Algebra and Functions
20%of exam
Financial Mathematics
20%of exam
Data Analysis and Statistics
15%of exam
Logic and Sets
15%of exam
Counting and Probability
10%of exam
Geometry
10%of exam
Triangles and QuadrilateralsCircle PropertiesSimilarity and PythagorasParallel and Perpendicular Lines
Numbers
10%of exam
Quick Facts
- Exam
- CLEP College Mathematics
- Owner
- College Board
- Questions
- About 60, some unscored
- Time
- 90 minutes
- Score scale
- 20-80 scaled
- ACE credit score
- 50 grants 3 semester hours
- Question mix
- 50% routine, 50% nonroutine
- Calculator
- On-screen TI-30XS MultiView
- Calculus
- None on this exam
- Wrong answers
- No penalty; never blank
- Retest wait
- 3 months, same title
- Fee
- $97 plus center fee
Order of Operations
PEMDAS: Parentheses, Exponents, Multiply/Divide, Add/Subtract
P: parentheses firstE: exponents nextMD: left to rightAS: left to right
Linear vs Exponential Growth
Linear
- Constant amount added
- Equal differences
- y=mx+b
Exponential
- Constant percent multiplied
- Equal ratios
- y=a(1+r)^t
Differences linear, ratios exponential
Model and Graph Shift
- Same amount added each step→Linear: y=mx+b
- Same percent each step→Exponential: y=a(1+r)^t
- Value falls by percent→Decay: y=a(1-r)^t
- Graph moves up→Add outside: f(x)+k
- Graph moves right→Subtract inside: f(x-h)
- Graph flips vertically→Negate outside: -f(x)
Linear Equations and Slope
- Slope formula
- (y2-y1)/(x2-x1)
- Slope-intercept form
- y=mx+b
- Point-slope form
- y-y1=m(x-x1)
- Standard form
- Ax+By=C
- Parallel lines
- Equal slopes
- Perpendicular lines
- Slopes multiply to -1
- Horizontal line
- Slope 0, form y=k
- Vertical line
- Undefined slope, form x=k
- System by substitution
- Solve one, substitute back
- System by elimination
- Add or subtract equations
- No solution system
- Same slope, different intercept
- Infinite solutions
- Identical equations
- Inequality sign flip
- Multiply or divide negative
Functions and Notation
- f(x) notation
- Input x, output f(x)
- Domain
- Allowed input values
- Range
- Resulting output values
- f(3) when f(x)=2x-1
- Equals 5
- Function types tested
- Linear, polynomial, radical, exponentialOfficial
- Also tested
- Logarithmic and piecewiseOfficial
- Zero of a function
- x where f(x)=0
- Composite f(g(x))
- Evaluate the inner first
- Vertical line test
- One output per input
- Piecewise function
- Rule depends on interval
Graph Transformations
- f(x)+k
- Shift up k
- f(x)-k
- Shift down k
- f(x-h)
- Shift right h
- f(x+h)
- Shift left h
- -f(x)
- Reflect across x-axis
- f(-x)
- Reflect across y-axis
- y-axis symmetry
- Even function, f(-x)=f(x)
- Origin symmetry
- Odd function, f(-x)=-f(x)
- x-axis symmetry test
- Replace y with -y
- a f(x) with a>1
- Vertical stretch
- a f(x) with 0<a<1
- Vertical shrink
Linear vs Exponential Growth
- Linear growth
- Constant amount added
- Exponential growth
- Constant percent multiplied
- Linear model
- y=mx+b
- Exponential model
- y=a(1+r)^t
- Exponential decay
- y=a(1-r)^t
- Equal differences
- Signals linear data
- Equal ratios
- Signals exponential data
- 100 at 5%, 3 years
- Grows to 115.7625
- Long run winner
- Exponential passes any linear
Simple Interest I=Prt
Interest equals principal times rate times time
P: starting principalr: annual decimal ratet: time in yearsI: interest only, not total
Simple vs Compound Interest
Simple interest
- I=Prt on principal only
- Same interest each year
- Straight-line growth
Compound interest
