Kaplan Math Source Concepts and Practice Questions

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What are real numbers?
Both rational and irrational numbers.
Rational Numbers
Numbers that can be expressed as a fraction in the form a/b, where a and b are integers and b does not = 0. Rational numbers can be classified into subsets.
Natural Numbers
Also known as the counting numbers. {1,2,3,4,5,6,7...} Positive, whole, without 0. These are rational numbers.
Whole Numbers
Natural numbers including 0. {0,1,2,3,4,5,6,7...} These are rational numbers. All positive except 0, which is neither positive or negative.
Integers
Whole numbers and their opposits. {...-3,-2,-1,0,1,2,3...} These are rational numbers. Negative, positive, include 0 (neither negative or positive)
Rules of integers
All numbers >0 are positive. All numbers <0 are negative.
Zero is neither positive or negative. Even intergers are divisible by 2 and includes 0. Odd integers are not divisible by 2.
Irrational numbers
Numbers that cannot be expressed as fractions such as sqrt 2, sqrt 3, sqrt 5, and pi. When expressed as decimals, they are non-terminating and non-repeating.
How do you define an even integer?
Divisible by 2
How do you define an odd integer?
Not divisible by 2
Is Zero positive or negative?
Neither
What is the order of operations used to evaluate mathematical expressions?
PEMDAS
What does PEMDAS stand for?
Parentheses (or other grouping symbols)-Exponents-Multipication & Division (left to right)-Addition & Subtraction (left to right)
What are grouping symbols?
parenthesis, brackets, absolute value symbol, and fraction bar
Exponents include what 2 symbols?
What is the commutative property of addition?
It states that changing the order of the addends in a sum does not change the sum.
What property does this equation represent?

a+b = b+a
What is the commutative property of multiplication
It states that changing the order of the factors in a product does not change the product.
What property does this equation represent?

a*b = b*a
Commutative Property of Multiplication
It states that changing the grouping (parenthesis or brackets) of addends in a sum does not change the resulting sum.
What property does this equation represent?

a+(b+c) = (a+b)+c
Associative Property of Multipication
It states that changing the grouping (parenthesis or brackets) of factors in a product does not change the resulting product.
What property does this equation represent?

a*(b*c) = (a*b)*c
Associative Property of Multiplication
The Distributive Property
It states that multiplication distributes over addition and subtraction
What does this equation represent?

a*(b+c) = a*b+a*c
The distributive property of multiplication over addition
What does this equation represent?

a*(b-c) = a*b-a*c
The distributive property of multiplication over division
What is the relationship between integers and the number line?
Each integer has a location on a real number line, where the sign of the number determines to which side of 0 the number is located.
What is the absolute value of a number?
It is the number of units a number is away from 0 on a number line.
How do you order integers?
When comparing integers, a number farther to the left of 0 on the number line will be less in value than a number to the right of 0 on the number line.
If the signs are the same add and keep the sign. If the signs are different subtract and take the sign of the number with the larger absolute value.
How do you subtract integers?
add the opposite of the number being subtracted and apply the sign of the greater number
How do you multiply negative integers?
If there is an even number of negatives in the problem, the solution will be positive. If there is an odd number of negatives in the problem, the solution will be negative.
How do you divide negative integers?
If there is an even number of negatives in the problem, the solution will be positive. If there is an odd number of negatives in the problem, the solution will be negative.
neg * neg = ?
pos
neg * pos = ?
neg
neg / neg = ?
pos
neg / pos = ?
neg
neg + neg = ?
neg
neg + pos = ?
which ever sign is larger
pos + pos = ?
pos
neg - pos = ?
neg
pos - neg = ?
larger value
neg - neg = ?
larger value
How do you know if a number is divisible by 2?
If the number is even.
How do you know if a number is divisible by 3?
If the sum of the individual digits of the number is divisible by 3.
How do you know if a number is divisible by 4?
If the last two digits, taken as a two-digit number, are divisible by 4.
How do you know if a number is divisible by 5?
If the last digit of the number is a 5 or a 0.
How do you know if a number is divisible by 8?
If the last three digits, taken as a three-digit number, are divisible by 8.
How do you know if a number is divisible by 9?
If the sum of the individual digits of the number is divisible by 9.
How do you know if a number is divisible by 6?
If it is divisible by both 2 and 3.
How do you know if a number is divisible by 12?
If it is divisible by both 3 and 4.
How do you know if a number is divisible by 10?
If the number ends in a 0.
How do you know if a number is divisible by 7?
If you remove the last digit of the number, double it, and subtract it from the remaining digits and the result is divisible by 7.
What is a factor?
