\mathbb{Z}$, consist of the natural numbers, their negative opposites, and zero, written as $\mathbb{Z} = \{-\infty, \dots, -3, -2, -1, 0, 1, 2, 3, \dots, +\infty\}$. In this set, zero is the center, positive numbers are to the right, and negative numbers are to the left, with all negative values being less than zero. Rational numbers, denoted by $\mathbb{Q}$, include all values that can be represented as the quotient of two integers ($\frac{a}{b}$, where $b \neq 0$), composed of a numerator, a quotient operator ($/$, $:$, or $\div$), and a denominator. The set is represented as $\mathbb{Q} = \{-\infty, \dots, -3:4, -1/2, 0, \dots, 33 \div 4, \dots, +\infty\}$. Finally, irrational numbers, symbolized by $\mathbb{I}$, are quantities that cannot be expressed as the quotient of two integers, characterized by infinite non-periodic decimals. Irrational numbers sub-classify into algebraic numbers, such as roots obtained from equations like $\sqrt{2}$ from $x^2 - 1 = 0$, and transcendental numbers, which come from transcendental functions, such as $\pi$ and $e$. The set is written as $\mathbb{I} = \{\dots, -\sqrt{2}, -\sin(30^{\circ}), \dots, 0, \dots, \pi, \dots\}$.

In the real number system, there are two primary operations: addition and multiplication. Subtraction is essentially the opposite of addition, and division is the opposite of multiplication. The Commutative Property of Addition states that the order of terms does not change the result ($a + b = b + a$). The Associative Property of Addition dictates that three or more terms can be grouped in any way without altering the sum ($a + b + c = (a + b) + c = a + (b + c)$). The Commutative Property of Multiplication establishes that the order of factors does not change the product ($a \times b = b \times a$). Similarly, the Associative Property of Multiplication allows for grouping factors in any manner ($a \times b \times c = (a \times b) \times c = a \times (b \times c)$). The Distributive Property is a hybrid property stating that the product of a number by a sum is equal to the sum of the products ($a \times (b + c) = a \times b + a \times c$). Additionally, the system includes Neutral Elements: zero for addition ($a + 0 = a$) and one for multiplication ($a \times 1 = a$).

RATIOS PROPORTIONS AND PERCENTAGES

A ratio is the result of comparing two quantities, indicating how many times one value corresponds to another. In a geometric ratio, the first term is the antecedent ($a$) and the second is the consequent ($b$), represented as $\frac{a}{b}$. For example, a classroom with 4 girls for every 6 boys has a ratio of $\frac{4}{6}$. A proportion is formed by equating two ratios ($\frac{ax}{bx} = \frac{c}{d}$). Proportions are classified into three types: Direct, where both quantities increase or decrease together; Inverse, where one quantity increases while the other decreases an equal number of times; and Composite, which involves a combination of both cases. To find a missing term in a proportion such as $\frac{4}{6} = \frac{x}{18}$, one must solve for the unknown ($x = \frac{4 \times 18}{6}$), resulting in $x = 12$.

Percentages represent a part of a total divided into 100 equal units, denoted by the symbol $\text{\%}$ and equivalent to a fraction with a denominator of 100 ($n\text{\%} = \frac{n}{100}$). For instance, $15\text{\%} = 0.15$ and $130\text{\%} = 1.3$. The basic formula for calculating the percentage ($P$) of a quantity ($A$) is $P\text{\%} \text{ of } A = \frac{P}{100} \times A$. To find what percentage one number ($B$) is of another ($A$), the formula is $\text{\%} = (\frac{B}{A}) \times 100$. In commercial applications, a price increase is calculated as $A \times (1 + \frac{P}{100})$, while a discount is $A \times (1 - \frac{P}{100})$. Successive increases or decreases follow the formula $\text{Result} = A \times (1 \pm \frac{P_1}{100}) \times (1 \pm \frac{P_2}{100})$. For example, a $\text{L. } 300$ product with a $60\text{\%}$ discount costs $\text{L. } 120$ after subtracting the discount of $\text{L. } 180$.