- Interest earns interest
- A=P(1+r/n)^nt
- Curved, accelerating growth
Compound grows faster over time
Pick the Interest Formula
- Interest on principal only→Simple: I=Prt(No compounding)
- Compounded n times yearly→A=P(1+r/n)^nt(n is periods)
- Compounded once a year→A=P(1+r)^t
- Compounded continuously→A=Pe^(rt)(Largest amount)
- Need value today→PV=FV/(1+r)^t(Discounting)
- Comparing two stated rates→Effective annual yield
- Percent off, then off again→Multiply the factors
Percent, Markup, Discount
- Percent of a number
- part=percent x whole
- Percent change
- (new-old)/old x 100
- Increase 40 to 50
- Up 25%
- Decrease 50 to 40
- Down 20%
- Markup on cost
- Cost x (1+rate)
- Discounted price
- Price x (1-rate)
- Sales tax total
- Price x (1+tax rate)
- 20% off then 10% off
- Total discount 28%
- Successive percents
- Multiply factors, never add
- Reverse a percent
- Divide by the factor
- Profit
- Revenue minus cost
- Loss
- Cost exceeds revenue
Percent Change Base
Change divided by the original, then times 100
Original is the baseIncrease: new above oldDecrease: new below oldNever divide by new
APR vs Effective Annual Yield
APR
- Stated nominal rate
- Ignores compounding frequency
Effective yield
- Actual yearly percent growth
- (1+r/n)^n - 1
- 6% monthly gives 6.1678%
Compare loans by effective yield
Interest Formulas
- Simple interest
- I=Prt
- Simple total amount
- A=P(1+rt)
- Compound amount
- A=P(1+r/n)^nt
- Annual compounding
- A=P(1+r)^t
- Continuous interest
- A=Pe^(rt)
- n for monthly
- n=12
- n for quarterly
- n=4
- n for daily
- n=365
- 1000 at 6%, 1 year simple
- Interest is 60
- Same, compounded monthly
- Amount 1061.68
- Same, compounded continuously
- Amount 1061.84
- More frequent compounding
- Higher amount, continuous caps
Present and Future Value
- Future value
- FV=PV(1+r)^t
- Present value
- PV=FV/(1+r)^t
- Discounting
- Moving money backward in time
- Compounding
- Moving money forward in time
- PV of 1000, 2 years, 5%
- About 907.03
- FV of 500, 4 years, 3%
- About 562.75
- Higher rate
- Lower present value
- Longer time
- Lower present value
- Rule of 72
- Doubling years about 72/rateEstimate
- Amortized loan
- Early payments mostly interest
APR vs Effective Yield
- Nominal rate
- Stated annual rate
- APR
- Annual percentage rate stated
- Effective annual yield
- Actual yearly percent growth
- Effective yield formula
- (1+r/n)^n - 1
- 6% compounded monthly
- Yield 6.1678%
- 6% compounded quarterly
- Yield 6.1364%
- 6% continuous
- Yield 6.1837%
- More frequent compounding
- Higher effective yield
- Annual compounding
- APR equals effective yield
- Comparing two loans
- Use the effective yield
Empirical Rule 68-95-99.7
68, 95, 99.7 percent within 1, 2, 3
1 deviation: about 68%2 deviations: about 95%3 deviations: about 99.7%Normal curve only
Mean vs Median
Mean
- Sum divided by count
- Pulled toward outliers
Median
- Middle of ordered data
- Resists outliers
Skewed data favors the median
Pick the Statistic or Graph
- Symmetric data, no outliers→Mean
- Skewed data or outliers→Median
- Most common category→Mode
- Simplest measure of spread→Range
- Typical distance from mean→Standard deviation(Concept only)
- Relating two variables→Scatterplot
- Parts of one whole→Circle graph
- Counts in numeric bins→Histogram
Numerical Summaries
- Mean
- Sum divided by count
- Median
- Middle of ordered list
- Even count median
- Average two middle values
- Mode
- Most frequent value
- Range
- Maximum minus minimum
- Mean of 2,4,4,6,9