A factor of a number X is a whole number that divides into X evenly without remainder.
What is a prime factor?
The most basic factors of a number, which are always prime.
What is the Greatest Common Factor (GCF)?
It is the largest common factor of 2 or more numbers. Determine this by finding all the prime factors of both numbers, then taking the prime factors that are in both (match them up) and multiply them.
What is the Least Common Multiple (LCM)?
It is the smallest number that two or more numbers will divide into evenly. Find this by multiplying the prime factors of both or all numbers together, however the Greatest Common Factors should only be multiplied once.
What is a fraction?
It is a rational number in the form a/b. a is the numerator and b is the denominator, which cannot be 0.
What is a numerator?
It is the number on the top half of a fraction.
What is a denominator?
It is the number on the bottom half of a fraction.
What is an equivalent fraction?
It is a fraction that is equal to another fraction, by a factor of a/a.
How do you find an equivalent fraction?
Multiply or divide the fraction by the number one, in the special form of a/a.
How do you multiply fractions?
Multiply the numerators with each other and the denominators with each other.
How can you make a fraction multiplication simpler?
Eliminate common factors between numerators and denominators.
How do you find x/y of t/v?
Multiply.
How do you divide fractions?
Multiply the first fraction by the recipricol of the second fration.
What is a reciprical fraction?
It is the inverse of a fraction.
What must you do first to add or subtract fractions?
You must find equivalent fractions for each fraction that have common denominators.
What is the least common denominator?
It is the least common multiple of the denominator of 2 fractions, used for addition or subraction of fractions.
Convert the fractions so they have an equivalent least common denominator, add numerators while keeping same denominator, and simplify the results.
How do you subtract fractions?
Convert the fractions so they have an equivalent least common denominator, subtract numerators while keeping same denominator, and simplify the results.
What is a mixed number?
It is a fraction that includes a whole number as in 2(2/5)
What is an improper fraction?
A fraction where the numerator is greater than the denominator and which can be expressed as a mixed fration.
How do you change a mixed number to an improper fraction?
A*(B/C) = (A*C+B)/C
What is a complex fraction?
Any fraction where either the numerator or the denominator is also a fraction.
What is the place value for decimals.
If the number is abc.def then a is hundreds, b is tens, c is ones, d is tenths, e is hundredths, f is thousandths.
How do you convert a decimal to a mixed fraction?
Keep the whole number and place the numbers past the decimal over the appropraite place value number.
How do you convert a fraction to a decimal?
Divide the numerator by the denominator.
What must you do to add or subtract decimals?
Line up the decimals, add trailing 0s if you need to maintain spacing.
How do you multiply decimals?
Multiply the entire numbers, then count the numbers to the right of both factors, then move the decimal over to the left of the same number as counted in the previous step.
How do you divide decimals?
Set up as long division, count how many digits are to the right of the decimal point in the divisor (number divding with), move decimal point in the dividend (number dividing into) by the amount in the previous step, raise the nely placed decimal point up to the quotient, divide as usual around decimal point
What is a ratio?
A comparison of two or more different quantities.
How can ratios be written?
a:b or a/b or a to b
How can you simplify ratios?
Remove common factors between the quantities and in the case of fractions, make improper, then divide.
What do you have to consider when looking at the actual numbers of a ratio?
What the numbers represent in terms of whole quantities or relationships between quantities. ie the ratio of girls to boys in the school is 3:2, but the ratio of girls to students in the school is 3:5 and the ratio of girls to students in the school is 2:5.
What is a rate?
A rate is a comparison of two types of units.
What is a unit rate?
A rate with a denominator of 1.
What is a proportion?
It is a comparison of two ratios.
What is an extreme when referring to a proportion?
When a/b = c/d, a and d are extremes.
What is a mean when referring to a proportion?
When a/b = c/d, b and c are means.
How do you solve a proportion problem?
Multiply the numerator from the first ratio by the denominator from teh second ratio and then the denominator from teh first ratio by the numerator of the second ratio, and set the values equal to each other.
What categories of proportions are there?
part to part or part to whole
What is a scale?
It is where a smaller number is used to represent a larger unit.
What does it mean when figures are called similar?
It means figures have corresponding sides that are in proprotion with one another and corresponding angles withe the same measure.
If two figures are known to be similar, how can you determine sides or angles?
By setting up proportions.
How can you determine the perimeter of a figure similar to another figure?
The ratio of the sides of two similar figures will be the same as the ratio of the perimeters of the same two similar figures.
What is a percent?
It is a special ratio that compares a numerical quantity to 100.
How do you change a percent to a decimal?
Remove the percent symbol and divide by 100
How do you change a decimal to a percent?
