DAT Math Section
amplitude
"A" value in the sine/cosine function -determines how far up on the y-axis the peaks reach EX: 2sin(x) means each peach will reach +2/-2
EX: Calculate the number of choices you have when deciding which four shirts to donate if you have 7 total, 1 pair of pants with 3 total, and 2 pairs of shoes if you have 4 total
# of choices made / # of choices needed *DO THIS SEPARATELY FOR EACH ARTICLE OF CLOTHING* 7 X 6 X 5 X 4 / 4 X 3 X 2 X 1 3 / 1 4 X 3 / 2 X 1 Then multiply all together for the final answer: 35 X 3 X 6 = 630
How can you calculate percent difference/increase
(original value - new value)/ original value x 100
Equation of a circle
(x-h)²+(y-k)²=r² *h and k are the coordinates for the center*
slope formula
(y₂- y₁) / (x₂- x₁)
Units digit problems
-multiply the units values of the numbers together (for large exponents) and the answer will be the same unit digit for the product -if given the "nth term" in a sequence, multiply the first five or six numbers to find the pattern
How to solve a problem with large exponents
-plug in a smaller exponent first to find the pattern
If a = b + 1/ b - 2 , what is b?
1. cross multiply 2. get each common factor on the same side of the equation
Geometry approach
1. draw the figure 2. label all the information that you know 3. write down the full equations needed 4. estimate values to save time
area of a triangle
1/2 a * b * sin(angle in between)
area of a triangle
1/2bh
Volume of a pyramid
1/3 x area of base x height
volume of a cone
1/3πr²h
Common Decimals
1/6 = .16 1/7 = .14 1/8 = .125 1/9 = .11
secant
1/cos
Cosecant (csc)
1/sin
EX: Calculate the value of 100!/98!
100 X 99 X 98! / 98! *the 98! will cancel* so the answer is 9900
What is the sum of all the interior angles of a triangle?
180
Interior angle for polygons
180 (n-2) / n
perimeter equation
2l+2w
If 2p is to 8 as 6 is to 0.4, then what is p?
2p/8 = 6/0.4
Circumference of a circle
2πr
Surface Area of a Cylinder
2πrh+2πr²
Surface area of a cylinder
2πrh+2πr²
What are the three Pythagorean triple triangles?
3-4-5 and 5-12-13 7-24-25
sum of parallelogram angles
360
Volume of a sphere
4/3πr³
Surface area of a sphere
4πr²
A soccer league has 1,000 members and a hockey league has 600 members. Among the players are 60 sibling pairs each consisting of one soccer player and one hockey player. If one player is randomly selected from each league, what is the probability that the two players will be siblings?
60/1,000 X *probability that the pair will be SAME sibling* 1/600
Surface area of a cube formula
6a^2
EX: Calculate the number of outfit choices you have if you have 7 shirts, 3 pants, and 4 pairs of shoes *Choosing one thing from multiple sources*
7 X 3 X 4 = 84
EX: Calculate the number of shirt choices you have for Monday Tuesday and Wednesday if you cannot repeat outfits *Permutations*
7 X 6 X 5 = 210
isosceles triangle
A triangle that has 2 equal sides.
Prime number
A whole number that has exactly two factors, 1 and itself.
Area of a Trapezoid Theorem
A=1/2h(b1+b2)
Area of Parrallelogram Formula
A=bh
Similarity polygons
ABC ~ DEF -the ratio of the sides is a constant *if this constant is B, then the perimeter of the larger triangle is B times greater and the area is B^2 times greater
Rate pie problems
Add rates together to get the total rate, and divide the total distance by the rates to get time EX: What time will two planes be 1800 miles apart? One travels 250mph and the other travels 350mph
arithmetic mean
An average that is calculated by adding up a set of quantities and dividing the sum by the total number of quantities in the set.