- Equals 5
- Median of that set
- Equals 4
- Mode of that set
- Equals 4
- Range of that set
- Equals 7
- Outlier effect
- Mean moves, median resists
- Skewed right
- Mean above median
- Skewed left
- Mean below median
Data Displays
- Bar graph
- Compares category counts
- Histogram
- Counts within numeric intervals
- Line graph
- Change over time
- Circle graph
- Parts of one whole
- Pie chart
- Same as circle graph
- Scatterplot
- Two variables per point
- Positive association
- Both rise together
- Negative association
- One rises, other falls
- Circle graph angle
- Percent x 360 degrees
- 25% sector
- 90 degree angle
Standard Deviation and Normal
- Standard deviation
- Typical distance from mean
- Larger deviation
- Data more spread out
- Deviation of 0
- All values identical
- Normal curve
- Symmetric and bell shaped
- Normal center
- Mean, median, mode equal
- 68-95-99.7 rule
- Within 1, 2, 3 deviations
- Mean 100, deviation 15
- 68% lie 85 to 115
- Same, two deviations
- 95% lie 70 to 130
- Tested depth
- Conceptual questions onlyOfficial
- Hand computation
- Deviation formula not required
Swap and Negate
Swap and negate: only contrapositive stays true
Converse: swap onlyInverse: negate onlyContrapositive: swap and negateContrapositive is always equivalent
Converse vs Contrapositive
Converse
- Swap p and q
- Not logically equivalent
Contrapositive
- Swap and negate both
- Always logically equivalent
Only contrapositive preserves truth
Which Statement Form
- Swap p and q→Converse(Not equivalent)
- Negate both parts→Inverse(Not equivalent)
- Swap and negate both→Contrapositive(Always equivalent)
- Disproving a general claim→Counterexample
- Elements in both sets→Intersection
- Elements in either set→Union
- Everything outside the set→Complement
Logic Connectives
- Conjunction
- AND: true only both
- Disjunction
- OR: true if either
- Negation
- NOT: reverses truth value
- Conditional
- IF p THEN q
- Biconditional
- p if and only if q
- Conditional is false
- Only p true, q false
- Disjunction is false
- Both parts false
- Conjunction is true
- Both parts true
- Truth table rows
- 2^n for n variables
- Three variables
- 8 rows
- Tautology
- True in every row
- Contradiction
- False in every row
Union vs Intersection
Union
- In A or B
- Add sizes, subtract overlap
Intersection
- In A and B
- Overlap region only
Or widens, and narrows
Conditional Statement Forms
- Conditional
- If p, then q
- Converse
- If q, then p
- Inverse
- If not p, not q
- Contrapositive
- If not q, not p
- Equivalent pair
- Conditional and contrapositive
- Other equivalent pair
- Converse and inverse
- Converse truth
- Not guaranteed by original
- Counterexample
- One case disproving the claim
- Negate all are
- Some are not
- Negate some are
- None are
- Negate if p then q
- p true and q false
- Logical equivalence
- Same truth table column
Set Operations and Relations
- Union
- In A or B
- Intersection
- In A and B
- Complement
- Everything outside the set
- Subset
- Every element also inside
- Proper subset
- Subset but not equal
- Disjoint sets
- Empty intersection
- Equal sets
- Same exact elements
- Empty set
- No elements, size 0
- Subsets of n elements
- 2^n subsets
- Set of 3 elements
- 8 subsets
- Size of a union
- Add sizes, subtract overlap
- Venn overlap region
- Elements in both sets
Order Means Permutation
Order matters: permutation. Order ignored: combination.
Ranking uses permutationsCommittees use combinationsnPr is largernCr=nPr/r!