ALGEBRAIC EXPRESSIONS AND PATTERNS

Algebraic expressions combine numbers, letters, and operation signs to model physical or economic situations. Common verbal-to-algebraic translations include: the double or duplo of a number ($2x$), the triple ($3x$), the quadruple ($4x$), half ($\frac{x}{2}$), a third ($\frac{x}{3}$), and a fourth ($\frac{x}{4}$). A value proportional to a set of numbers is written as $2x, 3x, 4x, \dots$. Squares and cubes are $x^2$ and $x^3$ respectively. An even number is represented as $2x$, an odd number as $2x + 1$, and consecutive numbers as $x, x + 1$. Consecutive even numbers are $2x, 2x + 2$, while consecutive odd numbers are $2x + 1, 2x + 3$. Operations involving 24 include a sum ($x + y = 24$), a difference ($x - y = 24$), a product ($xy = 24$), and a quotient ($\frac{x}{y} = 24$). Roots are written as $\sqrt{x}$ or $\sqrt[3]{x}$. Patterns are rules that order elements in a sequence, usually identified through three cases: addition/subtraction, multiplication/division, or a combination of both. For example, in the sequence 9, 17, 33, each term is generated by the pattern $2n - 1$ (e.g., $9 \times 2 = 18 - 1 = 17$).

LINEAR AND QUADRATIC EQUATIONS

Linear equations, also known as first-degree equations, take the general form $ax + b = c$ where $a \neq 0$. Solving these involves applying field properties to isolate the variable. For equations with identical denominators, such as $\frac{2}{1-x} + \frac{x}{1-x} = \frac{1}{1-x}$, one can cancel the denominators given the restriction that $x \neq 1$, resulting in $2 + x = 1$. Linear equations in two variables are graphed as lines with the form $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept. The slope is calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$. A negative slope results in an downward-leaning line, whereas a positive slope results in an upward-leaning line.

Quadratic equations, or second-degree equations, have a general form of $ax^2 + bx + c = 0$, where the highest exponent is 2. The associated graph is a vertical parabola that opens upward (if $a > 0$) or downward (if $a < 0$). Methods for resolution include: 1. Factorization by simple trial, searching for numbers $m$ and $n$ such that $(x \pm m)(x \pm n) = 0$. 2. Completing the square, adding/subtracting terms to achieve a form such as $(a \pm b)^2 = a^2 \pm 2ab + b^2$. 3. The General Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The discriminant $\Delta = b^2 - 4ac$ determines the nature of the roots: if $\Delta > 0$, there are two real roots; if $\Delta = 0$, there is one repeated real root; if $\Delta < 0$, the roots are complex. According to Cardano-Vieta properties, the sum of the roots is $\frac{-b}{a}$ and the product is $\frac{c}{a}$.

SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES

A system of linear equations is a collection of equations where all variables have an exponent of 1, often described as an $m \times n$ system (m equations with n unknowns). Common resolution methods include: 1. Equalization: Solving for the same variable in both equations and setting them equal ($3t + 12 = -3t + 36$). 2. Substitution: Solving for one variable in one equation and inserting the expression into the second ($y = 6 - 2x$\ into $4x + 3(6 - 2x) = 14$). 3. Reduction (Elimination): Adding or subtracting equations to cancel one variable, frequently requiring multiplication by a constant first. For example, a system $2x + 3y = -1$ and $3x + 4y = 0$ can be solved to find $x = -4, y = 3$.

Inequalities represent restrictions where one side is strictly less than ($<$, $>$) or less/greater than or equal to ($\leq$, $\geq$) the other. Key properties include: the Opuesto Property, where multiplying by a negative number inverts the inequality symbol (if $a < b$, then $-a > -b$), and the Reciprocal Property, where if $a < b$, then $\frac{1}{a} > \frac{1}{b}$ provided $a$ and $b$ have the same sign. Intervals are described as Open $( )$, where endpoints are not included; Closed $[ ]$, where endpoints are included; or Semi-open $[ )$ or $( ]$. Graphically, inequalities without a "=\" use a dotted line, while those with a "=\" use a solid line, with the solution set represented by a shaded region above or below the line.