The table above shows the number of students in three clubs at a local college. No student is in all three clubs, but 10 students are in both Civics and Debate, 5 students are in both Civics and Triathlon, and 6 students are in both Debate and Triathlon. How many different students compose the three clubs?
C = 40-10-5 D = 30-10-6 T = 25-5-6 Total = C + D + T + 10 + 5 + 6
Solving ratio problems
EX: a university science class consists of juniors and seniors in a ratio of 3 to 5, how many senior students are there if the total number of students in the class was 96? 3 juniors 5 seniors 8 total 12 multiply 12 multiply 12 multiply 36 total 60 total 96 total
age problems
EX: fifteen years ago Tom was three times as old as Steven. Today, he is only 6 years older than Steven. What was the sum of their ages 15 years ago? *make a system of equations*
basic probability
Favorable Outcomes/Total Possible Outcomes
"work" problems
Given the rates of two or more people to complete a "job" add the reciprocals of the time together and then take the reciprocal to get the total time if they would work together EX: worker one completes job in 2 hours and worker two completes job in 4 hours. How long if they work together? 1/2 + 1/4 = 6/8 or 1.33 hours
group grid
Group 1 Group 2 Total Group A Group B Total
Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
If 1 < x < 2, simplify |x − 1| + |x − 2|
If x > 1, then the quantity (x - 1) is positive, so |x - 1| = x - 1. If x < 2, the quantity (x - 2) is negative, so |x - 2| = -(x - 2) = 2 - x. Therefore, for any value of x between 1 and 2: |x − 1| + |x − 2| = (x − 1) + (2 − x) = 1
arcsinx
Inverse sin function used to solve for an angle when given the side lengths -still opposite/hypotenuse
How does the surface area change when the dimensions of a box are doubled?
It is clear from the above two solutions that the area of cuboid becomes n^2 times the previous area if each dimension raised to n times.
Perpendicular lines
Lines that intersect to form right angles *slopes are opposite reciprocals* EX: -1/2 and 2
A train travels at 200 mph during a 1,000 mile trip EXCEPT for one hour when it travels 150 mph. How long does it take the train to complete the trip?
Make a table for each scenario Rate Time Distance -set rate(time) + rate(time) = total distance
MADSPM Rules for Exponents
Multiply ADD Divide SUBTRACT Power MULTIPLY Remember bases need to be the same.
Please Excuse My Dear Aunt Sally
Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction
For graphing problems:
Plug in points to the equation or use points on the given graph
For problems with multiple variables:
Plug in your own numbers to solve!
Surface area of a box
S = 2 (lw + hw + lh)
Similar Triangles
Similar triangles have the same shape: corresponding angles are equal and corresponding sides are proportional -set up ratios to solve for missing side lengths
factorials
The factorial of n is the number of ways in which the n elements of a group can be ordered. The factorial of a number n represented by n! is the product of the natural numbers up to and including n. Example: 6! = 6 x 5 x 4 x 3 x 2 x 1
triangle inequality
The sum of the lengths of any two sides of a triangle is greater than the length of the third side. The difference of any two sides must be less than the length of the third side
In how many different ways can 6 books be arranged on a shelf?
There are 6 choices for the first book on the shelf, then 5 choices for the second book, 4 choices for the third, 3 for the fourth, and 2 for the fifth. The total number of choices is 6 × 5 × 4 × 3 × 2 × 1 = 6 × 5 × 4 × 6.
inverse of trig functions
Used to find the angle when sides are known.
