DAT Math Formulas

How many ways can 6 cards be arranged

6*5*4*3*2*1=720

Car travels 1/4 circle 10 miles from center Find the distance?

arc length= 90 radius = 10 90* (pi/180) = pi/2 radians S= pi*r = pi/2 * 10= 5 pi meters

variance definition

avg of the squared differences from the mean

Quadratic Equation

ax^2+bx+c (𝑥 + 𝑦)² = 𝑥2 + 2𝑥𝑦 + 𝑦2 (𝑥 − 𝑦)² = 𝑥2 − 2𝑥𝑦 + 𝑦2 (𝑥+𝑦)(𝑥−𝑦)=𝑥2 −𝑦2

data sets (union)

includes all data by excludes duplicates

data sets (intersection)

includes data present in all sets

Area of a regular polygon

(.5)(# of sides(N))(sin360/N)(s^2)

Combined Work ex) If Tom gets a job done in 4 hours and Jerry gets it done in 3 hours, how many hours does it take to get the job done working together?

(1/A)+(1/B)= (1/T)

x^2-1=

(x+1)(x-1)

Casey is selling 1 orange for x dollars. How many oranges can you buy for 80 cents? A. 10/x B. 4/5x C. 8/5x D. 6x/5

1 orange= x dollars 1 orange/x dollars * 0.8 dollars = 4/5x

standard deviation formula

1. Work out the Mean (the simple average of the numbers) 2.Then for each number: subtract the Mean and square the result. Then work out the mean of those squared differences. Take the square root of that √∑(x-xavg)²

Interquaritle Range 4,4,10,11,15,7,14,12,6

1. order the numbers 4,4,6,7,10,11,12,14,15 2. middle of the first hlaf (4+6)/2=5 3. middle of second half (12+14)/2=13 4. find the IQR by subtracting one from the other 13-5=8 which is the interquartile range

1meter=yd?

1.1 yards

Area of triangle

1/2bh

volume of a pyramid / square pyramid

1/3 (area base) h 1/3 s^2 h

volume of a cone

1/3 pi r^2 h

a^-n=

1/a^n

Combined Work Problems

1/t1+1/t2= 1/total time

How many ways can you arrange 7 people out of 10

10!/(10-7)! 7!= 10/3!7!= 10/3*2= 120

10 is 2y as 25x is to A. 5x B. 5xy C. 5x/y D. x/5y E. 5y/x

10/2y = 25x/ ? 10?=2y25x 10?=50xy ?=5xy

Numbers between 1-100 divisible by 3 and 4

100-=3*33 100=4*25 but both divisible by 3*4=12 100=12*8 33+25-8=50

Jacob is astonished to find the value of his home is $150,000. He calculates that the value of his car, increased by half, is 30% the value of his home. What is the value of his car? A. 40,000 B. 30,000 C. 20,000 D. 10,000 E. 5,000

150,000 * .30 = x+.5x 45,000= 1.5x x=30,000

Sum of interior angles

180(n-2)

Cos(A)

A/H

area of rhombus

A=bh

Area of a circle

A=πr²

A rectangular room is 3 meters wide, 4 meters long and 2 meters high. How far is it from the northeast corner at the floor to the southwest corner at the ceiling?

First we'll solve for the hypotenuse of a right triangle that goes from the northeast corner at the floor to the southwest corner at the floor: 3² + 4² = x² x=5 5²+2² = x² x= √29 or use 3d analysis in which √ 2² +3² +4² = √29

Four coins are tossed. What is the probability that three coins come up heads and one coin comes up tails?

The outcomes that satisfy this condition are HHHT, HHTH, HTHH, and THHH. Let's look at the probability of getting HHHT specifically. Because the instance of getting heads or tails is 1/2, the chance of getting heads on the first toss is 1/2. In addition, the probability of getting heads on the second and third toss would also be 1/2. Lastly, the chance of getting tails for the last flip is still 1/2. Using the multiplication rule, the total probability of getting HHHT would be: 1/2*1/2*1/2*1/2= 1/16 Using this same logic for the conditions HHTH, HTHH, and THHH, you will find that the probabilities of getting these outcomes would also be 1/16. Therefore, the total probability of getting any of these outcomes would be: 1/16+1/16+1/16+1/16=1/4