GEOMETRY ANGLES AND TRIANGLES

An angle is the portion of a plane between two rays (sides) sharing a common origin called the vertex. The opening is measured in degrees ($^{\circ}$), minutes ($'$, where $1^{\circ} = 60'$), and seconds ($''$, where $1' = 60''$). A bisector is the line that divides an angle into two equal parts. Angles are classified by measurement: Acute ($0^{\circ} < a < 90^{\circ}$), Right ($a = 90^{\circ}$), Obtuse ($90^{\circ} < a < 180^{\circ}$), Straight ($a = 180^{\circ}$), and Full ($a = 360^{\circ}$). In relation to other angles, they are Supplementary if they sum to $180^{\circ}$ and Complementary if they sum to $90^{\circ}$. Positional classifications include Consecutive (sharing vertex and one side), Adjacent (consecutive forming a straight line), and Opposite by the Vertex (equal angles). Parallel lines never intersect ($l_1 \parallel l_2$), and when cut by a secant, they form corresponding, alternate interior/exterior, and vertical angles.

Triangles are flat geometric figures with three sides, three vertices, and three internal angles that always sum to $180^{\circ}$. They are classified by sides as Equilateral (three equal sides), Isosceles (two equal sides), or Scalene (no equal sides). By angles, they are Rectangles (containing a $90^{\circ}$ angle), Oblicuangles (no right angle, subdivided into Obtusangles with one angle $> 90^{\circ}$ and Acutangles with all angles $< 90^{\circ}$). The Theorem of Pythagoras applies exclusively to right triangles, stating that the square of the hypotenuse ($h$) is equal to the sum of the squares of the catetos ($c_1, c_2$): $h^2 = c_1^2 + c_2^2$. Common Pythagorean triples include $(3, 4, 5)$, $(5, 12, 13)$, and $(7, 24, 25)$.

AREA PERIMETER AND VOLUME OF FIGURES

Perimeter is the total length of the sides or contour of a figure, measured in linear units ($m, cm$). Area is the 2D measure of the surface covered, measured in square units ($m^2, cm^2$). Common formulas include:

• Square: $P = 4L$; $A = L^2$.
• Rectangle: $P = 2b + 2h$; $A = b \times h$.
• Triangle: $P = a + b + c$; $A = \frac{b \times h}{2}$.
• Circle: $C = 2 \pi r$ (or $\pi d$); $A = \pi r^2$.
• Trapeze: $A = \frac{(B + b) \times h}{2}$.
• Regular Pentagon: $A = \frac{P \times ap}{2}$, where $ap$ is the apothem.

Volume measures the 3D space occupied by an object in cubic units ($m^3, cm^3$). Formulas include:

• Cube: $V = L^3$.
• Sphere: $V = \frac{4}{3} \pi r^3$.
• Cylinder: $V = \pi r^2 h$.
• Cone: $V = \frac{\pi r^2 h}{3}$.
• Piramid: $V = \frac{a^2 h}{3}$.
• Prisma: $V = (\frac{5b \times ap}{2}) \times h$ (for pentagonal).

STATISTICS AND MEASURES OF DATA DISPERSION

Measures of central tendency indicate the "center" of a data set. The Mean ($\mu$) is the arithmetic average obtained by summing all values ($\sum x_i$) and dividing by the total number of data points ($N$). The Median ($Me$) is the middle value when data is ordered; if $N$ is odd, it is the center value; if $N$ is even, it is the average of the two central values. The Mode ($Mo$) is the value with the highest frequency. Measures of dispersion quantify the variability of the data. The Range ($R$) is the difference between the maximum and minimum values ($R = x_{\max} - x_{\min}$). Variance ($\sigma^2$) represents variation relative to the mean: $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$. Standard Deviation ($\sigma$) is the square root of the variance ($\sigma = \sqrt{\sigma^2}$).

