GRE Math
If 10800 is invested at a simple interest rate of 4%, what is the value of the investment after 18 months?
$11,448
Hector invested $6000. Part was invested in account with 9% simple annual interest, and the rest in account with 7% simple annual interest. If he earned $490 in the first year of these investments, how much did he invest in each account?
$3,500 in the 9% and $2,500 in the 7%.
The number of strawberries 100 people ate per year is normally distributed with a mean of 29 strawberries, and a standard deviation of 4 strawberries eaten by each person. How many of the surveyed people ate more than 25 strawberries during the course of the year? A) 16 B) 48 C) 50 D) 68 E)84
%ppl eating more than mean 29 strawberries = 50% 68% of people fall within +/-1SD of the mean, so 1SD=34%. %ppl eating more than 25 strawberries = mean + 1SD = 50+34 = 84%. Answer E.
Negative bases
(-2)^4 = (-2)(-2)(-2)(-2) = 16 (-2)^5 = -32
f(x)=x-1 What happens when x is negative? Is (-x) -1 = -x-1? OR -(x-1) = -x+1?
(-x) -1 = -x-1
If (1/a)<(1/b)<0, then which of the following inequalities are true? A) a+b < ab B) |a| > |b| C) a < b D) (b/a) + (a/b) > 2
(1/a)<(1/b)<0 (1/a) and (1/b) must both be negative. Pick numbers: -1/2 < -1/4. If a=-2 and b=-4, then b<a. A) If a<0, b<0, then a+b<0 and ab>0, thus a+b<ab is TRUE. B) lal=l-2l=2, and lbl=4, so lbl>lal ; thus lal>lbl is false. C) a=-2 and b=-4, so b<a ; thus a<b is false. D) a^2+b^2 > 2ab, so (b/a)+(a/b)>2 is TRUE.
(12√15) / (2√5) =
(12/2) x (√15 / √5) = 6√3
What is the average of all the integers from 13 to 77?
(13 + 77) / 2 = 90/2 = 45
Simplify 4√21 X 5√2 / 10√7
(4x5/10) × √(21x2/7)=2√6
(6√3) x (2√5) =
(6 x 2)(√3 x √5) = 12√15
The Associative Property
(a + b) + c = a + (b+c) (a x b) x c = a x (b x c)
Simplify the expression (b² - c²) / (b - c)
(b + c)
Volume of a rectangular solid
(length)(width)(height)
The sum of of the angles in a polygon with n sides
(n-2) x 180
Simplify the expression p² - q²/ -5(q - p)
(p + q)/5
If |x - 3| < 2, what is the range of possible values of x?
(x -3) < 2 and (x-3) > -2 x < 5 and x > 1 So, 1 < x < 5
(x+y)²
(x+y)(x+y)
(x-y)²
(x+y)(x-y)
The lengths of two sides of a triangle are 22 and 11. A) 14 B) The length of the third side Is A or B bigger, C equal, D undetermined?
*The sum of the two shorter sides of a triangle cannot be less than the longest side, or else the triangle is not closed. Imagine 22 was the hypotenuse and base. Side x is greatest when the height is greatest, meaning a right triangle. 22^2 - 11^2 = 19^2, which is greater than 14. As the height shrinks, and the triangle becomes squatter/flatter. Side 11 is fixed, but Side X shrinks. The smallest that Side X can be is slightly > 22-11 = 11. Therefore 11< Side X < 19. Answer D.
Number is a multiple of 6
- Sum of digits is a multiple of 3 - the last digit is even.
Number is a multiple of 4
- last 2 digits are a multiple of 4. - eg. 3052 - 52 = 4*13
What is the maximum value for the function g(x) = (-2x²) -1?
-1
-3³
-27
6w² - w - 15 = 0
-3/2 , 5/3
For what values should the domain be restricted for the function f(x) = √(x + 8)
-8
X is the opposite of
-X
If a>b then
-a<-b
The ratio of the areas of two similar polygons is ...
... the square of the ratios of the corresponding sides.
How many digits are there between the decimal point and the first even digit in the decimal equivalent of 1/[(2^8)(5^3)]
0
1ⁿ
1
A cylinder has a surface area of 22pi. If the cylinder has a height of 10, what is the radius?
1
A cylinder has surface area 22pi. If the cylinder has a height of 10, what is its radius?
