DATOATQUANTITATIVEREASONINGOUTLINES2021-210705-175749.pdf
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DAT / OAT QUANTITATIVE REASONING OUTLINES Table of Contents 1 - Algebra Page 1 2 - Geometry and Trigonometry Page 11 3 - Probability and Statistics Page 14 4 - Data Analysis...
DAT / OAT QUANTITATIVE REASONING OUTLINES Table of Contents 1 - Algebra Page 1 2 - Geometry and Trigonometry Page 11 3 - Probability and Statistics Page 14 4 - Data Analysis Page 18 ChadsPrep.com 0 1 - Algebra Numbers Real numbers vs. Imaginary numbers Rational (can be expressed as a fraction) vs. Irrational numbers (can’t be expressed as a fraction) Integers (Counting numbers; can be positive, negative, or zero) Prime numbers (2,3,5,7,11,13,17,19,23,29…) Order of Operations (PEMDAS—Please Excuse My Dear Aunt Sallie) Fractions 2 3 1 1 3 + + = = 3 4 5 7 5 1 3 2 7 8 = = 3 4 7 9 3 2 3 2 3 4 +1 = 4 1 = 3 5 3 5 Decimals -CALCULATOR -Scientific Notation -Convert to fractions ChadsPrep.com 1 Comparing Fractions 1. Compare numbers above and below ½. 2. Compare numbers with equal differences between the numerator and denominator. 3. Get a common denominator to compare. 4. USE THE CALCULATOR! Which of the following is the largest? 4 9 12 13 11 , , , , 9 16 19 25 23 Percentages 1 = 50% Part Is 2 Percent = 100 or Percent = 100 Whole Of 1 = 33.3% 3 1 = 25% 4 1 What is 29% of 300? = 20% 5 1 = 12.5% 8 1 = 11.1% 9 1 = 10% 43 is what percentage of 7? 10 1 = 5% 20 1 = 4% 25 If Mark invests $3000 at 10% interest compounded annually, by how much will his investment have grown after 3 years? ChadsPrep.com 2 Scientific Notation 358 = 3.58 102 0.00358 = 3.58 10−3 ( 4.6 10 7 )(3.0 10 −4 ) = 1.7 10 −6 = 3.4 10 2 Roots (Square Roots and Other Roots) 12=1 13=1.0.0049 = 22=4 23=8 32=9 33=27 42=16 43=64 52=25 53=125 62=36 72=49 34=81 82=64.0.00049 = 92=81 102=100 3 0.000000064 = Dilutions (M1V1 = M2V2 or C1V1 = C2V2) How much water and how much of a 80% orange juice concentrate must be mixed to make 500ml of 20% orange juice? ChadsPrep.com 3 Conversions Temperature K = C + 273 and C = K – 273 Express 90C in Fahrenheit. 9 5 F = ( C ) + 32 and C = ( F − 32) 5 9 Kelvin Celsius Fahrenheit -40ºC -40ºC 273K 0C 32ºF Express 50F in Kelvin. 298K 25C 77ºF 373K 100C 212ºF Time (1hr = 60min, 1min = 60s, 1hr = 3600s) Terra 1012 Convert 100kph (kilometers per hour) to m/s. Giga 109 Mega 106 kilo 103 centi 10-2 milli 10-3 icro 10-6 nano 10-9 pico 10-12 femto 10-15 Weight Convert 160lbs to grams. Useful Conversions 1 in. = 2.54cm 1 yard = 3ft. 1 mile = 5280ft. 1 mile = 1.6km 1kg = 2.2 pounds Distance 1 pound = 16 ounces Covert 3200m to miles. 1 gallon = 4qts. 1qt. = 2pints = 32 fl. ounces 1pint = 2 cups ChadsPrep.com 4 Solving Equations Solve each of the following for x: x+2=9 3x = 15 x 5x =4 = 25 5 4 7x - 5 = 44 3x + 16 = 5x + 24 3 6 5 3 = = 2x 5 2x + 5 x Inequalities Solve each of the following for x: 2x + 1 < 5 -3x + 1 < 19 Absolute Values x−3 = 7 ChadsPrep.com 5 Exponents (xa)(xb) = xa+b (x3)(x4) = xa x3 = x a −b = x b x4 (xa)b = xab (x3)4 = (ax)(bx) = (ab)x (3x)(4x) = 6x ax a = x 10 x = (x y4 z) 2 3 = bx b 5x x5 y 2 z 4 Solve for x: 3x = 9x+1 Logs (Logarithms) log (0.001) = log (10-3) = -3 log(x) + log(y) = log(xy) log(2) + log(50) = log (0.01) = log (10-2) = -2 log (0.1) = log (10-1) = -1 log (1) = log (100) = 0 log (10) = log (101) = 1 x log (100) = log (102) = 2 log(x) - log(y) = log log(60) – log(6) = log (1000) = log (103) = 3 y log(xy) = ylog(x) log(105) = 2x Solve for x: log =3 5 ChadsPrep.com 6 Ratios and Proportions The ratio of men to women attending a wedding is 4 to 6. If the total attendance is 120, how many are men? 36 48 54 60 66 Quadratic Equations and Polynomials (x+5)(x+5) = x2+10x+25 (x–5)(x–5) = x2–10x+25 (x+5)(x–5)=x2–25 x2+25 is not factorable − b b 2 − 4ac , ax 2 + bx + c = 0 2a Factor the following: x2+6x+8 x2 –x–12 Solve