Kaplan DAT Math

In which of the four quadrants on a 2D axis does a circle with equation (x + 2)2 + (y - 2)2 = 4 have its center?

The equation is in the form (x - h)2 + (y - k)2 = r2, where the center of the circle is (h,k) *The center is at (-2,2), this is in the II quadrant*

PROBABILITIES OF INDEPENDENT EVENTS

The probability of independent events is the probability of both occurring P (A and B)

Cross section of a regular triangular prism is?

Triangle !!! think: area of triangle: A= 1/2 x b x h

How many permutations are there for the prime numbers less than 10?

Use the How many permutations Equation: # permutations = n! (order does matter) The prime numbers less than 10 = 2,3,5, and 7 = 4 prime numbers # permutations = 4! = 24

Volume of a prism?

Volume = area cross section x height V = A x h

Volume of a triangular prism?

Volume = area cross section x height V= (Area of triangle 1/2 x b x h)x (height of triangular prism)

How much water must be added to 10oz. of a 30% alcohol solution so that the resulting solution is diluted to 25% alcohol?

[(10*0.3)/(10 + x)] = 0.25

How many ounces of pure water must be added to 50 ounces of a 15% saline solution to make a saline solution that is 10% salt?

[(50*.15)/(x + 50)] = 0.1

how to solve simultaneous equations

combination (add or subtract equations to eliminate one of the variables)

Trig: 1 + cot^2x =

csc^2(x)

It the two variables x and y have different absolute values, and 2x - 3y = -5 which variable(s) cannot be negative? a.x b. y c. Both x and y d. x and y can both be negative e. Cannot be determined

d. x and y can both be negative If y and x are both negative, then any value above 3 for x will work, for example: 2(-4) - 3(-1) = -8 - -3 = -5

what are the sides of a 30/60/90 triangle?

hyp: 2x opp of 60: (x)sqrt(3) opp of 30: x

prime number?

is a whole number greater than 1, whose only two whole-number factors are 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29.

|n|

n if n≥0 and -n<0

how do you find the permutations of a set?

n!

A class of 10 students are waiting to visit the Headmistress in groups of 2. How many different groups of two students could possibly visit the headmistress if they are selected in groups at random and order does not matter? a. 90 b. 18 c. 9 d. 20 e. 45

n= total # of things/ppl (total # of items in the sample) r = # things that can be chosen (# items selected from sample) n= # of students r = # diff. groups of students (from n students) n = 10 and r = 2 Use Combinations formula: (because order DOES NOT MATTER AND SELECTED AT RANDOM. Remember if order Does MATTER= then use permutations formula) Combinations formula: C (n, r) = n! / (n-r)! x r! = 10! / (10-2)! x 2! = 10! / (8! )(2!) = 3628800 / (40320) (2) = 45

simple interest

principal x (interest rate) x (time)

Trig: tan^2x +1 =

sec^2(x)

Sin(5pi/6)

sin 150

x + y = 28 If x and y are two consecutive odd numbers, with x being the larger number, then what is the value of x?

x = x y = x-2 If x is the larger number, two consecutive odd numbers are 2 units apart, so y is equal to x-2 x + (x - 2) = 28 2x - 2 = 28 2x = 30 x = 15

Calculate x/y2 if x = 1/2 and y = 1/3

x/y2 = (1/2)/(1/3)2 = (1/2)/(1/9) = (1/2)(9/1) = 9/2

steps for standard deviation (adsar)

• calculate the average (average • determine the difference between each number and the average (difference) • square the differences (square) • add results and divide by number of terms (average) • take square root of average (root)

How do you find an angle that is intercepted to the arc?

(1/2)(major arc - minor arc) OR. if the arc is given already just do (1/2)(intercepted arc)

Volume of a sphere?

(4/3) π r^3

percent decrease

(amount of decrease)/(original whole) • 100%

percent increase

(amount of increase)/(original whole) • 100%

compound interest

(final balance) = (prinicpal) x (1 +(interest rate/# of annual compounds))

Equation for a circle?

(x - h)2 + (y - k)2 = r2, where the center of the circle is (h,k)

Combinations formula? Does order matter?

****ORDER DOES NOT MATTER!!!****** C (n, r) = n! / (n-r)! x r! n= total # of things/ppl (total # of items in the sample) r = # things that can be chosen (# items selected from sample)

0.2 is what percent of the fraction 4/5?

0.2 = 2/10 (2/10) / (4/5) * 100% = 25%

2/5

0.4

5/8

0.625

13/20

0.65

9/11

0.82

Which integer can be added to both the numerator and denominator of 1/3 to give 3/5?

1 + x /3+x = 3/5 1+x = 3 3+x= 5 x =2 x=2

probability of not event

1 - P(event)

13/52 = reduces to what fraction?

1/4

18 x 6 =

108

540 / 5 ?

108

1/8

12.5% = .125

4000/3 ?

1333

1/6

16.6...% = .1666...