1
Given d,e,f ≠0; d³ef⁵/2de³ / 3d²e³f⁷/6e⁵f²?
1
Evaluate 4/11 + 11/12
1 & 37/132
Probability of E not occurring:
1 - P(E)
Prime Numbers
1 is not prime, 2 is the lowest and only even prime number. 2, 3 , 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
n>0. The remainder when n is divided by 3 is 1, and the remainder when n+1 is divided by 2 is 1. A) The remainder when n-1 is divided by 6. B) 3 Is A or B bigger, C equal, D undetermined?
1) (n-1)/3 is an integer, so n could = 4, 7, 10, 13, 16, ... 2) (n+1-1)/2 = n/2, so n must be even. Revise 1) to include n = 4, 10, 16, ... A) What is remainder when (n-1)/6? If n=4, (4-1)/6=3/6=3. If n=10, (10-1)/6=9/6=1 3/6. So A) 3 and B) 3, therefore C is correct.
Perfect Squares 1-15
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Factorization
1. Find the Prime Factors 2. Add one to each of the exponents 3. Multiply together
PEMDAS
1. Parentheses 2. Exponents 3. Multiplication/Division 4. Addition/Subtraction
Write 10,843 X 10⁷ in scientific notation
1.0843 X 10¹¹
8.84 / 5.2
1.7
Fast Fractions
1/2 + 1/5 = 2 + 5/2 x 5 = 7/10
Work Rate
1/Total Work = 1 / Workrate 1+ 1/Workrate 2
a⁻¹/a⁵
1/a⁶
If [x^-2]([-y^2]x+1)=48, and if y^2=2, then what is a possible value for x^2? A. 34/70 B. 0.7655 C. 2/128 D. 0.9822 E. 12/63
1/x^2(-2x+1)=48 -2x+1=48x^2 0=48x^2 + 2x -1 = (6x+1)(8x-1) x=-1/6 or 1/8 x^2=1/36 or 1/64 C) 1/62 x 2/2 = 2/128
x⁻ⁿ =
1/xⁿ i.e. 5⁻³ = 1/(5³) = 1/125 = .008
Legs 6, 8. Hypotenuse?
10
A line has end points -2 and 10. What is the midpoint coordinate of the line?
10 - (-2) = 12. Midpoint = 12/2 = 6 Midpoint coordinate = 10 - 6 = 4
There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. In how many ways can the judges award the 3 prizes?
10! / (10-3)! = 720
There are 10 finalists for the school spelling bee. A first, second, and third place trophy will be awarded. How many different people can get the three prizes?
10! / 3!(10-3)! = 120
10<all primes<20
11, 13, 17, 19
From a box of 12 candles, you are to remove 5. How many different sets of 5 candles could you remove?
12! / 5!7! = 792
1/8 in percent?
12.5%
The perimeter of a square is 48 inches. The length of its diagonal is:
12sqrt2
Legs 5, 12. Hypotenuse?
13
What is the surface area of a cylinder with radius 5 and height 8?
130pi
A triangle is inscribed in a semi circle with legs 5 and 12. What is the circumfermence of the semicircle?
13pi / 2
What percent is 0.144?
14.4%
1/6 in percent?
16.6666%
Which is greater? 64⁵ or 16⁸
16⁸ 64⁵ = (4³)⁵ = 4¹⁵ 16⁸=(4²)⁸ = 4¹⁶
If the 80th percentile of the measurements is 72degrees, about how many measurments are between 69 degrees and 72 degrees? Round your answer to the nearest tenth
18
What is the ratio of the sides of an isosceles right triangle?
1:1:sqrt2
What is the ratio of the sides of a 30-60-90 triangle?
1:sqrt3:2
What is the coefficient of the x² term in the product of (x + 1)(x + 2)(x -1)?
2
Evaluate and write as a mixed number: 2/7 - 3/21 + 2 & 4/14
2 & 3/7
First 10 prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
In a triangle where the two legs are 4 and 3, what is the value of a line directly intersecting the middle coming from the meeting point of the two legs?
2.4. We calculate the area (6) and then turn the triangle on its side and use x as the height to calculate again. (5x)/2=6
A brick with dimensions 10. 15 and 25 weighs 1.5 kg. A second brick (same density) has dimensions 12, 18, and 30. What is the weight of the second brick?