the following for x (2 solutions): x2+6x+8 = 0 x2 –x–12= 0 Functions f(x) = 7x + 4 f(x) = x2–2x+5 f(2) = f(-3) = f(x) = x + 3 g(x) = 2x + 6 f(g(x)) = g(f(x)) = Solving Simultaneous Equations (Substitution or Linear Combination) Solve for x and y: x + 3y = 4 (-1,1) 3x + 2y = 5 (1,1) (-1,-1) (-5,3) (3,-5) ChadsPrep.com 7 Graphical Analysis Lines Slope-intercept equation of a line: y = mx + b m = slope and b = y-intercept Plot the following line: -3x + y = 2 Horizontal / Vertical Lines Parallel lines – lines that have the same slope Give the general formula of a line parallel to the following line: y = 3x – 1 Perpendicular lines – lines whose slopes are negative reciprocals on one another Give the general formula of a line perpendicular to the following line: y = -2x + 3 Inequalities Graph the following: -3x + y < 2 vs -3x + y ≤ 2 Absolute Values Graph the following: y = x − 3 Distance Formula d= (x 2 − x1 ) + (y 2 2 − y1 ) 2 (2-dimensional) d= (x 2 − x1 ) + (y 2 2 − y1 ) + (z 2 2 − z1 ) 2 (3-dimensional) Find the distance between the following points: (2,7) and (8,15) ChadsPrep.com 8 Parabolas y = x2 vs. y = -x2 Plot the graph of y = -x2 + 4 Circles (x – h)2 + (y – k)2 = r2 centered at (h,k) with radius = r x2 + y2 = r2 centered at (0,0) with radius = r Ellipses (x − h)2 (y − k)2 + =1 a2 b2 Centered at (h,k) with radii = a and b ChadsPrep.com 9 Common Applied Mathematics Problems Age Problems Robert is 8 years older than Sam. In 3 years, Robert will be twice as old as Sam. How old is Sam? 3 5 8 10 12 John is 15 years older than Alex. In 10 years John will be 1 year more than twice Alex’s age. How old is John? 12 16 19 21 25 Rate Problems Train #1 is heading north at 50mph and train #2 is heading south on the same track at 60mph. If they are separated by a distance of 275 miles, then how long before they’ll collide? 1.5hrs 2.0 hrs 2.5hrs 3.0hrs 3.5hrs John can paint a house inside and out in 4 days while Mike can do it in 6 days. How long would it take them to paint a house if they worked together? 1 day 1.6 days 2.0 days 2.4 days 3.2days ChadsPrep.com 10 2 - Geometry and Trigonometry Geometry Triangles 1 A= bh Sum of Interior Angles = 180 Sum of Exterior Angles = 360 2 (n – 2)(180) (for polygon of ‘n’ sides) -Acute vs Obtuse angles -Similar Triangles -Equilateral Triangles -Isosceles Triangles Right Triangles -Pythagorean Theorem (a2 + b2 = c2) -3:4:5 and 5:12:13 Right Triangles Rectangles and Squares Rectangle A = lw P = 2l + 2w Square A = s2 P = 4s The ratio of the length to the width of a rectangle is 3:2. If the perimeter of the rectangle is 100, what is its area? ChadsPrep.com 11 Circles A = r 2 C = 2r = d Asec tor = (r 2 ) ArcLength = (2r ) 360 360 If the circumference of a circle increases by 50%, by what percentage does the area increase? Uniform Solids V = Abase h Rectangular Solids V = l w h Cylinders V = r 2 h SurfaceAre a = 2rh + 2r 2 4 3 Spheres V = r SurfaceAre a = 4r 2 3 ChadsPrep.com 12 Trigonometry radians degrees sin cos tan SohCahToa 0 0 0 1 0 O 30 1 3 3 sin = H 6 2 2 3 45 2 2 1 A 4 cos = 2 2 H 60 3 1 3 3 2 2 O tan = 90 1 0 ∞ A 2 ChadsPrep.com 13 3 - Probability and Statistics Permutations (Distinction between selections…often order) n! Pk = n = # of options k = # of selections (n − k )! n Combinations (No distinction between selections) n! Ck = n = # of options k = # of selections k!