Permutation vs Combination
Permutation
- Order matters
- n!/(n-r)!
- 5P2 equals 20
Combination
- Order ignored
- n!/(r!(n-r)!)
- 5C2 equals 10
Ranking permutes, choosing combines
Order Matters or Not
- Order matters, no repeats→Permutation nPr
- Order does not matter→Combination nCr
- Repeats allowed each stage→Multiplication rule(n^k outcomes)
- Choosing a committee→Combination
- Assigning ranked places→Permutation
- Both events must happen→Multiply probabilities(If independent)
- Either event may happen→Add, subtract overlap
- At least one happens→1 minus complement
Counting Rules
- Multiplication rule
- Multiply choices per stage
- Permutation
- Order matters
- Combination
- Order does not matter
- nPr formula
- n!/(n-r)!
- nCr formula
- n!/(r!(n-r)!)
- 5P2
- Equals 20
- 5C2
- Equals 10
- Value of 0!
- Equals 1
- 4 shirts, 3 pants
- 12 outfits
- Choosing a committee
- Use combinations
- Ranked finish order
- Use permutations
- nCr symmetry
- nCr equals nC(n-r)
Independent vs Mutually Exclusive
Independent
- One does not affect other
- Multiply the probabilities
Mutually exclusive
- Cannot both happen
- Add the probabilities
Independent multiplies, exclusive adds
Probability Rules
- Basic probability
- Favorable over total outcomes
- Probability range
- 0 to 1
- Complement rule
- 1 minus P(A)
- Mutually exclusive
- Cannot happen together
- Addition, exclusive events
- P(A)+P(B)
- Addition, general
- Add, then subtract overlap
- Independent events
- One does not affect other
- Multiplication, independent
- P(A) x P(B)
- Conditional probability
- P(A and B)/P(B)
- Expected value
- Sum of value x probability
- Fair die above 4
- Probability 1/3
- Two coins, both heads
- Probability 1/4
- Win 10 with probability 0.5
- Expected value 5
Common Pythagorean Triples
3-4-5, 5-12-13, 8-15-17 and their multiples
6-8-10 doubles 3-4-59-12-15 triples itLongest side is cCheck a^2+b^2=c^2
Central vs Inscribed Angle
Central angle
- Vertex at the center
- Equals its arc
Inscribed angle
- Vertex on the circle
- Half the same arc
Inscribed is half the central
Pick the Geometry Formula
- Right triangle, two sides→a^2+b^2=c^2
- Area of any triangle→(1/2) x base x height
- Distance around a circle→C=2(pi)r
- Space inside a circle→A=(pi)r^2
- Slice of a circle→(angle/360) x area(Sector)
- Angle drawn on circle→Half the central angle
- Same shape, different size→Similarity ratio(Areas use k^2)
Triangles and Quadrilaterals
- Triangle angle sum
- 180 degrees
- Quadrilateral angle sum
- 360 degrees
- Triangle area
- (1/2) x base x height
- Rectangle area
- length x width
- Parallelogram area
- base x height
- Trapezoid area
- Average bases x height
- Pythagorean theorem
- a^2+b^2=c^2
- Legs 6 and 8
- Hypotenuse 10
- 3-4-5 triangle
- Right triangle, scales up
- Similar triangles
- Equal angles, proportional sides
- Similarity ratio k
- Areas scale by k^2
- Perimeter
- Sum of all sides
- Isosceles triangle
- Two equal sides, angles
Circle Properties
- Circumference
- C=2(pi)r
- Circle area
- A=(pi)r^2
- Diameter
- Twice the radius
- Central angle
- Vertex at the center
- Inscribed angle
- Half its central angle
- Angle in a semicircle
- Right angle, 90 degrees
- Sector area
- (angle/360) x area
- Arc length
- (angle/360) x circumference
- Radius 5 area
- 25(pi)
- Radius 5 circumference
- 10(pi)
- 90 degree sector
- One quarter of circle
- Parallel lines, transversal