2.592 kg
If a=-1 and b=3, what is the value of (4a³b² - 12a²b⁵) / 16a³b²?
20.5
20<all primes<30
23, 29
5 < sqrt(n) < 6 A) n^2 B) 500
25 < n < 36 625 < n^2 < 1296 A is larger.
Which is greater? 27⁻⁴ or 9⁻⁸
27⁻⁴
If 8 schools are in a conference, how many games are played if each team plays each other exactly once?
28. n = 8, k = 2. n! / k!(n-k)!
How many 3-digit positive integers are even and do not contain the digit 4?
288 (8 * 9 * 4)
Nine coins are tossed simultaneously. In how many of the outcomes will the fourth coin tossed show heads?
2^9 / 2 = 256
Circumference of a circle
2πr
Simplify 9^(1/2) X 4³ X 2⁻⁶?
3
x² = 9. What is the value of x?
3, -3
Pythagorean Triples
3-4-5 5-12-13 8-15-17 7-24-25 The multiple of a Pythagorean triple is also a triple
30< all primes<40
31, 37
In a Regular Polygon, the measure of each exterior angle
360/n
In any polygon, all external angles equal up to
360°
The sum of the angles in any quadrilateral is
360°
3/8 in percent?
37.5%
200<x<300. How many values of x are divisible by 5 & 8?
3: 200, 240, 280
Divisibility
3: Sum of digits divisible by 3 4: The last two digits of the number are divisible by 4 5: The last digit is either a 5 or a 0 6: Even number and sum of digits is divisible by 3 8: Last 3 digits are divisible by 8 9: Sum of the digits is divisible by 9
If r, t, s & u are distinct, consecutive prime numbers, less than 31, which of the following could be an average of them (4, 4.25, 6, 9, 24, 22, 24)
4.25, 6, 22
Evaluate (4³)²
4096
40 < all primes<50
41, 43, 47
How many integers are there from 73 through 419, inclusive?
419 - 73 + 1 = 347
A company places a 6-symbol code on each product. The code consists of the letter T, followed by 3 numerical digits, and then 2 consonants (Y is a conson). How many codes are possible?
441000 = 1 * 10 * 10 * 10 * 21 * 21
If 4500 is invested at a simple interest rate of 6%, what is the value of the investment after 10 months?
4725
Employee X is paid 19.50 per hour no matter how many a week. Employee Y earns 18 for the first 40 and 1.5 the hourly wage for every hour after that. If both earned the same amount and worked the same in one week, how many did each work?
48
In similar hexagons, the ratio of the areas is 16:25. What is the ratio of their corresponding sides?
4:5
For similar triangles, the ratio of their corresponding sides is 2:3. What is the ratio of their areas?
4:9. The ratio of the areas of two similar triangles equals the square of the ratio of the corresponding sides.
Simplify (a² + b)² - (a² - b)²
4a²b
What is the side length of an equilateral triangle with altitude 6?
4sqrt3. The triangle can be divided into two equal 30-60-90 triangles with side 6 as the side in which 6 = xsqrt3. So x =2sqrt3...
Legs: 3, 4. Hypotenuse?
5
25^(1/2) or √25 =
5 OR -5
If Madagascar's exports totaled 1.3 billion in 2009, and 4% came from China, what was the value in millions of the country's exports to China?
52
50 < all primes< 60 and mult
53, 59 note 57=3x19
What is the area of a regular hexagon with side 6?
54√3. (divide the hexagon into 6 congruent equilateral triangles.
What percent of 40 is 22?
55%
10⁶ has how many zeroes?
6
Reduce: 4.8 : 0.8 : 1.6
6 : 1 : 2
60 < all primes <70
61, 67
5/8 in percent?
62.5%
What are the smallest three prime numbers greater than 65?
67, 71, 73
Convert 0.7% to a fraction.
7 / 1000
What is the third quartile of the following data set: 44, 58, 63, 63, 68, 70, 82
70
70 < all primes< 80
71, 73, 79
What is the measure of an exterior angle of a regular pentagon?
72
What number between 70 & 75, inclusive, has the greatest number of factors?
72
What is the ratio of the surface area of a cube with an edge of 10 to the surface area of a rectangular solid with dimensions 2, 4, and 6?