(n − k )! n William won 4 front-row tickets to a concert: seats 1A, 1B, 1C, and 1D. Unfortunately he has 5 siblings and can only take 3 of them. If William chooses the aisle seat (1A) for himself, how many possibilities are there for the seating arrangement for the remaining three seats? William won 4 tickets to a concert. Unfortunately he has 5 siblings and can only take 3 of them. How many possibilities are there for which siblings attend the concert with him? ChadsPrep.com 14 Mike chooses 3 of his 10 new books to take with him on vacation. How many possible sets of 3 books can he take with him? A class of 12 students is divided into groups of 4. How many groups of 4 can be formed? A pre-dental club with 8 members is selecting a president, vice president, and treasurer. How many different sets of officers are possible? Jack and Jill have to choose a main color and a color for the trim of their house among 20 possible paint colors. How many possibilities exist for the outcome of their house painting project? (Assume two different colors are chosen.) ChadsPrep.com 15 Probability Marbles A bag contains 6 red marbles, 4 green marbles, and 2 blue marbles. What is the probability of pulling out 2 green marbles without replacement? Dice What is the probability that the sum of two dice rolled will add up to 5? Deck of Cards What is the probability of pulling 4 face cards out of a deck of cards (52 cards) without replacement? Coin Toss What is the probability of getting exactly 3 tails when flipping a coin 5 times? What is the probability of getting at least 3 tails when flipping a coin 5 times? Chance of Rain If there is a 25% chance of rain for each of the next 4 days, then what is the probability that it will rain the first day only? If there is a 25% chance of rain for each of the next 4 days, then what is the probability that it will rain exactly 2 out of the 4 days? ChadsPrep.com 16 Statistics Mean = average Median = middle value Mode = most common value Population Standard Deviation ∑(𝑥 − 𝑥̅ )2 √ 𝑁 Sample Standard Deviation ∑(𝑥 − 𝑥̅ )2 √ 𝑁−1 For the following set of data, determine the mean, median, and mode. 60, 70, 70, 75, 90, 95, 100 Find the population standard deviation in the following set of data. 2, 2, 3, 4, 5, 5, 6 If Jennifer has scored 75, 86, and 64 on her first 3 exams, what score does she need on the 4th exam to have an average of 80? ChadsPrep.com 17 4 - Data Analysis Variable – a characteristic that can vary for a population Variables can be quantitative/numerical like age or height Variables can be categorical/nonnumerical, like hair color or race Frequency and frequency distribution Relative frequency (expressed in percents, fractions, or decimals) Tables and Bar Graphs Grade Number of Grade Percentage Students of Students F 5 F 11 D 6 D 13 C 16 C 36 B 11 B 24 A 7 A 16 Grade Distribution Grade Distribution 18 40 16 Percentage of Students 35 Number of Students 14 30 12 25 10 20 8 15 6 4 10 2 5 0 0 F D C B A F D C B A Grades Grade How many students earned a C or D? What percentage of students had a grade of C or higher? ChadsPrep.com 18 Grade Juniors Seniors F 4 1 D 4 2 C 6 10 B 4 7 A 3 4 Grade Distribution Grade Distribution 12 18 Number of Students Number of Students 16 10 14 8 12 10 6 8 4 6 4 2 2 0 0 F D C B A F D C B A Grade Grade Juniors Seniors Juniors Seniors How many seniors had a grade lower than a B? What percentage of students had a grade higher than a C? What percentage of juniors had a grade higher than a C? ChadsPrep.com 19 Histograms Grade Number of Students 0-60 5 60-70 6 70-80 16 80-90 11 90-100 7 What fraction of students had a grade from 90- 100? Circle Graphs (Pie Charts) What fraction of students received a B or higher? ChadsPrep.com 20 Scatterplots Sales of Bottled Water vs Temperature $1,200 $1,000 $800 Sales $600 $400 $200 $0 50 55 60 65 70 75 80 Temperature (°F) What would be the approximate expected sales on a day where the temperature reaches 90F based upon the presented data? A. $800 B. $1100 C. $1400 D. $1700 E. $2000 Line Graphs Advertising Budget $3,500 Advertising Budget $3,000 $2,500 $2,000 $1,500 $1,000 $500 $0 2011 2012 2013 2014 2015 2016 2017 2018 2019 Year In which year did the advertising budget experience the greatest increase? In which year did the advertising budget experience the greatest increase? ChadsPrep.com 21
