Wonderlic Mathematics Complete Review

Most of the math problems on the Wonderlic are in the form of word problems.  With word problems, the difficulty is rarely in performing the calculations themselves.  Instead, the difficulty is in determining what calculations need to be performed, which typically involves translating the word problem into an equation or set of equations.  The following sections will go over the computation skills and concepts you'll need for the test, but the more important skill is speed in interpreting the word problems.  For that, there's nothing that will substitute for lots of practice.

Numbers and their Classifications
Numbers are the basic building blocks of mathematics.  Specific features of numbers are identified by the following terms:
Integers – The set of whole positive and negative numbers, including zero. 

Integers do not include fractions , decimals (0.56), or mixed numbers .
Prime number – A whole number greater than 1 that has only two factors, itself and 1; that is, a number that can be divided evenly only by 1 and itself.
Composite number – A whole number greater than 1 that has more than two different factors; in other words, any whole number that is not a prime number.  For example:  The composite number 8 has the factors of 1, 2, 4, and 8.
Even number – Any integer that can be divided by 2 without leaving a remainder.  For example:  2, 4, 6, 8, and so on.
Odd number – Any integer that cannot be divided evenly by 2.  For example:  3, 5, 7, 9, and so on.
Decimal number – a number that uses a decimal point to show the part of the number that is less than one. Example: 1.234.
Decimal point – a symbol used to separate the ones place from the tenths place in decimals or dollars from cents in currency.
Decimal place – the position of a number to the right of the decimal point.  In the decimal 0.123, the 1 is in the first place to the right of the decimal point, indicating tenths; the 2 is in the second place, indicating hundredths; and the 3 is in the third place, indicating thousandths.
 
The decimal, or base 10, system is a number system that uses ten different digits (). 

An example of a number system that uses something other than ten digits is the binary, or base 2, number system, used by computers, which uses only the numbers 0 and 1.  It is thought that the decimal system originated because people had only their 10 fingers for counting.
 
Rational, irrational, and real numbers can be described as follows:
Rational numbers include all integers, decimals, and fractions. 

Any terminating or repeating decimal number is a rational number.
Irrational numbers cannot be written as fractions or decimals because the number of decimal places is infinite and there is no recurring pattern of digits within the number. 

For example, pi (π) begins with 3.141592 and continues without terminating or repeating, so pi is an irrational number.
Real numbers are the set of all rational and irrational numbers.

Operations
There are four basic mathematical operations:

Addition increases the value of one quantity by the value of another quantity. Example:
. 

The result is called the sum. 

With addition, the order does not matter.
.

Subtraction is the opposite operation to addition; it decreases the value of one quantity by the value of another quantity. Example:
. 

The result is called the difference. 

Note that with subtraction, the order does matter. 
.
 
Multiplication can be thought of as repeated addition. 

ne number tells how many times to add the other number to itself.
Example:
. 

With multiplication, the order does not matter. 
 or
.

Division is the opposite operation to multiplication; one number tells us how many parts to divide the other number into.
Example:
; if 20 is split into 4 equal parts, each part is 5. 

With division, the order of the numbers does matter.
.

An exponent is a superscript number placed next to another number at the top right.  It indicates how many times the base number is to be multiplied by itself.  Exponents provide a shorthand way to write what would be a longer mathematical expression.
Example:
4 is read as '8 to the 4th power,' or '8 raised to the power of 4.' 

A negative exponent is the same as the reciprocal of a positive exponent.

Example:
.

Parentheses are used to designate which operations should be done first when there are multiple operations.
Example: 4 – (2 + 1) = 1; the parentheses tell us that we must add 2 and 1, and then subtract the sum from 4, rather than subtracting 2 from 4 and then adding 1 (this would give us an answer of 3).
 
Order of Operations is a set of rules that dictates the order in which we must perform each operation in an expression so that we will evaluate at accurately.  If we have an expression that includes multiple different operations, Order of Operations tells us which operations to do first.

 The most common mnemonic for Order of Operations is PEMDAS, or "Please Excuse My Dear Aunt Sally." 

PEMDAS stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. 

It is important to understand that multiplication and division have equal precedence, as do addition and subtraction, so those pairs of operations are simply worked from left to right in order.

Example: Evaluate the expression
 using the correct order of operations.
P:  Perform the operations inside the parentheses,
.
E:  Simplify the exponents,
.
The equation now looks like this:
.
MD:  Perform multiplication and division from left to right,
; then
.
The equation now looks like this:
.
AS:  Perform addition and subtraction from left to right,
; then
.