75:11
5/6 in percent?
83.333%
7/8 in percent?
87.5%
-3²
9
Ratio of ages of Anna and Emma is 3:5 and of Emma and Nicolas is 3:5. What is the ratio of Anna to Nicholas' ages?
9 : 25
The four angles around a point measure y, 2y, 35 and 55 respectively. What is the value of y?
90
Find the surface area of a cylinder with radius 3 and height 12.
90pi
2³×7³
= (2x7)³ = 21³
Given: original price, new price Find: % increase / decrease ex: If a lamp increases from $80 to $100, what is the percent increase?
= (actual increase or decrease/original amount) x 100% = 20/80 x 100% = 1/4 x 100% = 25%
2⁵x2³
= 2⁵⁺³ = 2⁸
2⁵/2³
= 2⁵⁻³ = 2²
3√4 - √4?
= 3(2) - 2 = 6-2 = 4 - Yes, like radicals can be added/subtracted.
(2²)³
=2²*³ =2⁶
1:√3:2 is the ratio of the sides of what kind of triangle?
A 30-60-90 triangle.
What is a central angle?
A central angle is an angle formed by 2 radii.
What is a chord of a circle?
A chord is a line segment joining two points on a circle.
What transformation occurs if point C is reflected over the x-axis and then the y-axis?
A reflection about the axis.
What is it called when a point is reflected to the quadrant opposite it (i.e. I to III or II to IV)?
A reflection about the origin.
What is a finite set?
A set with a number of elements which can be counted.
What is the empty set?
A set with no members, denoted by a circle with a diagonal through it.
What is the name for a grouping of the members within a set based on a shared characteristic?
A subset.
What is a tangent?
A tangent is a line that only touches one point on the circumference of a circle.
Define a "term",
A term is a numerical constant or the product (or quotient) of a numerical constant and one or more variables. (3x, 4x² and 2a/c)
A) (41)^2 - (21)^2 B) (41-21)^2 Is A or B bigger, C equal, D undetermined?
A) a^2 - b^2 = (a+b)(a-b) 41^2 - 21^2 - (41+21)(41-21)=62x20 B) (41-21)^2 = 20x20 A is bigger. No need to multiply out.
A) sqrt(2) B) cuberoot(3) Is A or B bigger, C equal, D undetermined?
A) sqrt(2) and B) cuberoot(3) Square both sides: A) 2 and B) 3^(2/3) Cube both sides: A) 8 and B) 3^2 =9 B is bigger!
Let m = odd integer, n = even integer. List all the following that must be even. A) nm^2 B) n+m C) 3n - 2m D) n - m E) nm
A, C, E A number is even only when it has a factor or 2.
The length of rectangle B is 10% less than the length of rectangle A. The width of rectangle B is 10% greater than the width of rectangle A. A) Area of A B) Area of B
A: l x w = area B: length of rectangle B is 10% less width of rectangle B is 10% greater B: 0.9l x 1.1w = 0.99area A is greater.
Alan is 3 times as old as Betsy. If Alan were 11 years older and Betsy were 1 year older, then Alan would be 5 times as old as Betsy. How old is Alan right now?
A=3B (A+11)=5(B+1) A = 9 years If having trouble writing an equation, try numbers: B=2, A=3B=6. A+11=17 B=3, A=9. A+11=20 B=4, A=12. A+11=23 B=5, A=15. A+11=26 B=6, A=18.. A+11=29 Which of these is 5(B+1)=A+11?
Given five letters, A B C D E, how many ways can you select 3 of five letters?
ABC ABD ABE ACD ACE ADE BCD BCE BDE CDE 10 ways Number of ways to order all five: n! = 5! Number of ways to pick and order 3 for 3 positions: n!/(n-k)! = 5!/2! = 5x4x3x2!/2! = 5x4x3 = 60 Number of ways to pick any 3 for 3 positions: n!/(n-k)! k! = 5!/2!3! = 5x4x3!/2x1x3! = 20/2 = 10
Multiplying numbers with exponents (same base)
Add them. i.e. 5⁷ * 5³ = 5¹⁰
What are the integers?
All numbers multiples of 1.
What are the rational numbers?
All numbers which can be expressed as a ratio of two integers. (All integers and fractions.) (-2, 1, .25, 1/2)
What are the irrational numbers?