BIOLOGY GENETICS AND THE MOLECULAR BASIS OF LIFE

Genetics is the science studying heredity and how traits are passed from parents to offspring. DNA (Deoxyribonucleic Acid) functions as the "instruction book" of life. Its structure is a double helix made of nucleotides containing deoxyribose sugar, a phosphate group, and nitrogenous bases: Adenine (A), Thymine (T), Cytosine (C), and Guanine (G), following the complement rule A-T and C-G. It is primarily located in the cell nucleus within chromosomes. RNA (Ribonucleic Acid) transmits DNA's info to synthesize proteins. It is single-stranded, contains ribose sugar, and replaces Thymine with Uracil (U), resulting in A-U and C-G pairings. Refined types include mRNA (Messenger), tRNA (Transfer), and rRNA (Ribosomal).

Human genetics involves 46 chromosomes (23 pairs). The first 22 pairs are autosomes (general characteristics), while the 23rd pair determines sex (XX for female, XY for male). Fundamental concepts include Genes (DNA segments for traits), Alleles (versions of a gene), Genotype (allele combination), and Phenotype (visible characteristic resulting from genotype and environment). Inheritance patterns include Dominant (one allele required), Recessive (two identical alleles required), and Sex-Linked (located on the X chromosome, affecting males more frequently due to their single X chromosome). The Central Dogma of Molecular Biology states that genetic flow moves from DNA to RNA to Proteins.

HUMAN ANATOMY AND PHYSIOLOGY

Life is organized at chemical and biological levels. Cells are the basic unit of life, classified as Prokaryotic (no nucleus, e.g., bacteria) or Eukaryotic (with nucleus, animal/plant). Tissues group similar cells. The Digestive System transforms food into nutrients via a process: 1. Mouth (mastication, amilase for carbs), 2. Esophagus (peristalsis), 3. Stomach (acid and pepsina for proteins), 4. Small Intestine (duodenum receives bile from liver and pancreatic juice for full digestion; vellosities absorb nutrients), 5. Large Intestine (water absorption), 6. Rectum/Anus. The Respiratory System facilitates gas exchange ($O_2$ and $CO_2$) via Fosas Nasales, Trachea, Bronchioles, and Alveoli (where hematosis occurs). Breathing mechanics involve the Diaphragm: contracting for inspiration and relaxing for expiration. The Skeletal System provides support and protects organs ($206$ bones), and the Circulatory System transports blood via heart, arteries, veins, and capillaries through double circulation (pulmonary and systemic).

GENERAL CHEMISTRY AND THE SCIENTIFIC METHOD

Chemistry studies the composition, structure, and changes of matter. The Atom is the unit of matter, featuring a Nucleus (positive Protons and neutral Neutrons) and a Crust (negative Electrons). The Periodic Table organizes elements into Groups (columns with similar properties) and Periods (rows with energy levels). Chemical Bonding includes Ionic (electron transfer), Covalent (electron sharing), and Metallic (free electron cloud). pH measures acidity or basicity on a scale from 0 to 14, where 7 is neutral, $< 7$ is acidic ($H^+$), and $> 7$ is basic ($OH^-$). Inorganic compounds include Oxides, Hydrides, Acids, Peroxides, Hydroxides, and Salts. Agrochemicals are specific chemical tools: Herbicides (kill weeds), Insecticides (kill insects), Fungicides (kill fungi), and Nematicides (kill nematodes).

The Scientific Method is the systematic process for investigation. The steps are: 1. Observation (using senses or instruments), 2. Research Question (specific inquiry), 3. Hypothesis (provisional, falsifiable explanation), 4. Experimentation (manipulating a variable independent to measure a variable dependent while keeping constants), 5. Analysis of Results (statistical data), 6. Conclusion (accept or reject hypothesis), and 7. Communication (sharing with the scientific community).