Volume of a cone
V=1/3πr²h
Volume of a cylinder
V=πr²h
unit circle
a circle with a radius of 1, centered at the origin
tangent line
a line that just touches a circle at exactly one point
z-score
a measure of how many standard deviations you are away from the norm (average or mean)
What does a negative amplitude value do to sine/cosine functions?
a negative amplitude indicates a flip over the x-axis...so instead of starting upwards the graph begins downward or in the (-)x direction
30-60-90 triangle
a right triangle that consists of a right angle, a 30 degree angle, and a 60 degree angle *equilateral triangle cut in half*
scalene
a triangle with no congruent sides
Sector formula
arc length/2pi r = 90/360= sector area/pi r^2
quadratic equation
ax² + bx + c = 0
Pythagorean Theorem
a²+b²=c²
parallel lines
coplanar lines that do not intersect *slopes are the same*
Distance between two points
d = √(x₁-x₂)² + (y₁-y₂)² -draw a right angled triangle and use Pythagorean Theorem
bell curve
distribution of scores in which the bulk of the scores fall toward the middle, with progressively fewer scores toward the "tails" or extremes
How can you get an even integer?
even x even OR even x odd -2 x even -2 x odd
Probability "and"
means multiply
When solving inequalities...
need to create two equations 1. equaling a positive integer 2. equaling a negative integer with a flipped inequality
When taking the square root...
need to worry about the POSITIVE and the NEGATIVE answer
How can you get an odd integer?
odd x odd ONLY
percent difference
percent change = difference / original value X 100
Sin and Cos curves
plug in the given points to the equations in the answers to check
How to find the area of a circle given the center and a point?
plug the points into the circle formula and take the square root to calculate the radius, then find the area
Data Sufficiency strategy
questions ask you to assess whether or not the information provided in two statements is sufficient enough to answer the question -focus on what each answer choice gives you *individually* -re-evaluate to see if both answer choices together could answer the question
radians to degrees conversion
radians x 180/π
Cotangent
reciprocal of tangent
Pythagorean Identity
sin²θ + cos²θ = 1 *for any angle value this is always true*
Graphing inequalities
solve inequalities for variable same way you solve regular equations -shading above the line is indicative for a "greater than" -shading below the line is indicative for "less than" Solid line = equal to Dashed line = not equal to
How do you find the median of an even set of numbers?
take the average of the two middle numbers *numbers must be in numerical order*
variance
the difference between a value and the mean
period
the distance between peaks -period values are equal to 2pi/B *to find B you must set equal to the period from the graph* EX: sin(2x) means the period is equal to 2pi/2 or just pi
vertex
the lowest or highest point on a parabola
The average of six numbers is 36. After one of the numbers is removed, the average of the remaining numbers is 39. What number was removed?
the number that was removed or added is the difference between the averages of the two data sets
standard deviation
the square root of the variance
vertical shift value
the value in a sine/cosine function that determines the midline value of the graph...the whole graph shifts up or down according to this value EX: sin(x) + 4 means the midline of the curve must start at 4
What are the "roots" of a parabola
they are two points that are equidistant from the vertex
horizontal shift value
this is the value that determines the horizontal starting point of a sine/cosine function set X + # = 0 EX: sin(x-pi/2) means that the graph will start at pi/2 and have a period of 2pi
Probability "or"
to add
average speed
total distance divided by total time
group formula
total= (group1) + (group2) - (both) + (neither)
EX: What is the value of sin (75)
use sin (A + B) = sin (A) X cos (B) + sin (B) X cos (A)
Quantitative comparison
uses numeric adjustments based on supported adjustments. -"cannot be determined" answer choice means that sometimes A is greater or sometimes B is greater
If a particular point on a line is known then...
we can figure out the y-intercept value for the equation of the line
supplementary
when two angles add up to 180 degrees
complementary angles
when two angles add up to 90 degrees
vector
when two line segments or rays meet at a point
Inequalities
whenever you multiply or divide when solving, you MUST flip the sign in the inequality * this also applies when you factor out a negative number *
45-45-90 triangle
x, x, x√2 *cutting a square in half*
quadratic graph
y=x^2
EX: x + 3y = 4 and x - 3y = 6, what does x equal?
you can STACK these equations!! add or subtract to get the 'y' variable to cancel out -you can also multiple one or both of these equations to get variables that can cancel!
Area of a circle
πr²