Multiply by 100
How do you change a fraction to a percent?
Use the proportion x/111 = a/b and cross multiply to solve for the variable x.
How do you convert a percent into a fraction?
Put the given percent over 100, remove the percent symbol (%) and simplify if needed.
What type of a relationship is a percentage?
part - whole
The word 'of' in mathematics usually means what?
Multiply
The word 'is' in mathematics usually means what?
Equals
How is percent increase or decrease calculated?
change/original amount = percent/100
How is simple interest calculated?
I = prt where I is the interest charged or paid out, p is the principle amouth that is saved or borrowed, r is the percentage rate written as a decimal, and t is the time in years
How do you calculate the total amount of money that is either earned or owed using simple interest?
Total = p + prt
What are the common measures of central tendency?
mean, median, mode, range
What is the definition of the measures of central tendency?
They are values that are examined from a set of data in order to make predictions and draw conclusions about that set of data as a whole
What is the definition of the mean of a set of numbers?
It is the sum of the numbers in the set divided by the total number of values in the set.
What is the definition of the median of a set of numbers?
It is the middle number when the values are listed in order from smallest to largest.
How do you determine the median of an odd number of values?
It will be the middle number if the values are listed from smallest to largest
How do you determine the median of an even number of values?
It will be the average of the two middle values if the values are listed from smallest to largest
What is the definition of the mode of a set of numbers?
It is the number that occurse the most in a data set
What does it mean if a set is 'no mode'?
In a set of numbers there is not a number that occurs more than any other in the list
What does it mean if a set is 'bimodal'?
In a set of numbers there is more than one value that occurs as much as another number. It is also possible to have more than two modes.
What is the definition of the range of a set of numbers?
It is the difference between the largest value in the list and the smallest value in the list.
What are bar graphs for?
They are used to compare data and can be drawn using horizontal or vertical bars
What are line graphs for?
They are used to show a trend that occurs over time
What are circle graphs for?
They are used to show parts of a whole and percents are often used; always make sure that any percents used in one always add up to 100%
What are scatter plots for?
They are used to show trends in data. The data points are not connected and the general trend of the data is examined
What are the three most common conclusions that can be made based on the graph?
positive correlation, negative correlation, and no correlation
What does a positive correlation look like on a scatter plot?
As the x-values increase there is also an increase in the y-values. The points look like a line rising from the lower left to the upper right.
What does a negative correlation look like on a scatter plot?
As the x-values increase there is a decrease in the y-values. The points look like a line descending from upper left to lower right.
What does a no correlation look like on a scatter plot?
There is no trend in the values across the y-values or x-values. The points look random or cloud-like.
What is a stem and leaf plot?
It is a way to organize a list of numbers. There are typically two columns. the first column, known as the stem, often represents the first digits in the number and the secnod column, known as the leaf, represents the last digit of the number
In a stem and leaf plot, how can you tell the total number of values in the list?
Count the leaves.
In a stem and leaf plot, how can you tell if a number repeats?
If the stem has a leaf of a particular number that repeats
A stem and leaf plot can make it easy to determine what measures?
mode, median, and range
What is a histogram used for?
The bars represent number values or intervals of values. Visually the bars are touching each other. Stem and leaf plots are particularly helpful for these.
What is probability used for?
Calculating the chance that these and other events could happen.
What is the definition for the probability of an event (E)?
P(E) = (the number of ways event E can occur)/(the total number of possible outcomes)
What is the outcome set?
A list of all possible outcomes
What is the sample space?
A list of all possible outcomes
What is the probability of an event that can't happen or an impossible event?
0
What is the probability of an event that is certain to happen?
1
What are the probability of an event that can possibly happen?
between 0 and 1
What is simple probability?
The likelihood of a single event
What is compound probability?
The likelihood of more than one outcome to occur. The key word is often 'or'.
What is the formula for calculating the compound probability of event A and event B?
P(A or B) =
P(A) + P(B) - P(A and B)
What are mutually exclusive events?
Situations that cannot occur at the same instance
If an event is mutually exclusive then P(A and B) are equal to what?
0
In probability, what are independent events?
Two or more events that will occur without the outcome of one affecting the outcome of any other.
How do you calculate the probability of two or more independent events occuring is succession?
Multiply the probability of each event happening independently.
In probability, the phrase 'replace' refers to what type of event?
Independent
In probability, what are dependent events?
Situations where the outcome of the first event does affect teh probability of the second event.
In probability, the phrase 'without replacement' refers to what type of event?
Dependent
How do you calculate the probability of two or more dependent events occuring is succession?