Three fair coins are tossed at once. What is the probability of getting 2 or more heads

There are 3 coins with 2 sides = 2*2*2= 8 possibilities Probability of getting 2 heads = 3!/(3-2!)2! = 3 Probability of getting 3 heads = 3!/ (3-3!) 3! = 1 Therefore there is a total of 4 possibilities of getting 2 or 3 heads and 8 total possibilities 4/8=1/2

The mean and standard deviation of a set of data is given by m and s, respectively. If 10 additional data points are added exactly at the mean, which of the following is true a. m stays the same s decreases b. m stays the same s increases c. m decreases s stays the same d. both stay the same

a. m stays the same and s decreases

If Corey gave 70 stickers to Jared and Jared gave 10 stickers to Megan then each of them would have the same amount. How many more stickers does Corey have than Megan?

To solve this we need to know the net change of each person Corey -70 Jared (+70-10)=+60 Megan +10 C-70=M+10 C-M=80 Corey has 80 more stickers than Megan

avg. rate

Total distance travelled/total time

A person travels to work at an average speed of 40 mph, and returns home at 60 mph. What, in mph, is the average speed for the entire trip?

V = D/t don't know distance so 2D ÷ (D/60) + (D/40) = 2D÷5D/120 2D * 120/5D = 240/5 = 48

Volume of a cylinder

V=πr²h

Midpoint

[(x1+x2/2),(y1+y2/2)]

dilution

c1v1=c2v2

card problems w/o replacement

each card picked decreases total and increases number of cards of interest

parallel line slopes

equivalent

Volume of a rectangle

l*w*h

log(675)=2x+3y

loga^2+logb^3 25*27 5 and 3

z score

measure of standard deviation

perpendicular line slopes

negative reciprocal

Area of ellipse

pi a b

a<2 Quantity A= 5a Quantity B = 2a+7 Which quantity is bigger

subtract 2a from each quantity A=5a-2a=3a B= 7 Quantity B is bigger

card problem w/replacement

total number of carbs stays the same

The numbers (1,2,3,6) have an average mean of 3 and variance of 3.5. What is the average mean and variance of (3,6,9,18)?

3+6+9+18 / 4 = 9 variance = (9-3)² + (9-6)² + (9-18)² / 4 = 126/4 = 31.5

How to find an exterior angle

360/n= angle

How many ways can words of APPALOOSA be arranged?

3 A's 2 P's 2 O's 9 total letters 9!/3!2!2! = 15120 9*8*7*6*5*4*3*2*1/3*2*2*2

At what point of the graph does 3x-4y=12 cross the x axis

3x-4(0)=12 3x=12 x=4 (4,0)

If 3 liters of 40% orange juice and 1liter of 50% orange juice are mixed, what is the percentage of the organe juice in the mixture?

3+1= 4 liters total (.5)(1)+(.4)(3) = 1.7 1.7/4 * 100= 42.5%

Volume of a sphere

4/3πr³

Area of a cylinder

(2pi*r*h) + (2pi* r^2)

Exponent Rules (X^b)(Y^b) (X^a)(X^b) (x^a)^b X^a/X^b X^b/y^b

(X^b)(Y^b) = (XY)^b (X^a)(X^b) = X^a+b (x^a)^b = x ^ a*b X^a/X^b = X^a-b X^b/Y^b = (X/Y)^b

Equation of a circle

(x-h)^2 + (x-k)^2 = r^2

percent change

(x2-x1/x1) x 100

Gaussian distribution 1 standard deviations 2 standard deviations 3 standard deviations

1 standard deviations= 68% 2 standard deviations+= 95% 3 standard deviations= 99.7%

1kg=ibs?

2.2 pounds

1in= cm?

2.5 centimeters

surface area of a rectangle

2lw+2wh+2lh

Area of hollow cylinder

2pi r^2

1Ib.=g?

454 grams

Area of a sphere

4πr^2

Liklihood of pulling 3 spades out of a deck of cards without replacement

52 cards total 13 cards in each suit 4 suits 13/52* 12/51 * 11/50 = 33/2550

1mile=ft?

5280 ft

Jill has six different books. She will select one book on Monday and a different one to read on Wednesday. In how many ways can Jill select two different books?

