Grammar Hero's Mathematics Reference Sheet PDF

Grammar Hero’s Reference Sheet Slope of a Line Triangle Sum Theorem Let (x1, y1) and (x2, y2) be two points on the line. The sum of the three interior angles in a triangle is always...

Grammar Hero’s Reference Sheet Slope of a Line Triangle Sum Theorem Let (x1, y1) and (x2, y2) be two points on the line. The sum of the three interior angles in a triangle is always 180°: ∠a + ∠b + ∠c = 180° change in y 𝑦2 − 𝑦1 Slope = m = change in x = 𝑥2 − 𝑥1 Quadratic Formula −𝑏 ± √𝑏 2 − 4𝑎𝑐 𝑥= 2𝑎 Note: ax2 + bx + c = 0 and a ≠ 0 Pythagorean Theorem Equations of Lines In any right triangle, the sum of the squares of the legs is equal to the square of the hypotenuse: a2 + b2 = c2 Standard Form: Ax + By = C Slope-Intercept Form: y = mx + b where m = slope and b = y – intercept Point-Slope Form: y – y1 = m (x – x1) Distance Formula 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 Area of a Triangle 1 d = distance between points (x1, y1) and (x2, y2). 𝐴= 𝑏ℎ 2 Midpoint Formula 𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑀=( , ) 2 2 M = point halfway between points (x1, y1) and (x2, y2). b = the base of any triangle h = perpendicular height of any triangle Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Simple Interest Formula Distance, Rate, and Time Formula 𝐼 = 𝑝𝑟𝑡 𝑑 = 𝑟𝑡 p = Principal (i.e., loan or investment amount) d = distance, I = Interest earned r = rate r = Rate of interest per year in decimal form t = time t = Time in terms of years Rate and time must be in proportional units (e.g., if rate is given in terms of miles per hour, Order of Operations (PEMDAS) time must be in terms of hours) Order of operations refers to the order in Percent Change which calculations are performed to evaluate an expression. The acronym PEMDAS is useful 𝑁𝑒𝑤 𝑉𝑎𝑙𝑢𝑒 − 𝑂𝑙𝑑 𝑉𝑎𝑙𝑢𝑒 for remembering it. 𝑃𝐶 = 𝑥 100 𝑂𝑙𝑑 𝑉𝑎𝑙𝑢𝑒 P Parenthesis PC = Percent Change E Exponents If PC is positive, there was an increase. M Multiplication and Division (Left to If PC is negative, there was a decrease D Right) A Addition and Subtraction (Left to Right) S Statistics Note: Multiplication and division have equal precedence, so their calculations are Sum of all Data Points performed as they appear in the expression 𝐌𝐞𝐚𝐧 = Number of Data Points from left to right. Likewise, addition and subtraction have equal precedence, so their Range = Maximum Value – Minimum calculations are performed as they appear Value from left to right. Mode = The value in the data set that occurs the most often Perimeter To find the perimeter of any polygon, Median = The value in the middle of the excluding circles, you simply add up all of its data set sides. For example: To find the median of a data set, arrange the observations in order from smallest to largest value. If there is an odd number of observations, the median is the middle value. If there is an even number of observations, the median is the average of the two middle values. Note: The perimeter of a circle is its circumference. Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Circles Area of a Circle: Circumference of a Circle: r = radius of a circle 𝐴 = π𝑟2 𝐶 = π𝑑 or 𝐶 = 2π𝑟 d = diameter of a circle π = 3.14 or 22/7 Converting Units Weight and Mass Larger unit → smaller unit Multiply 1 Ton (T) 2,000 pounds Smaller unit → Larger unit Divide 1 pound (lb) 16 ounces (oz) Linear Units Time 12 inches (in) 1 foot (ft) 1 day 24 hours 3 feet 1 yard (yd) 1 hour (hr) 60 minutes (min) 36 inches 1 yard 1 minute 60 seconds (sec) 63,360 inches 1 mile (mi) 1 year (yr) 365.25 days 5,280 feet 1 mile 1 week 7 days 1,760 yards 1 mile 1 year 12 months (mon) 1440 minutes 1 day 3600 seconds 1 hour Capacity 8 fluid ounces 1 cup Polygons 2 cups 1 pint (pt) Shape Number of Sides Sum of Interior Angles 2 pints 1 quart (qt) Triangle 3 180 degrees 4 quarts 1 gallon Quadrilateral 4 360 degrees Pentagon 5 540 degrees Hexagon 6 720 degrees Heptagon or 7 900 degrees Septagon Octagon 8 1080 degrees Any Polygon n S = 180(n - 2) Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Multiplication Rules The product and the quotient of one and any number is that number. 