The laws of exponents are as follows:
1)      Any number to the power of 1 is equal to itself:  .
2)      The number 1 raised to any power is equal to 1: .
3)      Any number raised to the power of 0 is equal to 1: .
4)      Add exponents to multiply powers of the same base number: .
5)      Subtract exponents to divide powers of the same number; that is .
6)      Multiply exponents to raise a power to a power: .
7)      If multiplied or divided numbers inside parentheses are collectively raised to a power, this is the same as each individual term being raised to that power:
;
.
Note: Exponents do not have to be integers. 

Fractional or decimal exponents follow all the rules above as well.
Example:
.
 
A root, such as a square root, is another way of writing a fractional exponent.  Instead of using a superscript, roots use the radical symbol (th root of a.' 

The relationship between radical notation and exponent notation can be described by this equation:
. 

The two special cases of n = 2 and n = 3 are called square roots and cube roots.  If there is no number to the upper left, it is understood to be a square root (). 

Nearly all of the roots you encounter will be square roots. 

A square root is the same as a number raised to the one-half power. 

When we say that a is the square root of b (), we mean that a multiplied by itself equals b: ().
 
A perfect square is a number that has an integer for its square root. 

There are 10 perfect squares from 1 to 100: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 (the squares of integers 1 through 10).

Scientific notation is a way of writing large numbers in a shorter form. 

The form  is used in scientific notation, where a is greater than or equal to 1, but less than 10, and n is the number of places the decimal must move to get from the original number to a.
Example: The number 230,400,000 is cumbersome to write.

To write the value in scientific notation, place a decimal point between the first and second numbers, and include all digits through the last non-zero digit ().
To find the appropriate power of 10, count the number of places the decimal point had to move (). 

The number is positive if the decimal moved to the left, and negative if it moved to the right.  We can then write 230,400,000 as
. 

If we look instead at the number 0.00002304, we have the same value for a, but this time the decimal moved 5 places to the right (
). 

Thus, 0.00002304 can be written as
. 

Using this notation makes it simple to compare very large or very small numbers. 

By comparing exponents, it is easy to see that
 is smaller than
, because 4 is less than 5.

Positive and Negative Numbers
A precursor to working with negative numbers is understanding what absolute values are. A number's Absolute Value is simply the distance away from zero a number is on the number line. The absolute value of a number is always positive and is written
.
 
When adding signed numbers, if the signs are the same simply add the absolute values of the addends and apply the original sign to the sum.

For example,
 and
.

When the original signs are different, take the absolute values of the addends and subtract the smaller value from the larger value, then apply the original sign of the larger value to the difference.

For instance,
 and
.
 
For subtracting signed numbers, change the sign of the number after the minus symbol and then follow the same rules used for addition.

For example,
.
 
If the signs are the same the product is positive when multiplying signed numbers.

For example,
 and
.

If the signs are opposite, the product is negative.

For example,
 and
.

When more than two factors are multiplied together, the sign of the product is determined by how many negative factors are present. If there are an odd number of negative factors then the product is negative, whereas an even number of negative factors indicates a positive product. For instance,
 and
.
 
The rules for dividing signed numbers are similar to multiplying signed numbers.

If the dividend and divisor have the same sign, the quotient is positive. If the dividend and divisor have opposite signs, the quotient is negative.

For example,
.

Factors and Multiples
Factors are numbers that are multiplied together to obtain a product.  For example, in the equation
, the numbers 2 and 3 are factors. 

A prime number has only two factors (1 and itself), but other numbers can have many factors. 
A common factor is a number that divides exactly into two or more other numbers. 

For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, while the factors of 15 are 1, 3, 5, and 15.  The common factors of 12 and 15 are 1 and 3.
A prime factor is also a prime number. 

Therefore, the prime factors of 12 are 2 and 3.  For 15, the prime factors are 3 and 5.

The greatest common factor (GCF) is the largest number that is a factor of two or more numbers. 

For example, the factors of 15 are 1, 3, 5, and 15; the factors of 35 are 1, 5, 7, and 35. 

Therefore, the greatest common factor of 15 and 35 is 5.

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. 

For example, the multiples of 3 include 3, 6, 9, 12, 15, etc.; the multiples of 5 include 5, 10, 15, 20, etc.  Therefore, the least common multiple of 3 and 5 is 15.

Fractions, Percentages, and Related Concepts
A fraction is a number that is expressed as one integer written above another integer, with a dividing line between them (
).  It represents the quotient of the two numbers 'x divided by y.'  It can also be thought of as x out of y equal parts.
 
The top number of a fraction is called the numerator, and it represents the number of parts under consideration. 

The 1 in
 means that 1 part out of the whole is being considered in the calculation. 

The bottom number of a fraction is called the denominator, and it represents the total number of equal parts. 

The 4 in
 means that the whole consists of 4 equal parts. 

A fraction cannot have a denominator of zero; this is referred to as 'undefined.'

