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