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

Essential Math Formulas and Concepts

Volume of Shapes

(a) Cube:

• Formula: V = s³ (where s is the length of a side)
• Example: A cube with side length 4 cm

V = 4³ = 64 cm³

(b) Cylinder:

• Formula: V = πr²h (where r is the radius and h is the height)
• Example: A cylinder with radius 2 cm and height 6 cm

V = π(2²)(6) = 24π cm³ (approximate value)

Surface Area of Shapes

(a) Cube:

• Formula: SA = 6s² (where s is the length of a side)
• Example: A cube with side length 3 cm

SA = 6(3²) = 54 cm²

(b) Cone:

• Formula: SA = πr(r + √(r² + h²)) (where r is the radius and h is the height)
• Example: A cone with radius 5 cm and height 8 cm

SA = π(5)(5 + √(5² + 8²)) = 129.74 cm² (approximate value)

Stem and Leaf Plot

(a) Mean:

• Example: Data: 5, 7, 8, 9, 10, 12, 15

Mean = (5 + 7 + 8... Continue reading "Essential Math Formulas and Concepts: A Quick Reference" »

Discrete Math Concepts

Binary Relations

Cartesian Product

The Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B.

Example: If A = {0, 1}, the Cartesian product A × A is {(0,0), (0,1), (1,0), (1,1)}.

Properties of Binary Relations

A binary relation R on a set A can have several properties:

1. Reflexive: For all x ∈ A, (x, x) ∈ R.
2. Symmetric: For all x, y ∈ A, if (x, y) ∈ R, then (y, x) ∈ R.
3. Transitive: For all x, y, z ∈ A, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.
4. Irreflexive: For all x ∈ A, (x, x) ∉ R.
5. Antisymmetric: For all x, y ∈ A, if (x, y) ∈ R and (y, x) ∈ R, then x = y.

Examples of Relation Properties

Consider a set A = {0, 1}.

• The relation R = A

Cost of the Building: Land+Construction

When was the building bought? YTD amort.construction 2020 / construction P&L --> it has been amortized x years until 2020

At which price did the company sell the building? ((Land + Construction) - YTD amort.constr) + Profit from selling

useful life of the building? construction / construction P&L

Occupancy: 33%. VERY LOW !! ===> 70 rooms x 600 ADR x occupancy x 365 days = 5.059.000 €

F&B Menu: 7 € ==> VERY LOW & doesn't match with an 600 ADR !! [(80% of 100 people for lunch) + (80% of 100 people per Dinner) ] * Menu Price * 365 days = 409.000 €
F&B Breackfast: 18€/person. Reasonable with 600 ADR 80% of 1,5 clients x 33% occupancy * 70 rooms * 365 days * Breackfast Price... Continue reading "Hotel Financial Analysis and Recommendations" »

Stock Valuation and WACC Examples

Problem 1: Constant Dividend Growth

Thomas Brothers is expected to pay a $0.35 per share dividend at the end of the year (D1 = $0.35). The dividend is expected to grow at a constant rate of 6% a year. The required rate of return is 13%. What is the stock’s current price?

P0 = D1 / (r – g) = 0.35 / (0.13 - 0.06) = $5.00

Problem 2: Current Stock Price with Constant Growth

A stock just paid a dividend of D0 = $1.50. The required rate of return is r = 10%, and the constant growth rate is g = 4.0%. What is the current stock price?

