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The Basics PCAT Review July 9-10, 2011 Laws of Operation Operations, addition and multiplication, are both commutative. You can switch the order of the numbers and not affect the result. a+b=b+a a*b=b*a Operations, addition and multiplication, are both associative. You can group numbers any way and not affect the result. (a+b)+c=a+(b+c) (a*b)*c=a*(b*c) NOTE: Division and Subtraction are NOT commutative or associative! The Distributive Law allows you to “distribute” a factor over a group of numbers being added or subtracted. Example: 2*(3x-4y)=2*(3x)-2*(4y) The acronym PEMDAS (or Please Excuse My Dear Aunt Sally) guides you in the correct order of operation. Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right) *Note: If there are parentheses within parentheses, work from the innermost, then outermost. Example: 30 5 4 (7 3) 2 8 30 5 4 4 8 30 5 4 16 8 First perform Parentheses Next, do Exponents 30 20 2 Next, multiply and divide in order from left to right 10 2 Finally, add and subtract from left to right 12 Fractions For the fraction 1 , 1 is the numerator and 2 is the denominator 2 A mixed number is a whole number with a fraction. The Basics Example: To convert 2 PCAT Review July 9-10, 2011 3 3 8 3 11 to one fraction, we do 2 4 4 4 4 4 Adding/Subtracting Fractions For fractions with different denominators, change the fractions so that their denominators agree, then add/subtract the numerators. Example: 1 1 5 4 1 4 5 20 20 20 Multiplying Fractions When multiplying two fractions, multiply the numerators together and multiply the denominators together. Example: 5 2 10 5 6 3 18 9 Dividing Fractions When dividing two fractions, reverse the numerator and denominator of the dividing fraction (2nd fraction), and multiply this to the divided fraction (1st fraction). Example: 5 2 5 3 15 5 1 1 6 3 6 2 12 4 4 Multiplying Decimals Example: 0.342 0.25 1. Multiply the two numbers, ignoring the decimals. 0.342 0.25 8550 2. Find the number of digits behind the decimal in each number and add them together. 3 digits (0.342) + 2 digits (0.25) = 5 digits 3. Take the result of your answer in Step 1 and move the decimal to the left by the number of digits given in the result of Step 2. 0.08550 (since there are only four digits in 8550, we need to add a 0 behind the decimal) The Basics PCAT Review July 9-10, 2011 Dividing Decimals Example: 0.475 0.25 1. Find the number of digits behind the decimal in each number and pick the larger of the two. 3 digits (0.475), 2 digits (0.25). Therefore, 3. 2. Move both decimals the number of places from Step 1. 0.475 0.25 will be changed to 475 250 3. Use long division to determine the decimal answer: 475 250 19 . Decimals to Fractions Example: 0.1337 1. Find the place value of the last digit in the decimal. 2. Write the digits from the decimal in the numerator and the place value of the decimal in the denominator. 01337 . 1337 10,000 Decimals to Percents To convert a decimal to a percent, multiply the decimal by 100. This is equivalent to moving the decimal 2 places to the right in the decimal. Example: 01337 . (01337 . 100)% 1337% . Percents x % is equivalent to the fraction x % of a = x a 100 x Part (or Percent ) 100 Whole The Basics Examples: PCAT Review July 9-10, 2011 72 0.72 100 1. 72% 2. 60% of 20 = 60 20 12 100 Common Percents to Know Fraction Decimal Percent 1/1 1.0 100 3/4 0.75 75 2/3 0.66 66.6 1/2 0.5 50 1/3 0.33 33.3 1/5 0.25 25 1/5 0.2 20 1/8 0.125 12.5 1/10 0.1 10 1/20 0.05 5 Percent Formulas %Increase=Amount of increase/Original whole * 100 %Decrease=Amount of decrease/Original whole *100 New Whole= Original whole Amount of Change Exponents The Basics PCAT Review July 9-10, 2011 Multiplying powers with the same base: Keep the base the same and add exponents x 3 x 4 x 3 4 x 7 Example: Dividing power with the same base: Keep the base the same and subtract exponents x10 x 3 x103 x 7 Example: Raising a power to an exponent: keep the bas the same and multiply the exponents Example: ( x 3 ) 5 x 35 x15 Multiplying powers with same exponent: Multiply the base and keep exponent the same Example: (3 x )( 4 x ) (3 4) x 12 x Dividing powers with the same exponent : Divide the base and keep the exponent the same Example: 10 x 2x 5x Parallelograms and Rectangles Four sided figure, opposite sides parallel Area = base/width • height Perimeter (P) is the sum of the lengths of the sides Triangles Area = 1 1 • base • height = bh 2 2 Sum of the three angles equal 180º The Basics Similar Triangles Two triangles that have the exact same shape. Corresponding angles are equal between the triangles. Corresponding sides are proportional between the triangles. Right Triangles One angle is 90º The sides of the triangle satisfy the Pythagorean Theorem, a2 + b2 = c2 , where c is the longest side of the triangle Common Pythagorean Theorem values: (3,4,5) and (5,12,13) Isosceles Triangles One side is the base, the other two sides have equal length The two angles at the base are equal Equilateral Triangles Each side has the same length. Each angle is 60º PCAT Review July 9-10, 2011 The Basics Circles r is the radius of a circle d is the circumference Area = r 2 ( 314 . ) Circumference = 2r or d Slice of a Circle Slice of a circle, radius r and angle x Area of a slice = x ( r 2 ) 360 Length of an arc = x (2 r ) 360 Uniform Solid Figure Each cross-section is identical Volume = Ah = area of the base • height Examples: PCAT Review July 9-10, 2011 The Basics Suppose the cube has side s = 2. 1. What is the total surface area and volume? 2. What are the lengths of a and d? Solution to #1: The area of one side is s 2 2 2 4 The total surface area is made up of 6 sides: SA = 6 s 2 6 4 24 The volume of the cube is s 3 2 3 8 Solution to #2: From the two right triangles in the cube, we have 2 2 2 2 a 2 8 So a 8 2 2 We also have a 2 2 2 d 2 8 4 12 So d 12 2 3 Spheres Volume = 4 r3 3 Surface Area = 4 r 2 Cones PCAT Review July 9-10, 2011 The Basics Volume = 1 r 2h 3 Total Surface Area = ( r 2 ) + ( rs ) Due to right triangle and Pythagorean Theorem, s 2 r 2 h 2 Pyramids Area of the base = A A depends on the shape Volume = 1 Ah 3 PCAT Review July 9-10, 2011