Unit 1 Whole Numbers and Fractions MATH-0910 Review Concepts (Haugen) Exam 1 Sections 1.5, 1.6, 1.7, 1.8, 2.1, 2.2, 2.3, 2.4, and 2.5 Dividing Whole Numbers Equivalent ways of expressing division: a b, b a, a b, or a b Components of a division problem: Dividend, divisor, quotient, and remainder Division involving 0 or 1 Exponents and Exponential Expressions Expanded form vs exponential form Evaluating an exponential expression Evaluate = find the value of Order of Operations: P.E.M.D.A.S. P = Parentheses, brackets [ ], set braces { }, or a fraction bar Multiply or Divide from left to right as the operators occur in the expression Example: 40 5 4 = 8 4 = 32, not 40 20 = 2 Add or Subtract from left to right Example: 15 6 + 3 = 9 + 3 = 12, not 15 9 = 6 Rounding Whole Numbers Using the Principle of Estimation to approximate sums, differences, products, and/or quotients Each rounded number in the approximation should have only one nonzero digit Fractions Numerator vs denominator Proper vs improper fractions Reducing fractions Factor trees help us identify common factors Divide out common factors Mixed Numbers Mixed numbers have a whole number component and a (proper) fractional component Convert from an improper fraction to a mixed number and vice versa Multiplying Fractions and/or Mixed Numbers Cancel common factors (if possible) and then multiply straight across Convert mixed numbers to improper fractions before multiplying (or dividing) Dividing Fractions and/or Mixed Numbers Convert division to multiplication using the reciprocal of the divisor: a c a d = b d b c
Unit 2 Adding and Subtracting Fractions Exam 2 Sections 2.6, 2.7, 2.8, and 2.9 Least Common Denominator (LCD) Goal is to identify the smallest number that is a multiple of both denominators Creating Equivalent Fractions Adding and Subtracting Fractions Case 1: Like Denominators Case 2: Unlike Denominators Adding and Subtracting Mixed Numbers* Case 1: Like Denominators Case 2: Unlike Denominators *Sometimes subtracting mixed numbers involves borrowing. This can be avoided by converting the mixed numbers to improper fractions and then subtracting. Unit 3 Decimals Exam 3 Sections 3.1, 3.2, 3.3, 3.4, 3.5, 3.6, and 3.7 Writing a Word Name for a Decimal Using the Place Value System Comparing Two or More Decimals Rounding Decimals Identify the round off place and then apply rounding rules Adding and Subtracting Decimals Helps to add or subtract vertically, keeping the decimal points aligned Multiplying and Dividing Decimals Writing the Decimal Equivalent of a Fraction Should be familiar with three scenarios: 1. Terminating decimal (zero remainder) 2. Non-terminating repeating decimals (we use a bar over the digit or digits that repeat) 3. Non-terminating non-repeating decimals (when this happens, we usually round to a specified place)
Unit 4 Rates, Ratios, and Proportions Exam 4 Sections 4.1, 4.2, 4.3, and 4.4 Ratios and Rates Ratios and rates are comparisons of two quantities If the quantities have the same units, the comparison is called a ratio If the quantities have different units, the comparison is called a rate Three Ways of Expressing Ratios: a to b, a : b, and a b Remember when writing as a fraction to reduce to simplest form Writing a Rate in Simplest Form Unit Rates A unit rate is a rate whose denominator is 1 180 miles 60 miles Ex. or 60 miles per hour 3 hours 1 hour Proportions A proportion is a statement that two rates or two ratios are equal Use cross products to determine if an equation is a proportion Solve Equations of the form a n = b Divide both sides of the equation by a (the coefficient of the variable n) Solve Proportions Use cross products to convert the proportion to an equation of the form a n = b Unit 5 Percents Working with Percents Converting percents to decimal numbers Drop the percent symbol and then move the decimal point two places to the left Converting decimal numbers to percents Move the decimal point two places to the right and then write the percent symbol at the end of the number Converting percents to fractions Two options: 1. Use the definition of percent 2. Convert the percent to a decimal and then convert the decimal to a fraction Converting fractions to percents Convert the fraction to a decimal and then convert the decimal to a percent
Unit 5 Percents (continued) Solve Percent Problems Two options: 1. Use the equation: amount = percent x base 2. Use the percent proportion: Solve Applied Percent Problems amount base = percent number 100 Sales tax calculations Sales Tax = Sales Tax Rate x List Price Discount problems Discount Amount = Discount Rate x List Price Commission problems Commission = Commission Rate x Value of Sales Percent of increase/decrease problems Percent of Increase = Amount of Increase / Original Amount Percent of Decrease = Amount of Decrease / Original Amount Simple interest calculations Interest = Principal x Rate x Time or I = P x R x T * *note: the units of time on R and T must match in these problems Unit 6 Signed Numbers Exam 6 Sections 9.1, 9.2, 9.3, and 9.4 Adding Signed Numbers Geometric approach: draw vectors on a number line Vectors representing positive numbers should point toward the right Vectors representing negative numbers should point toward the left Alternative approach is based on absolute values: If the addends have the same sign, add the absolute values of the addends. The sign of the sum will be the same as the sign of the addends. If the addends have opposite signs, find the absolute value of the addends. Subtract the smaller absolute value from the larger absolute value. The sign of the sum will be the same as the sign of the addend with the larger absolute value. Subtracting Signed Numbers In the expression a b, a is called the minuend and b is called the subtrahend Subtraction can be converted to addition using the opposite of the subtrahend: a b = a + ( b)
Unit 6 Signed Numbers (continued) Multiplying and Dividing Signed Numbers The product (or quotient) of two numbers with the same sign will always be positive ( + )( + ) = + ( )( ) = + ( + ) + = + ( ) = + The product (or quotient) of two numbers with opposite signs will always be negative ( + ) ( ) = ( )( + ) = ( ) + = ( + ) =