Calculate the probability of the first event and multiply it by the probability of the second event, remembering that the original set might decrease.
In probability, what is the counting principle?
A strategy to determine the number of combinations created from the number of choices.
In probability, what is a permutation
It is the number of possible arrangements for that set of objects.
What is a factorial?
For a whole number it is the product of that whole number and each of the natural numbers less than the number.
n! = n*(n-1)*(n-2)*...*1
What does n! mean?
Factorial.
n! = n*(n-1)*(n-2)*...*1
How do you calculate the number of permutations of n objects taken n at a time?
nPn = n!
How do you calculate the number of permutations of n objects you have to choose from and r is the number of objects actually selected to be arranged? (also when not all of the objects are considered for each different arrangement)
nPr = (n!)/(n-r)!
In probability, what is a combination?
It is the total number of groupings of a set of objects
What is the formula for calculating a combination?
nCr = (n!)/[r!(n-r)!]
Note: nCn will always be equal to 1 since there is only one way to select all the components to a combination.
An exponent is also known as a what?
Power
A power is also known as a what?
Exponent
What is an exponent?
Total number of times a base is used as a factor.
What is the base number?
The number that will be the factor in the exponent
What is the exponent in an equation?
It tells you how many times to use the base number as a factor.
If a number has no apparent exponent, what is the exponent?
1
If a base is raised to the 0 power, what is it equal to?
1
How do you multiply exponents with like bases?
Add the exponents and maintain the same base.
a^m * a^n = a^(m+n)
How do you divide exponents with like bases?
Subtract the exponents and maintain the same base.
a^m / a^n = a(m-n)
How do you raise an exponent to another exponent?
Multiply the exponents.
(a^m)^n = a^(m*n)
How do you raise a product to a power?
multiply the exponents for each base
((a^m)*(b^n))^t = (a^(m*t))*(b^(n*t))
How do you raise a fraction to a power?
apply the exponent to both the numerator and denominator
(a/b)^m = (a^m)/(b^m)
How do you simplify an expression with a negative exponent?
Turn it into a fraction with the numerator as 1 and the base number as the denominator to a positive power.
a^(-m) = 1/(a^m)
What is scientific notation?
It is used to write very large or very small numbers more efficiently.
How do you write a number in scientific notation?
Place a decimal within the non-zero digits of the number to create a number between 1 and 10. Multiply this number by a factor of 10, where the exponent to the 10 is the number of decimal places the decimal point has moved.
What are perfect squares?
Numbers that are the result of multiplying two of the same integers together.
In a square root, the number within the radical sign is a what?
How can you write a square root as an exponent?
the exponent must be a fraction where the denominator is the root or index
What is the equation for how fractional exponents can be written in terms of square roots?
a^(x/y) = (y sqr root (a))^x
How do you simplify square roots that are not perfect squares?
Look for the perfect square factors of the number in the radical, and factor out the perfect square.
The radicals must have the same radicand for all terms. If the randicands are not the same and all are in the simplest form, the radicals cannot be combined.
How do you multiply or divide radicals?
How can you simplify an expression with a radical in the denominator?
Rationalize the denominator by multiplying both the numerator and the denominator by the same radical that is in the denominator to eliminate the denominator.
How do you know if a radical is simplified?
There are no perfect square factors of the radicand other than 1, there are no fractions under the radical sign, there are no radicals in the denominator of a fraction.
What is a point?
A position in space, it has no length or width.
What is a line?
The shortest distance between two points. It has no width, and an infinite length.
What is a line segment?
A piece of a line and includes two points called endpoints and all points in between
What is a plane?
Any three non-colinear points determine a plane and has an infinite width and length
What is a ray?
One half of a line. It includes an endpoint and all of the collinear points on one side of this endpoint.
What is congruent in geometry terms?
When two geometric figures have the same measure.
What is a segment bisector?
It is a line or line segment that divides the segment into two congruent segments.
What is the midpoint of a segment?
It is where the bisector intersects the segment.
What are parallel lines?
Lines in the same planes that do not intersect. Also applies to line segments, rays, and planes
What are perpendicular lines?
Lines in the same plane that intersect at one point and form four 90 degree angles. Also applies to line segments, rays, and planes
What are skew lines?
Lines in different planes that do not intersect.
What is an angle?
It is defined by two distinct rays that have the same endpoint.
What is the vertex of an angle?
The common endpoint of the two rays that form an angle
What is an angle bisector?
It is a ray in the interior of the angle that divides the angle into congruent angles with the same measure
What is an acute angle?
It is an angle whose measure is between 0 degrees and 90 degrees
What is a right angle?
It measures exactly 90 degrees.
What is an obtuse angle?