2x = y/4 and z = 2y/3 Which of the following is equal to z in terms of x?

16x/3 2x = y/4 Solve for y: y = 8x Substitute: z = 2(8x)/3 z = 16x/3

Formula for the sum of interior angles?

180(n-2) n= # sides of polygons

What is the SA of a cylinder?

2(pi)rh +2(pi)r^2

1/5

20% = .2

1/4

25% = .25

A rectangle has area A. If the length is doubled and the width is tripled then what will be the new area in terms of A?

L*W = A 2L*3W = 6A

What is the sum of the interior angles of an octagon?

Sum of interior angles equation= 180(n - 2) 180(8 - 2) 180(6) = 1080o

52/2 = ?

26

Trig: sin(2x) =

2sin(x)cos(x)

If 2x -2 = x-16 What is x?

2x -2 = x-16 2x - x = -16 + 2 x = -14

Square root of 12?

3.5

1/3

33...% = .3333...

3/8

37.5% = .0375

four types of "special" right triangles

3:4:5 5:12:13 1:1:√2 (-45º-45º-90º) 1:√3:2 (-30º-60º-90º)

52/13 = ?

4

184 divided by 4?

46

1/2

50% = .5

17 X 3 ?

51

180 x 3

540

2.5 x 2.5

6.25

3/5

60% = .6

2/3

66...% = .666...

What is the empirical value?

68% - 95% - 99.7%. So 68% would lie within 1SD of the mean (above and below) #21 in Math PT4 also #2 in Math PT1

2 x 3.5?

7

Rachel drives the same route to and from work every day. Rachel's drive to work in the morning is always 10 minutes faster than her return journey at night. If the distance to and from work is the same, 30miles, and her average speed for the journey to work is 40mph, how much slower his her average speed for the drive home?

7.3mph Strategy: make your Rate x Time = Distance Table with Person 1 and Person 2 rows fill in each section of table given and solve blanks with t=d/r convert min = into hours solve with R = d/t R= 30/ (11/2) R= 32.7 Answer question: 40 mph- 32.7 mph = 7.3 mph

Two hundred liters of a punch that contains 35% fruit juice is mixed with 300 liters (L) of another punch. The resulting fruit punch is 20% fruit juice. Find the percent of fruit juice in the 300 liters of punch.

70 + 300x = 100 x = 10%

3/4

75% = .75

What are some special triangle sides?

7:24:25 6:8:10

4/5

80% = .8

5/6

83.3...% = .83...

7/8

87.5% = .875

In 5 years time, Simon will be twice the age that his brother Peter is now. If Peter is now four years old, how old is Simon now?

If x is equal to Simons age now, and y is equal to Peter's age now: x + 5 = 2y x + 5 = 2(4) x + 5 = 8 x = 3

A right triangle has height 3 and base length 2.5, what is the length of the third side? a. 3.9 b. 15.25 c. 2.3 d. 2.75 e. 5.5

Clue: think right triangle = pythag. theorem equation look for c^2!! a^2 + b^2 = c^2 c^2 = (3)2 + (2.5)2 c^2 = 9 + 6.25 c^2 = sqrt 15.25 c = 3.9 (approximate that sqrt(16) = 4

Find the distance between the following points: (1,2,3) and (-4,-5,-6)? a. 12.4 b. 9.6 c. 4.6 d. 21.0 e. 196.0

Formula: x1 = 1 , x2 = -4 y1 = 2, y2 = -5 z1 = 3, z2 = -6 Square root of 144 is 12 and square root of 169 is 13 so the distance is between 12 and 13, we can guess the answer is 12.4

At a sporting event, the stadium holds 15,000 spectators and there are 650 empty seats. If the number of male spectators is 3 times the number of women, and 8 times the number of children, how many children are present? a. 9874 b. 1230 c. 3702 d. 423 e. 3,388

Let # children = x # male = 8x # female = (8/3)x *for female remember your working backwards: you have to put it in terms of children variable so male spectators = 8x *and you divide that by 3 because "male spectators is 3 times the number of women" remember that you have male/women so 8x/3 = (8/3)x *opposite of multiplying is dividing which is what we do when we divide 8x men by 3 # of women Last step: Place 3 variables above into child x equation to equal Solve for x = spectators - empty seats x + 8x + (8/3)x = 15000 - 650 11x (round to 11) = 14350 x = 1230 (rounded answer is 1195 which is closest real answer)

You have x. It increased by 20%, what is the new value?

Literally (x)(1.2) OR x(0.2) + x. Same results (QR destroyer #113)

Probability formula of 2 independent events occurring? example of when to use: Two cards are taken randomly from a normal deck without replacement. What is the probability that both cards are hearts?