6*5 = 30

1 standard deviation of the mean

68% of data

The length of symptoms of the flu is normally distributed with an average of 7 days and a standard deviation of 2 days. What is the probability someone with the flu will experience symptoms for less than 5 days?

7 days is the normal and the standard deviation is 2 days so if someone experiences symptoms for less than 5 days they are 1 standard deviation from the mean so 68% of the population is 1 standard deviation from the mean and the person is therefore 32% below and above the mean but you only want the value for what is below therefore it is 16% another way to look at this 1-.68/2=16%

Probability of 3 out of 8 heads

8!/3!5! = 8*7*6/3*2= 56

In a given course a student receives preliminary examination grades of 81,85, and 95. The final examination is weighed for one-third and the average of the preliminary grades is weighed as 2/3 of the final grade. What should the final examination grade be for a semester average of 90?

90 = 2/3 (81+85+95/3) + 1/3x 90= 2/3(87) + 1/3x 32= x/3 x= 96

2 standard deviations of the mean

95% of data

3 standard deviations of the mean

99.7%

J, K, L, M, N are consecutive whole numbers. When is JxKxL>12? 1. J ≥ 2 2. J is odd A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient C. BOTH Statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D. EACH statement ALONE is sufficient E. Statements 1 and 2 TOGETHER are NOT sufficient.

A. From statement 1, if J =2 then 2x3x4 = 24 > 12. Any other value of Jgreater than 2 will satisfy the condition. Statement 1 is sufficient. From statement 2, if J = 1 or J = 3, the condition is not satisfied, so statement 2 is not sufficient. The answer is choice A

Tim invests $1000 in an account that accrues 12% interest per year, compounded quarterly. After 15 months, how much interest has Tim earned?

A= P (1+r/n)^nt 15 months/12 months = 1.25 1000 * (1+.12/4)^4*1.25 1000 (1.03) ^ 5 = 1159.27 so he accrued $159.27 interest *** if you do not remember formula could do 1000*.12+1000=1120 so it would be something slightly larger than 120 since its 15 months not a year

If 40% of the population owes a bike, 10% of the population owes a skateboard, and 5% of the population owns both, what percent of the population owns either a skateboard or a bike or both?

Assume the population is out of a 100 since 5 people own both you can assume that out of the equation so 40-5=35 and 10-5=5 35+5+5=45 45/100=45%

A = odd numbers from 1 to 25 B= prime numbers from 1 to 25 A= median of A B= interquartile range of B A. A is greater than B B. B is greater than A C. A and B are equal D. Cannot determine the relationship

B. A={1,3,5,7,9,11,13,15,17,19,21,23,25} B={2,3,5,7,11,13,17,19,23} The median (middle number). of set A is 13 . The interquartile range is given by: Q3 - Q1 where Q1 is the median of the lower 50% of the data and Q3 is the median of the upper 50% of the data. 3+5/2 = 8 17+19/2=18 The IQR for set B is then 18 - 4 = 14, which is greater than the median of set A.

A. area of a square with side a B. area of a triangle with side 2a A. A is greater than B B. B is greater than A C. A and B are equal D. Cannot determine the relationship

B. Square= a² Triangle= s²√3 ÷ 4 = 4a²√3÷4 = a²√3 a²√3 > a²

Isoceles triangle fact

B=h

x² + y² = 100 value of x+y=? 1. y>x 2. y=6 A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient C. BOTH Statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D. EACH statement ALONE is sufficient E. Statements 1 and 2 TOGETHER are NOT sufficient.

C. Statement 1 is not sufficient to find the value of x+ y . Statement 1 only tells us is that y > x , thus there is not enough information to determine a unique solution for x and y. Statement 2 leads us to not one, but two solutions: 36 + x² = 100 x=±8

y>10 A= 35% of y% of 5 B= Y% of 5% of 35 A. A is greater than B B. B is greater than A C. A and B are equal D. Cannot determine the relationship

C. A= 35/100 * y/100 * 5 B= y/100 * 5/100 * 35

Diameter= 20 m Subtended arc= 50 Whats the distance

C= 2pir= 2pi*10= 20 pi (50/360) * 20pi = 2.8 pi

Circle A and Circle B are respectively defined in standard coordinate plane by equations (x+3)^2 + ( y -2) ^2 = 21 and (x+12)^2 + (y+2)^2 = 16 what is the shortest distance between the two centers of the circles?