- Corresponding angles equal
- Perpendicular lines
- Meet at 90 degrees
Rational vs Irrational
Rational
- Ratio of two integers
- Repeating or terminating decimal
- sqrt(9) equals 3
Irrational
- No integer ratio
- Nonrepeating, nonterminating decimal
- sqrt(2) and pi
Judge by the decimal pattern
Number Sets and Properties
- Natural numbers
- 1, 2, 3, and onward
- Whole numbers
- Naturals plus 0
- Integers
- Whole numbers and negatives
- Rational number
- Ratio of two integers
- Irrational number
- Nonrepeating, nonterminating decimal
- Real numbers
- Rationals plus irrationals
- sqrt(2)
- Irrational
- sqrt(9)
- Rational, equals 3
- 0.333 repeating
- Rational, equals 1/3
- Absolute value
- Distance from zero
- Value of |-7|
- Equals 7
- Commutative property
- Order does not matter
- Distributive property
- a(b+c)=ab+ac
Number Theory and Measurement
- Prime number
- Exactly two distinct factors
- Composite number
- More than two factors
- The number 1
- Neither prime nor composite
- Smallest prime
- 2, the only even
- Fundamental theorem
- Unique prime factorizationOfficial
- Prime factors of 60
- 2^2 x 3 x 5
- GCF
- Largest shared factor
- LCM
- Smallest shared multiple
- GCF of 12 and 18
- Equals 6
- LCM of 12 and 18
- Equals 36
- Divisible by 3
- Digit sum divisible by 3
- Scientific notation
- a x 10^n with 1<=a<10
- 4500 in notation
- 4.5 x 10^3
- 0.0072 in notation
- 7.2 x 10^-3
- Numerical precision
- Significant digits you keep
- Unit conversion
- Multiply by a unit ratio
Common Traps
Percent Change Denominator
Divide by the original value ≠ Not by the new value
Successive Discounts
20% then 10% is 28% ≠ Not a flat 30%
Nominal Is Not Effective
6% monthly yields 6.1678% ≠ APR alone understates growth
Converse Is Not Equivalent
Contrapositive preserves truth ≠ Converse and inverse do not
Mean Is Not Median
Outliers pull the mean ≠ Median holds its place
Standard Deviation Depth
Conceptual questions only ≠ No hand computation required
Geometry Scope Limit
Official list: perimeter and area ≠ Volume is not listed
Order in Counting
5P2 equals 20 ≠ 5C2 equals only 10
Zero Factorial
0! equals 1 ≠ It does not equal 0
Inscribed Angle Halves
Inscribed is half central ≠ The two are not equal
Roots Can Be Rational
sqrt(9) equals 3, rational ≠ sqrt(2) stays irrational
One Is Not Prime
1 is neither prime nor composite ≠ 2 is the smallest prime
Retest Wait Is Real
Three months, same exam title ≠ Early retest voids the score
Never Leave Blanks
No penalty for wrong answers ≠ Blank scores the same as wrong
Last Minute
- 1.About 60 questions, 90 minutes
- 2.Scaled 20-80; ACE credit at 50
- 3.Algebra and Financial Math: 20% each
- 4.Data and Logic: 15% each
- 5.Counting, Geometry, Numbers: 10% each
- 6.Half routine, half nonroutine problems
- 7.On-screen TI-30XS MultiView, no calculus
- 8.Simple interest: I=Prt
- 9.Compound: A=P(1+r)^t
- 10.n periods: divide r, multiply t
- 11.Continuous: A=Pe^(rt)
- 12.Percent change divides by original
- 13.Multiply successive percent factors
- 14.Contrapositive equivalent, converse is not
- 15.Order matters means permutation nPr
- 16.Expected value sums value x probability
- 17.a^2+b^2=c^2 for right triangles
- 18.Inscribed angle halves the central
- 19.Standard deviation is conceptual only
- 20.No penalty for wrong answers
- 21.Retest wait: three months, same title
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