Fractions can be manipulated, without changing the value of the fraction, by multiplying or dividing (but not adding or subtracting) both the numerator and denominator by the same number.  If you divide both numbers by a common factor, you are reducing or simplifying the fraction.  Two fractions that have the same value, but are expressed differently are known as equivalent fractions. 

For example,
 are all equivalent fractions. 

They can also all be reduced or simplified to
.
 
When two fractions are manipulated so that they have the same denominator, this is known as finding a common denominator.  The number chosen to be that common denominator should be the least common multiple of the two original denominators.

Example:
 the least common multiple of 4 and 6 is 12. 

Manipulating to achieve the common denominator:
.
 
If two fractions have a common denominator, they can be added or subtracted simply by adding or subtracting the two numerators and retaining the same denominator.
Example:
.  If the two fractions do not already have the same denominator, one or both of them must be manipulated to achieve a common denominator before they can be added or subtracted.

Two fractions can be divided flipping the numerator and denominator of the second fraction and then proceeding as though it were a multiplication.
Example:
.

A fraction whose denominator is greater than its numerator is known as a proper fraction, while a fraction whose numerator is greater than its denominator is known as an improper fraction.  Proper fractions have values less than one and improper fractions have values greater than one.
 
A mixed number is a number that contains both an integer and a fraction. 

Any improper fraction can be rewritten as a mixed number.
Example:
. 

Similarly, any mixed number can be rewritten as an improper fraction.
Example:
.
 
Percentages can be thought of as fractions that are based on a whole of 100; that is, one whole is equal to 100%.  The word percent means "per hundred."

Fractions can be expressed as percents by finding equivalent fractions with a denomination of 100.
Example:
;
.

To express a percentage as a fraction, divide the percentage number by 100 and reduce the fraction to its simplest possible terms.
Example:
;
.
 
Converting decimals to percentages and percentages to decimals is as simple as moving the decimal point.
To convert from a decimal to a percent, move the decimal point two places to the right.
To convert from a percent to a decimal, move it two places to the left.
Example: 0.23 = 23%; 5.34 = 534%; 0.007 = 0.7%; 700% = 7.00; 86% = 0.86; 0.15% = 0.0015.

It may be helpful to remember that the percentage number will always be larger than the equivalent decimal number.
 
A percentage problem can be presented three main ways:  ·    

1. Find what percentage of some number another number is.
Example: What percentage of 40 is 8?·   

2. Find what number is some percentage of a given number.

Example: What number is 20% of 40?·   

3. Find what number another number is a given percentage of. 

Example: What number is 8 20% of?
 
The three components in all of these cases are the same: a whole (W), a part (P), and a percentage (%).  These are related by the equation:
.  This is the form of the equation you would use to solve problems of type (2).
To solve types (1) and (3), you would use these two forms:
 and
.
 
The thing that frequently makes percentage problems difficult is that they are most often also word problems, so a large part of solving them is figuring out which quantities are what. 

Example:  In a school cafeteria, 7 students choose pizza, 9 choose hamburgers, and 4 choose tacos.  Find the percentage that chooses tacos.
To find the whole, you must first add all of the parts: 7 + 9 + 4 = 20.  The percentage can then be found by dividing the part by the whole
):
.
 
A ratio is a comparison of two quantities in a particular order.
Example: If there are 14 computers in a lab, and the class has 20 students, there is a student to computer ratio of 20 to 14, commonly written as 20:14.  Ratios are normally reduced to their smallest whole number representation, so 20:14 would be reduced to 10:7 by dividing both sides by 2.

A proportion is a relationship between two quantities that dictates how one changes when the other changes.  A direct proportion describes a relationship in which a quantity increases by a set amount for every increase in the other quantity, or decreases by that same amount for every decrease in the other quantity.
Example: Assuming a constant driving speed, the time required for a car trip increases as the distance of the trip increases.  The distance to be traveled and the time required to travel are directly proportional.

Inverse proportion is a relationship in which an increase in one quantity is accompanied by a decrease in the other, or vice versa.
Example: the time required for a car trip decreases as the speed increases, and increases as the speed decreases, so the time required is inversely proportional to the speed of the car.

Systems of Equations
Systems of Equations are a set of simultaneous equations that all use the same variables. A solution to a system of equations must be true for each equation in the system. Consistent Systems are those with at least one solution. Inconsistent Systems are systems of equations that have no solution.
 
To solve a system of linear equations by substitution, start with the easier equation and solve for one of the variables. Express this variable in terms of the other variable. Substitute this expression in the other equation, and solve for the other variable. The solution should be expressed in the form (x, y). Substitute the values into both of the original equations to check your answer.

Consider the following problem.