Circle A's center is -3,2 Circle B's center is -12,-2 distance formula = sqr ( x2- x1) ^2 + (y2-y1) ^2 = sqr (-12-(-3))^2 + (-2-2)^2 = sqr (-9^2 + - 4^2) = sqr 97

Sin(A)=

Cos(90-A)

X is a real number and it is greater than 0 A= x B = x² A. A is greater than B B. B is greater than A C. A and B are equal D. Cannot determine the relationship

D. if x= 1 x²=x if x=2 x²>x if x= 1/2 x²<x

X<y x≠0 y≠0 A. (1/x)² B. (1/y)² A. A is greater than B B. B is greater than A C. A and B are equal D. Cannot determine the relationship

D. if x=3 y=4 --> 1/9 > 1/16 but if x=-3 y=-4 --> -1/9< -1/16

Rectangles perimeter = 96 inches What are the dimensions of the rectangle ? A= area is 572 B= W is 4 less than the length A. Statement 1 ALONE is sufficient, but statement 2 alone is not sufficient B. Statement 2 ALONE is sufficient, but statement 1 alone is not sufficient C. BOTH Statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient D. EACH statement ALONE is sufficient E. Statements 1 and 2 TOGETHER are NOT sufficient.

D. A.2L + 2W = 96 576=LW 2 unknowns we can solve for! B. W= L-4 again 2 unknowns we can solve for!

Distance Formula

D=(rate)(time)

C to F conversion

F=9/5(C)+32

Simple Interest Formula

FV=PV(1+r)^t FV: Future value r: Interest rate t: Time the money is invested for

Compound Interest Formula

FV=PV(1+r/n)^nt FV: Future value r: Interest rate t: Time the money is invested for n: Number of times interest is compounded per unit "t"

Jaime is 6 years older than Tyroine now. In 2 years, Jaime will be one year less than twice Tyroines current age. What is Jaime's current age?

J= T + 6 (T=J-6) J+2= 2T-1 J+2= 2 (J-6) -1 J + 2 = 2J - 12 -1 2= J - 13 J= 15

degrees C to K

K=C+273

Tan(A)

O/A

Sin(A)

O/H

A bag is filled with 20 pieces of candy A and 10 pieces of candy B. If you choose 2 pieces of candy at random without replacement, what is the probability of picking one piece of candy of each type?

Scenario where A is chosen first= 20/30*10/29=20/87 Scenario where B is chosen first = 10/30*20/29=20/87 Both scenarios together 20/87+20/87=40/87

Cos(A)=

Sin(90-A)

Variance

Stand. Dev. Squared

log rules log(x) = b log(mn) log(m/n) b(loga)(X)

log(x) = b: x=10^b log (mn) = log(m) + log(n) log (m/n) = log(m) - log(n) log (m^n) = nlog(m) nlog(x) = Log(x^n) b(loga)(X) = loga(X^b)

Combination Formula

n! / (n-k)! * k! - n is the total number and k is the sample number - order doesn't matter

Permutation Formula

n!/(n-k)! - n is the total number and k is the sample number - order does matter

Permutation

order does matter

Combination

order doesnt matter

Perpendicular lines are y= -5 +6

reciprocal slopes y= x/5 + 3

Convex polygon with n number of sides what is the sum of angles (s)

s= (n-2) (180)

Area of a equilateral triangle

s^2 * root 3/ 4

sin(x)/cos(x)

tan(x)

How many ways can you arrange 4 identical footballs and 3 identical basketballs on a shelf?

total=7 7!/ 4! 3!= 35

Parabola

vertex = -b/2a

What is greater Quantity A = 5x^2-20x+19 Quantity B = -2

x vertex = -b/2a 20/2(5)=2 5(2)^2-20(2)+19=-1 Quantity A is greater

√x^(4/5) = 4

x^(4/5) = 16 x^(4/5)*^(5/4) = 16 ^ ( 5/4) x= ⁴√16⁵= 32

linear equation

y=mx+b y1-y2=m(x1-x2): y2-y1/x2-x1

Dist between two points

√(x2-x1)²(y2-y1)²