Multiplication rule!! P(A and B) = P(A) x P (B) 13 cards per suit Probability that the first card is a heart = 13/52 52 cards - 1 = 51 cards total for second probability 13 hearts possible- 1 = 12 heart cards possible Probability that the second card is a heart = 12/51 Probability of both these events occurring: (13/52)*(12/51) = reduce fractions to: (1/4)*(4/17) = 1/17

Find the value of y given that 2x - y = 6 3x + 2y = 23

Multiply the first equation by 2: 2(2x - y = 6) 3x + 2y = 23 4x - 2y = 12 3x + 2y = 23 Add the two equations together to eliminate y: 4x - 2y = 12 +(3x + 2y = 23) 7x = 35 x = 5 Now substitute x = 5 to find y: 2x - y = 6 2(5) - y = 6 10 - y = 6 10 - 6 = y y = 4

Surface area of a cube?

SA= 6(a^2)

A rectangle has length 10 and width 6. What is the length of the diagonal formed by connecting two opposite corners?

Strategy see diagonal think pythag theorem: c2 = a2 + b2 c2 = 102 + 62 c2 = 100 + 36 c2 = sqaure root 136 c = 11.7

Chris runs 10 yards in 0.8 seconds. At the same speed, how many meters can Chris run in 10 seconds? a. 114 meters b. 11.1 meters c. 120.5 meters d. 90 meters e. 70.7 meters

Strategy: 1.) Convert yards ---->ft------->meters *****Remember: 3ft = 1 yd 1m = 3.28 ft 2.) Place numbers into a cross multiply ratio meters/seconds = x meters/ seconds 3.) Solve for x meters 10yd/1 x 3tf/1yd x 1m/3.28 ft = 9.15 m 10 yd = 9.15 m 9.15 m = x ------- ------------ .8 seconds 10 seconds 9.15 x 10 = 91.5 = 91.5 / .8 seconds = 114 meters

What is the surface area of a sphere with diameter 12? a. 144 π b. 36 π c. 576 π d. 48 π e. 27π

Strategy: 1.) Diameter is the full line so half of that is radius Therefore if diameter is 12, then radius is 6. d = 12 r = 6 2.) Plug into equation *****Remember: Surface Area of Sphere = (4/3) (π) r^2 = (4/3) (π) r^2 = (4/3) (π) 6^2 = (4/3)(36) π = 144 (π)

At a rock concert the ratio of males to females is 5:4. If there are 6300 people in the crowd, then how many males are there?

Strategy: 3 steps to ratio problems Step 1: Add ratios together 5+4 = 9 parts Step 2: Divide Total quantity of people given/ by total # parts (from step 1) 6300/ 9 = 700 groups Step 3: Multiply answer of step 2 x #males = 700 x 5 = 3500 To check: add up males + females = total quantity given 3500+ 2800 = 6300

Susie is 2 years younger than her brother Tim. In 8 years time the sum of their ages will be 56. How old is Tim now? a. 21 b. 23 c. 27 d. 24 e. 26

Strategy: Let x equal Tim's age Let x-2 = Susie's age Step 1.) Add 8 years to both new equations together Tim: (x + 8) Susie: (x - 2 + 8) Step 2.) Add both equations to each other and equal it to combined age of 56 (x - 2 + 8) + (x + 8) = 56 2x + 14 = 56 2x =42 x =21 Therefore Tim is 21.

If a rectangle has width equal to three-quarters its length and the perimeter is 140cm, what is the width of the rectangle rounded to the nearest whole number? a. 28cm b. 56cm c. 30cm d. 35cm e. 21cm

Strategy: Perimeter of a rectangle = 2 (L + w) w = 3/4 x L 2 (L + 3/4 L)= 140 cm 2L+ 6/4L = 140 cm 14/4 L= 140 (4/14) L = 40 Final step: plug L back into w equation w = 3/4 x L w= 3/4 x 40 w= 30

A quadrilateral has width x - 4 and length 2x + 5. If the perimeter is 80, what is the length? a. 44 b. 19.5 c. 15.5 d. 28.3 e. 31

Strategy: Use Perimeter equation P= 2(l+w) first add width and length given then multiply by 2 and equal it to perimeter given. 2x+5 + (x-4) =3x+1 2(3x+1) = 80 6x + 2 = 80 6x = 78 x = 13 Next, plug x into length equation: 2x+5 2(13) + 5 = 26+ 5 = 31

Which of the following fractions has the smallest value? a. 1/3 b. 2/5 c. 5/8 d. 9/11 e. 13/20

Strategy: cross multiply with first fraction, compare fraction to every single fraction with opposite number over top of it a. 1/3

What percent is 1/3 of 3/4? a. 25% b. 33.3% c. 44.4% d. 13.3% e. 60%

Strategy: is/ of (1/3) / (3/4) = 1/3 x 4/3 =.4444 =44.4%

What percentage of a foot is 5 centimeters?

Strategy: is/of = x% /100 5cm/1ft = x% /100 Step 1: convert cm ----->ft Gameplan: cm---->inch--->ft 5cm x (1 in. /2.54cm) x (1ft/ 12 in.) =.164 = 16.4%