PHYSICS KINEMATICS ENERGY AND WORK

Physics studies matter, energy, space, and time through mathematical laws. Branches include Mechanics (motion), Thermodynamics (heat), Optics (light), and Electromagnetism. Physical magnitudes are classified as Fundamental (Masa in $kg$, Tiempo in $s$, Longitud in $m$) or Derived ($Velocity$ in $m/s$, $Force$ in $N$). They are also Scarlar (magnitude only) or Vectoral (magnitude, direction, and sense). Kinematics in one dimension studies motion: Position ($x$), Velocity ($v$), and Acceleration ($a$). Constant velocity ($a = 0$) characterizes Uniform Rectilinear Motion (MRU): $x_f = x_i + v \times t$. Uniformly Varied Rectilinear Motion (MRUV) involves constant acceleration: $v_f = v_i + a \times t$ and $x_f = x_i + v_i \times t + \frac{1}{2} a \times t^2$.

Work ($W$) is energy transfer through force and displacement: $W = F \times d \times \cos(\theta)$. There are two main mechanical energies: Kinetic ($E_c = \frac{1}{2} m v^2$), based on motion, and Potential ($E_p = m \times g \times h$), based on position or height. The Law of Conservation of Energy states energy is neither created nor destroyed, only transformed. Renewable sources include Solar (photovoltaic/thermal), Wind (aerogenerators), Hydraulic (reservoirs), Geothermal (Earth's internal heat), Biomass (organic matter/biogas), and Mareomotriz (tides/waves).

SOCIAL SCIENCES AND THE HISTORY OF HONDURAS

The history of Honduras is defined by several pivotal periods. Independence from Spain occurred on September 15, 1821, followed by the Federal Republic of Central America (1823–1839), a union involving Guatemala, El Salvador, Honduras, Nicaragua, and Costa Rica. The “Banana Enclave” was an economic model where foreign US companies controlled production and export, significantly influencing politics, leading to events like the 1954 Banana Strike. In 1990, during the mandate of Rafael Leonardo Callejas, Neoliberalism was systematically implemented through “paquetazos económicos” involving currency devaluation, tax increases (ISV), and privatization.

Sociology studies society, relationships, and power structures. Stratification theories include Marxist (economic ownership), Weberian (wealth, prestige, and power), and Functionalist (meritocracy). The Constitution of the Republic of Honduras is the supreme legal document defining the state as free, sovereign, and representative. Human Rights are inherent freedoms characterized as Universal, Indivisible, and Inalienable. The guarantee of Habeas Corpus ensures physical integrity and prevents unlawful detention. Philosophy remains the rational search for existence, knowledge, and behavior, while Anthropology explores human origins via Creacionista (supernatural) or Evolution (Darwinian) theories and recognizes the multicultural reality of Honduras (Lencas, Garífunas, Miskitos, among others).

QUESTIONS AND DISCUSSION

Questions in the study process often involve practical analysis of scenarios. For example, if a notebook and pencil cost $\text{L. } 25$, and two notebooks plus a pencil cost $\text{L. } 40$, using a system of equations reveals the notebook costs $\text{L. } 15$ and the pencil $\text{L. } 10$. In a health scenario where alcoholism is studied alongside schizophrenia and drug abuse, identifying variables is key; here, alcoholism is the independent variable being manipulated, while psychiatric disorders are the dependent variables being measured. Ethical considerations are noted regarding the implementation of AI, highlighting the need for gender equity and inclusion to prevent biased systems. Historical context is also queried, such as identifying if a figure like Lempira belonged to the Lencas (which is correct), or calculating arrival times across time zones. For a flight from Honduras ($\text{GMT-6}$) to Madrid ($\text{GMT+1}$) leaving at $\text{8:00 p.m.}$ with a 10-hour flight, the time difference and duration must be combined: $\text{8:00 p.m.}$ ($\text{Honduras time}$) is $\text{3:00 a.m.}$ next day ($\text{Madrid time}$) with a 7-hour difference; adding 10 flight hours results in a $\text{1:00 p.m.}$ arrival.