It measure is between 90 degrees and 180 degrees.
What is a straight angles?
It measures exactly 180 degrees.
What are complementary angles?
Any two angles whose combined measures equal 90 degrees. It is not necessary for these angles to share a side.
What are supplementary angles?
Any two angles whose combined measures equal 180 degrees. It is not necessary for these angles to share a side.
What is a linear pair?
Two supplementary angles that share a common side and no common interior points. They will form a straight angle or a line.
What is a transversal?
A line that cuts through two parallel lines, eight angles are formed.
What are vertical angles?
They are formed when two line intersect and share a common vertex but no common sides and are congruent.
What are corresponding angles?
Two angles on the same side of the transversal, but one is in the interior of the parallel lines and one is in the exterior of the parallel lines and are congruent.
What are altenate interior angles?
Two angles on different sides of the transversal, both in the interior of the parallel lines and are congruent.
What are alternate exterior angles?
Two angles on different sides of the transversal, boh in the exterior of the parallel lines and are congruent.
What are polygons?
Closed geometric figures made up of line segments and angles.
What is a triangle?
A 3 sided, 3 angled polygon.
What is an acute triangle?
A triangle with three acute angles.
A four sided polygon.
What is a pentagon?
A 5 sided polygon.
What is a hexagon?
A 6 sided polygon.
What is an octagon?
An 8 sided polygon.
What is an obtuse triangle?
A triangle with one obtuse angle and two acute angles.
What is a right triangle?
A triangle with exactly one right angle and two acute angles.
What is a scalene triangle?
A triangle with 3 sides of different lengths.
What is an isosceles triangle?
A triangle with two sides of the same length. Two angles are also the same; the angles opposit the sides of the same length.
What is an equilateral triangle?
A triangle with all three sides the same length. Three angles are also the same and always equal 60 degrees
What do the 3 angles in a triangle add up to?
180 degrees
What is equal to the measure of an exterior angle of a triangle?
The sum of the two remote interior angles.
In a right triangle, what do the two acute angles equal up to?
90 degrees, they are complementary.
In a triangle, the length of the longest side must be what?
Less than the sum of the two shorter sides.
What is a trapezoid?
A quadrilateral with one pair of parallel sides.
What is a parallelogram?
A quadrialteral with two pairs of parallel sides.
What is the sum of the angles of a quadrilateral?
360 degrees
In a parallelogram, the opposits are what?
The opposit sides are parallel, and the opposit angles are congruent.
What is an isosceles trapezoid?
A trapezoid with non-parallel sides that are equal in length.
In an isosceles trapezoid, the angles are what?
2 pairs of congruent angles, and also 2 diagonals.
In a rectangle, what are the angles?
All are 90 degrees, and opposite sides are congruent as are diaganols.
What is a rhombus?
A parallelogram with four congruient sides and perpendicular diaganols.
What is a rectangle?
A parallelogram with four right angles.
In a rhombus, what are the angles?
Opposite angles are congruent as are opposite sides. The diaganol is perpendicular.
What is a square?
It is both a rhombus and a rectangle and a parallelogram with four 90 degree angles and all sides are congruent. The diagonals of a square are both congruent and perpendicular.
What is a perimeter?
It is the distance around a polygon.
How do you calculate the perimeter of a square?
P = 4x where x is the length of one side
How do you calculate the perimeter of a parallelogram?
P = 2y + 2x where y and x are non-congruent sides.
What is the area of a geometric shape?
It is the amount of square units needed to cover the shape.
How do you determine the height of a polygon?
It is the length of the segment that is perpendicular to a side of the polygon (called the base)
What is the radius of a circle?
It is the segment whose endpoints are the center of the circle and any point on the circle
What is the diameter of a circle?
It is a segment that passes through the center of the circle and whose endpoints are both on the circle.
How do you calculate the circumference of a circle?
C = 2*pi*r = pi*d
What is the perimeter of a circle called?
A circumference.
How do you calculate the area of a circle?
A = pi*r^2
What is a prism?
It is a three-dimensional solid with two congruent bases, and any number of other faces that are all rectangles.
What is a cylinder?
It has two congruent circular bases, with one rectangular face wrapped around the bases.
How do you calculate surface area for three-dimensional solids?
It is the sum of the areas of its faces.
How do you calculate the volume of a three-dimensional solid?
It is the area of one of the bases multiplies by the height h.
What is the volume of a rectangular solid?
V = l*w*h
What is the volume of a cylinder?
V = pi*r^2h
What is the volume of a pyramid?
V = (1/3)*l*w*h
What is the volume of a cone?
V = (1/3)*pi*r^2*h

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