MATH-0910 Review Concepts (Haugen)
Unit 1 Whole Numbers and Fractions
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
, or a b
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
a
Three Ways of Expressing Ratios: a to b , a : b , and
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:
amount percent number
=
base
100
Solve Applied Percent Problems
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
(+)(–) = –
( – )( + ) = –
(+) = (-)
(-) = (+)