Mathematics Essentials

Cheatsheet Content

Percent & Discount Discount: Amount deducted from the original price. Percentage: Part of the total number, calculated as Base $\times$ Rate. Rate: Ratio expressed as a percent (%). It always comes with the percent symbol. Discount Rate: $\text{Discount Amount} / \text{Original Price}$. Interest: Amount earned for the use of money. Principal: The total amount on which interest is calculated. Discounted Price (Sale Price): Original Price - Discount Amount. Formulas $\text{Percentage} = \text{Base} \times \text{Rate}$ $\text{Rate} = \text{Percentage} / \text{Base}$ $\text{Base} = \text{Percentage} / \text{Rate}$ $\text{Simple Interest} = \text{Principal} \times \text{Rate} \times \text{Time}$ Ratio & Proportion Ratio: Comparison of two quantities (e.g., $a:b$, $a$ to $b$, or $\frac{a}{b}$). Proportion: An equation stating that two ratios are equal. Cross-Multiplication: For a proportion $\frac{a}{b} = \frac{c}{d}$, then $a \cdot d = b \cdot c$. Geometry: Polygons Polygon: A closed 2D shape with straight sides. Congruent Polygons: Polygons with the same shape and size. Sides and Names: 3 sides: Triangle 4 sides: Quadrilateral 5 sides: Pentagon 6 sides: Hexagon 7 sides: Heptagon 8 sides: Octagon 9 sides: Nonagon 10 sides: Decagon Area of a Trapezoid: $A = \frac{1}{2}(b_1 + b_2)h$, where $b_1, b_2$ are bases and $h$ is height. Area of a Parallelogram: $A = \text{base} \times \text{height}$. Geometry: Angles Angle: Formed by two rays with a common endpoint. Types of Angles: Acute Angle: $ Right Angle: $= 90^\circ$ Obtuse Angle: $> 90^\circ$ and $ Straight Angle: $= 180^\circ$ Reflex Angle: $> 180^\circ$ and $ Parallel Lines: Lines in a plane that do not meet. Geometry: Triangles Types of Triangles (by sides): Equilateral: All 3 sides are equal. Isosceles: 2 sides are equal. Scalene: No sides are equal. Number Series/Patterns Identify the rule (addition, subtraction, multiplication, division, or a combination). Apply the rule to find subsequent numbers. Example Calculations 1. Finding Percentage Mm got 80% of 60 items correct. How many? $\text{Correct Items} = 60 \times 0.80 = 48$ 2. Finding Missing Element in Proportion $24/8 = N/17$ $8N = 24 \times 17$ $8N = 408$ $N = 51$ 3. Rate of Discount Book from ₱250 to ₱175. Rate of discount? $\text{Discount Amount} = 250 - 175 = 75$ $\text{Rate} = 75 / 250 = 0.30 = 30\%$ 4. Area of Trapezoid Upper base 7m, lower base 9m, height 4m. $A = \frac{1}{2}(7 + 9) \times 4 = \frac{1}{2}(16) \times 4 = 8 \times 4 = 32\,\text{m}^2$ 5. Original Price with Discount Shoes discounted by ₱450 at 30%. Original price? $0.30 \times \text{Original Price} = 450$ $\text{Original Price} = 450 / 0.30 = 1500$ 6. Population Calculation 357 thousand is 50% of Ilocos Sur population. Total population? $0.50 \times N = 357$ $N = 357 / 0.50 = 714\,\text{thousand}$ 7. Percent Not Harvested 67% of crops harvested. What percent not harvested? $100\% - 67\% = 33\%$ 8. Work Rate Problem 9 eggs for 4 omelets. How many for 12 omelets? $\frac{9}{4} = \frac{N}{12}$ $4N = 9 \times 12$ $4N = 108$ $N = 27$