P0 = D0(1+g) / (r – g) = 1.50(1 + 0.04) / (0.10 - 0.04) = $26.00

Problem 3: Expected Stock Price in One Year

Using your answer for problem 2, what is the expected stock price in one year... Continue reading "Calculating Stock Prices and Weighted Average Cost of Capital (WACC)" »

Geometry

Angles

Adjacent angles - that have a common vertex and side, but no common interior points

Angle - formed by two rays with a common endpoint called the vertex

Complementary angles - whose measures add up to 90°

Congruent angles - that have the same measure

Supplementary angles - two angles whose measures add up to 180°

Vertical angles - the nonadjacent angles formed by two intersecting lines

Lines

Midpoint - the point that divides a segment into two congruent segments

Parallel lines - in a plane that never meet

Perpendicular lines - that intersect at 90° angles

Transversal - a line that intersects any two or more other lines

Triangles

Triangle Inequality Theorem - the sum of the lengths of any two sides of a triangle is greater than the length... Continue reading "Essential Geometry and Algebra Glossary" »

Statistics: Key Concepts and Methods

1. Statistics is a collection of methods for planning experiments, obtaining data, and then organizing, summarizing, presenting, analyzing, interpreting, and drawing conclusions.

2. Statistical methods consist of a series of procedures for managing, analyzing, and collecting qualitative and quantitative research data.

3. A population includes all objects of interest, whereas a sample is only a portion of the population.

Example: The population could be all students of a postgraduate program, and a representative sample of this population could be 400 students from different studies of the program.

Understanding Variables in Statistics

4. A variable is any characteristic, number, or quantity that can be measured... Continue reading "Key Concepts and Methods in Statistical Analysis" »

Unit 6: Costing and Production Systems

1. Production Systems

The two common cost accumulation systems are job order costing and process costing. The election to use one or the other of these systems depends primarily on the nature of the product and the manufacturing process by which it is created. Process production and job-order production are the extreme possibilities when considering the heterogeneity of products. The difference between order accounting and section accounting is the aim of the cost system, and the difference between process costing and job costing is the method to assign the cost to products.

1.1. Job Order Costing System

Job order costing suits the processes where it is necessary to assign costs to a specific amount of production... Continue reading "Costing and Production Systems: Job Order and Process Costing" »

Total Cost = w₁x₁ + w₂x₂ + … + w_nx_n

Isocost function:

x₂ = -w₁x₁/w₂ + tc/w₂ where the slope is: -w₁/w₂ ratio of relative factor prices with negative sign

Cost minimization mate: -w₁/w₂ = -MP₁/MP₂ (isoquant curve)

Cost minimization implies producing a certain amount of output (y > 0) with the lowest possible cost. As with the maximization of profits, the firm is getting the most out of the resources it is using.

What is the difference between cost minimization and profit maximization? Cost minimization is a necessary condition for the maximization of profits, but not the other way around. For example, if you bring costs down to zero then output and profits will be zero too, failing to maximize profits while minimizing costs. As before, when... Continue reading "Cost Minimization and Profit Maximization in Economics" »

Pocket money isn’t a controversial topic nowadays.

If there have been a lot of debates over pocket money.

Pocket money used to be a controversial topic.

T

Pocket money has always been a controversial topic.

T

The debate regarding pocket money nowadays is about how much money should be given.

If today’s debate rages about something entirely different.

The debate regarding pocket money nowadays isn’t about how much money should be given.

T

The debate regarding pocket money nowadays is about at what age money should be given.

If

Twelve-year-old will be able to use the new debit card.

A new debit card has just been launched for teenagers as young as 13.

Thirteen-year-olds won’t be able to use the credit card.

Teenagers over 13 will be able to use the... Continue reading "The Future of Pocket Money: Debit Cards and Financial Management" »

Key Statistical Concepts

Variables

Categorical (Qualitative) Variables: Represent characteristics or qualities. Examples include race and gender.

Quantitative Variables: Represent numerical values that differ in magnitude.

Scales of Measurement

• Nominal: Unordered categories.
• Ordinal: Ordered categories.
• Interval: Equal intervals between values.

Data Visualization

Frequency: Displays the possible values of a variable and the number of times each occurs.

Histogram: A bar graph of frequencies or percentages. Shapes can be bell-shaped, skewed, or bimodal.

Box Plot: Displays the distribution of data, including median, quartiles, and outliers.

Scatter Plot: Shows the relationship between two variables.

Distribution

• For symmetric distributions, the mean equals