7x1=7 100 ÷ 1 = 100 Zero times any number equals zero. 0×2=0 858 x 0 = 0 Zero divided by any nonzero number is zero. 0÷3=0 Dividing a number by zero is undefined. 5 = undefined 0 When multiplying or dividing with positives and negatives, use the signs charts. Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Quadrilaterals Square Perimeter: P = 4s Area: A = s2 Note: To find the perimeter of any quadrilateral, you simply add up all of its sides. Rectangle Perimeter: P = 2l + 2w Area: A = lw Note: To find the perimeter of any quadrilateral, you simply add up all of its sides. Parallelogram Perimeter: P = 2a + 2b Area: A = bh Note: To find the perimeter of any quadrilateral, you simply add up all of its sides. Trapezoid Perimeter: P= a + b1 + c + b2 1 Area: A = (b1 + b2 ) ∙ h 2 Note: To find the perimeter of any quadrilateral, you simply add up all of its sides. Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Formulas for Volume (V) and Surface Area (SA) Cube V = a3 SA = 6a2 Rectangular Solid V= lxwxh SA = 2(l x w) + 2(w x h) + 2(h x l) l = length w = width h = height Cylinder V = π𝑟2 h SA = 2π𝑟h + 2π𝑟2 Sphere 4 V= 𝜋𝑟 3 3 SA = 4π𝑟2 Rectangular Pyramid 1 𝑉= 𝑎𝑏ℎ 3 Note: ab is the area of the base of the pyramid Cone 1 𝑉= 𝜋𝑟 2 ℎ 3 Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Basic Probability 1. For any event A: 0 ≤ P(A) ≤ 1 2. P(impossible event) = 0. 3. P(sure event) = 1. Desired outcome 4. P(A) = Total number of outcomes 5. P(not A) = 1 - P(A) 6. P(A and B) = P(A) x P(B) Independent (Replacement) vs. Dependent Events (No Replacement) 7. P(A or B) = P(A) + P(B) (Exclusive Events ) 8. P(A or B) = P(A) + P(B) – P(A and B) (Non-Exclusive Events) Prime Numbers Times Table Perfect Squares 12 = 1 62 = 36 112 = 121 162 =256 22 = 4 72 = 49 122 = 144 172= 289 32 = 9 82 = 64 132= 169 182= 324 42 = 16 92= 81 142= 196 192= 361 52 = 25 102= 100 152= 225 202= 400 Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Laws of Exponents Zero-Exponent Rule a0 = 1 Anything raised to the zero power is 1. Power Rule (am)n = amn To raise a power to a power you need to multiply the exponents. Negative Exponent Rule Negative exponents in the numerator get moved to the denominator and become positive exponents. Negative exponents in the denominator get moved to the numerator and become positive exponents. Product Rule am ∙ an = am + n To multiply two exponents with the same base, you keep the base and add the powers. Quotient Rule To divide two exponents with the same base, you keep the base and subtract the powers. Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Writing Equations of Lines Given the following, you can write equations of lines using these steps: 1. The slope of the line and the y-intercept a. Slope = m b. Y-intercept = b c. Plug the m and b values into y = mx + b 2. The slope of the line and a point that lies on the line a. Slope = m b. Substitute the point (x, y) in for x and y in the equation y = mx + b and solve for b. c. Plug the m and b values into y = mx + b OR a. Slope = m b. Substitute the slope and the point (x, y) in for x1 and y1 in the point-slope equation: y - y1 = m(x - x1) c. Solve the equation for y. 3. Two points that lie on the line change in y 𝑦2 − 𝑦1 a. Using the slope formula: m = change in x = , find the slope. 