Solve the system using substitution:

 Solve the first equation for x:
 

Substitute this value in place of x in the second equation, and solve for y:



 

Plug this value for y back into the first equation to solve for x:
 Check both equations if you have time:

Therefore, the solution is (9.6, 0.9). 

To solve a system of equations using elimination, begin by rewriting both equations in standard form
.

Check to see if the coefficients of one pair of like variables add to zero. If not, multiply one or both of the equations by a non-zero number to make one set of like variables add to zero.

Add the two equations to solve for one of the variables. Substitute this value into one of the original equations to solve for the other variable. Check your work by substituting into the other equation.

Next we will solve the same problem as above, but using the addition method.

Solve the system using elimination:



If we multiply the first equation by 2, we can eliminate the y terms:

 

Add the equations together and solve for x:

 

Plug the value for x back into either of the original equations and solve for y:

 

Check both equations if you have time:

Therefore, the solution is (9.6, 0.9).

Polynomial Algebra
To multiply two binomials, follow the FOIL method. FOIL stands for:·    
First: Multiply the first term of each binomial·    
Outer: Multiply the outer terms of each binomial·    
Inner: Multiply the inner terms of each binomial·    
Last: Multiply the last term of each binomial
 
Using FOIL,
.

 
Sample Math Question:
Three coins are tossed up in the air.  What is the probability that two of them will land heads and one will land tails?
A. 0 B. 1/8 C. 1/4 D. 3/8

Let's look at a few different methods and steps to solving this problem.


1. Reduction and Division
Quickly eliminate the probabilities that you immediately know.  You know to roll all heads is a 1/8 probability, and to roll all tails is a 1/8 probability.  Since there are in total 8/8 probabilities, you can subtract those two out, leaving you with 8/8 – 1/8 – 1/8 = 6/8.  So after eliminating the possibilities of getting all heads or all tails, you're left with 6/8 probability.  Because there are only three coins, all other combinations are going to involve one of either head or tail, and two of the other.  All other combinations will either be 2 heads and 1 tail, or 2 tails and 1 head.  Those remaining combinations both have the same chance of occurring, meaning that you can just cut the remaining 6/8 probability in half, leaving you with a 3/8ths chance that there will be 2 heads and 1 tail, and another 3/8ths chance that there will be 2 tails and 1 head, making choice D correct.

2. Run Through the Possibilities for that Outcome
You know that you have to have two heads and one tail for the three coins.  There are only so many combinations, so quickly run through them all.
You could have:
H, H, H
H, H, T
H, T, H
T, H, H
T, T, H
T, H, T
H, T, T
T, T, T
Reviewing these choices, you can see that three of the eight have two heads and one tail, making choice D correct.

3. Fill in the Blanks with Symbology and Odds
Many probability problems can be solved by drawing blanks on a piece of scratch paper (or making mental notes) for each object used in the problem, then filling in probabilities and multiplying them out.  In this case, since there are three coins being flipped, draw three blanks.  In the first blank, put an 'H' and over it write '1/2'.  This represents the case where the first coin is flipped as heads.  In that case (where the first coin comes up heads), one of the other two coins must come up tails and one must come up heads to fulfill the criteria posed in the problem (2 heads and 1 tail).  In the second blank, put a '1' or '1/1'.  This is because it doesn't matter what is flipped for the second coin, so long as the first coin is heads.  In the third blank, put a '1/2'.  This is because the third coin must be the exact opposite of whatever is in the second blank.  Half the time the third coin will be the same as the second coin, and half the time the third coin will be the opposite, hence the '1/2'.  Now multiply out the odds.  There is a half chance that the first coin will come up 'heads', then it doesn't matter for the second coin, then there is a half chance that the third coin will be the opposite of the second coin, which will give the desired result of 2 heads and 1 tail.  So, that gives 1/2*1/1*1/2 = 1/4.

But, now you must calculate the probabilities that result if the first coin is flipped tails. 

So draw another group of three blanks.  In the first blank, put a 'T' and over it write '1/2'.  This represents the case where the first coin is flipped as tails.  In that case (where the first coin comes up tails), both of the other two coins must come up heads to fulfill the criteria posed in the problem.  In the second blank, put an 'H' and over it write '1/2'.  In the third blank, put an 'H' and over it write '1/2'.  Now multiply out the odds.  There is a half chance that the first coin will come up 'tails', then there is a half chance that the second coin will be heads, and a half chance that the third coin will be heads.  So, that gives 1/2*1/2*1/2 = 1/8.
Now, add those two probabilities together.  If you flip heads with the first coin, there is a 1/4 chance of ultimately meeting the problem's criteria.  If you flip tails with the first coin, there is a 1/8 chance of ultimately meeting the problem's criteria.  So, that gives 1/4 + 1/8 = 2/8 + 1/8 = 3/8, which makes choice D correct.