𝑥2 − 𝑥1 b. Substitute the slope and either point (x, y) in for x1 and y1 in the point-slope equation: y - y1 = m(x - x1) c. Solve the equation for y. Fractions Adding and Subtracting Fractions 𝑎 𝑐 (𝑎 ∙ 𝑑) + (𝑐 ∙ 𝑏) + = 𝑏 𝑑 𝑏 ∙𝑑 Multiplying Fractions 𝑎 𝑐 𝑎 ∙𝑐 ∙ = (Multiply straight across) 𝑏 𝑑 𝑏 ∙𝑑 Dividing Fractions 𝑎 𝑐 𝑎 ∙𝑑 ÷ = (Keep, Change, Flip) 𝑏 𝑑 𝑏 ∙𝑐 Converting Mixed Numbers to Improper Fractions 𝑏 (𝐴 ∙ 𝑐) + 𝑏 𝐴 = 𝑐 𝑐 Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Combined Rates & Work Step 1: A problem involving combined rates or work can be solved using the formula: T = time working together A = the time it takes for one person or thing to complete the task by themselves B = the time it takes for one person or thing to complete the task by themselves Step 2: Adjust the formula in accordance with the problem and solve Example: Walter and Helen are asked to paint a house. Walter can paint the house by himself in 12 hours and Helen can paint the house by herself in 16 hours. How long would it take to paint the house if they worked together? Step 1: A problem involving work can be solved using the formula: Step 2: Solve the equation created in the first step. This can be done by first multiplying the entire problem by the common denominator and then solving the resulting equation. In this case, the least common denominator is 48. Step 3: Answer the question asked of you in the problem and be sure to include units with your answer. Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Angle Types & Special Angle Pairs Acute Angle: An angle whose measure is less than 90 degrees. Right angle: An angle whose measure is 90 degrees. Obtuse angle: An angle whose measure is bigger than 90 degrees but less than 180 degrees. Straight angle: An angle whose measure is 180 degrees. Thus, a straight angle look like a straight line. Complementary angles: Two angles that have a sum of 90 degrees. ∠ 1 + ∠2 = 90° Supplementary angles: Two angles that have a sum of 180 degrees. ∠ 1 + ∠2 = 180° Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Properties of Radicals Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Factoring Guide Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Factorials n! = n x (n – 1) x (n – 2) x … x 1 0! = 1 1! = 1 2! = 2 x 1 = 2 3! = 3 x 2 x 1 = 6 4! = 4 x 3 x 2 x 1 = 24 5! = 5 x 4 x 3 x 2 x 1 = 120 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 7! = 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5,040 Prime Numbers & Divisibility Rules A number is divisible by… Divisible Not Divisible 2 – if the last digit is even (0, 2, 4, 6, or 8). 3,978 4,975 3 – if the sum of the digits is divisible by 3. 315 = 3 + 1 + 5 = 9 139 = 1 + 3 + 9 = 13 4 – if the last two digits are divisible by 4. 8,512 7,511 5 – if the last digit is 0 or 5. 14,975 10,999 6 – if the number is divisible by both 2 and 3. 48 20 9 – if the sum of the digits is divisible by 9. 711 = 7 + 1 + 1 = 9 93 = 9 + 3 = 12 10 – if the last digit is 0. 15,990 10,536 Since a number is considered prime is if it is only divisible by one and itself, you can use these divisibility rules to quickly determine if a number is NOT prime. In other words, if a number is divisible by 2, 3, 4, 5, 6, 9, or 10, then it cannot be prime. For example: 1. 10,995 – this number is not prime because its last digit is 5, which means it is divisible by 5 (i.e., by something other than one and itself). 2. 988 – this number is not prime because its last digit is 8, an even number, which means it is divisible by 2 (i.e., by something other than one and itself). 3. 333 – this number is not prime because the sum of its digits (3 + 3 + 3 = 9) is divisible by 3, which means it is divisible by 3 (i.e., by something other than one and itself). Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Reciprocals of Fractions, Whole Numbers, and Mixed Numbers To find the reciprocal of a fraction, simply swap the numerator and denominator. 2 5 (The reciprocal of 2/5 is 5/2) 5 2 To find the reciprocal of a whole number, we rewrite whole number as a fraction by placing it over one and then swap the numerator and denominator. 5 1 5=1 (The reciprocal of 5 is 1/5) 5 To find the reciprocal of a mixed number, we rewrite the mixed number as an improper fraction and then swap the numerator and denominator. 2 7 5 15 = 5 (The reciprocal of 1 2/5 is 5/7) 7 Like Terms Like terms have the same letter variables and are raised to same powers. Like terms can be combined into a single term. Like Terms Not Like Terms 2x and -5x 6x and 6y 2a2 and -5a2 y and 6y2 -2xy2 and 8xy2 X and 7 Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Adding and Subtracting Polynomials When adding polynomials, you simply combine like terms. For example: (x2 – x + 5) + (6x2 + 2x – 10) x2 – x + 5 + 6x2 + 2x – 10 7x2 + x – 5 When subtracting polynomials, you rewrite subtraction as addition by distributing the negative sign to every term in the second polynomial and then combine like terms. For example: (3x2 – 8x + 7) – (2x2 – 6x + 12) = (3x2 – 8x + 7) + (-2x2 + 6x - 12) 3x2 – 8x + 7 + -2x2 + 6x - 12 x2 - 2x - 5 Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  The Fundamental Counting Principle The Fundamental Counting Principle is a way to quickly determine the total number of ways different events can occur. Event M can occur m number of ways The total number of ways they can occur is: Event N can occur n number of ways mxn For example: A new restaurant opened, and it offers lunch combos for $5.00. With the combo meal, you get one choice of a sandwich, one choice of a side, and one choice of a drink. How many different combinations can you make for lunch? The choices are below. Sandwiches: Chicken Salad, Turkey, Grilled Cheese Sides: Chips, French Fries, Fruit Cup Drinks: Soda, Water Using a tree diagram to answer this question is slow and inefficient! Applying the Fundamental Counting Principle to answer it takes 30 seconds or less: 3 choices of sandwiches x 3 choices of sides x 2 choices of drinks 3 x 3 x 2 = 18 possible combinations Example 1 Sarah goes to a local deli, which offers a soup, salad, and sandwich lunch combo. There are 3 soups, 3 salads, and 6 sandwiches from which to choose. How many different lunches can be formed? 3 soups × 3 salads × 6 sandwiches = 54 different lunch combos Example 2 A website offers 5 sizes, 8 colors, and 25 logos for their t-shirts. How many different t-shirts can be created, given these options? 5 sizes × 8 colors × 25 logos = 1,000 combinations of shirts Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel  Converting Celsius (C) to Fahrenheit (F) and Fahrenheit (F) to Celsius (C) For the ASVAB, you generally do not have to memorize these formulas. Rather, you must understand that, algebraically, you can use either of these formulas to convert Celsius to Fahrenheit as well as Fahrenheit to Celsius. 9 𝐹 = 𝐶 + 32 5 5 𝐶 = (𝐹 − 32) 9 For example: Convert 30 degrees Celsius to Fahrenheit using both formulas. Using the First Formula Using the Second Formula 9 5 𝐹 = 𝐶 + 32, 𝐶 = 30 𝐶= (𝐹 − 32), 𝐶 = 30 5 9 9 5 𝐹 = (30) + 32 30 = (𝐹 − 32) 5 9 9 30 9 5 9 𝐹= ⋅ + 32 ⋅ 30 = ⋅ (𝐹 − 32) 5 1 5 9 5 270 𝐹= + 32 9 30 5 9 5 ⋅ = ⋅ (𝐹 − 32) 5 1 9 5 𝐹 = 54 + 32 270 = 𝐹 − 32 5 𝐹 = 86 54 = 𝐹 − 32 54 + 32 = 𝐹 𝐹 = 86 Copyright © 2020 by Grammar Hero Tutoring. This reference sheet is for personal use only. Unauthorized replication, especially for commercial purposes, is strictly prohibited. All rights reserved. Grammar Hero’s YouTube Channel