All real numbers which can't be expressed as a ratio of two integers, positive and negative (pi, -sqrt3)
What are the real numbers?
All the numbers on the number line (negative, rational, irrational, decimal, integer). All the numbers on the GRE are real. (-2, 1, .25, 1/2, pi)
The distance from (b,2) to (-2,2) is 3. A) b B) 0
Always be careful when 0 is an option. Check -ve. On a graph, b is somewhere on the line y=2. b could be 1 or -5. Ans D.
Define an "expression".
An algebraic expression is a combination of one of more terms. Terms in an expression are separated by either addition or subtraction signs. (3xy, 4ab, -5cd, x² + x - 1)
What is an exterior angle?
An angle which is supplementary to an interior angle.
What is an arc of a circle?
An arc is a portion of a circumference of a circle.
Define a "monomial"
An expression with just one term (-6x, 2a²)
What is the name of set with a number of elements which cannot be counted?
An infinite set.
Intergers
Any counting number including negative numbers
Polygon
Any figure with three or more sides
Power of O
Any nonzero number raised to the power of 0 is 1
Perfect Square
Any number that can be multiplied by the two same numbers to create the value. (Exp. 7x7 = 49 therefore the square root of 49 = 7)
Formula to calculate arc length?
Arc length = (n/360) x 2πr where n is the number of degrees.
A square is inscribed in a circle. If the area of the square is 4, what is the radius of the circle?
Area of square = 4, therefore side =2. The diagonal of the square = diameter of the circle. Also, it is a right triangle with side ratio 1:1:sqrt2, therefore 2:2:2sqrt2 and diameter is 2sqrt2, therefore radius = sqrt2.
Which of the following inequalities have at lease one positive and at least one negative solution for x? Indicate all posibilities. A) (5/3)x < x B) x^3 < x C) x-6 < x-7
B
Triangle ABC has points C=(0,0), B=(6,0), and A=(x,y). The area of ABC is 18. A) y B) 6 Is A or B bigger, C equal, D undetermined?
Base = Bx-Cx = 6-0=6 Area=1/2 x base x height = 18 Height = 36/6 = 6 = l Ay - x-axis I = lAy - 0l = l6l y could be -6 or 6. D undetermined.
X@Y = X^2 - Y A) (3 @ 2) @ -1 B) 50
Be careful with negative and subtraction! X@Y = X^2 - Y (3 @ 2) @ -1 3^2 - 2 = 9 - 2 = 7 7^2 - (-1) = 49 + 1 = 50 C.
The Length of the longest side of a triangle
Can never be greater than the sum of the two other sides
How to find the circumference of a circle which circumscribes a square?
Circumference = Diameter(pi). Use pythagorean theorem to find the diagonal of the square (the diameter).
How many 8" squares will it take to fill a rectangular floor measuring 18 feet x 24 feet?
Convert both lengths into inches. It is not enough to do 18 x 24 x 12 because area is length^2. 18x12 = 216" 24x12 = 288" Area = 216 x 288 = 62,208 # of 8" tiles needed = 62,208/64 = 972 tiles
x cannot equal y. x@y=(x+y)/(x-y) If q<0<p: A) p@q B) q@p Is A or B bigger, C equal, D undetermined?
D If lql = lpl, then both p@q and q@p = 0, meaning equal. The numerators are the same. Examine the denominators. Since q<0<p, A) p-q >0, and B) q-p <0. Not equal. A would be bigger than B.
n is an integer greater than 0. Option A) The number of different prime factors of 9n. Option B) The number of different prime factors of 8n. Is A or B bigger, C equal, D undetermined?
D) it depends what n is. If n=2: 9n=18 -> (1, 18; 2, 9; 3, 6) = 2,3 8n=16 -> (1, 16; 2, 8; 4, 4) = 2 If n=3: 9n=27 -> (1, 27; 3, 9) = 3 8n=24 -> (1, 24; 2, 12; 3, 8; 4, 6) = 2,3
If x is an integer, which of the following must be even? A) x^2 -x -1 B) x^2 -4x +6 C) x^2 -5x +5 D) x^2 +3x +8 E) x^2 + 2x +10
D) x^2 +3x +8
Time
Distance / Rate
Rate
Distance / Time
A triangle ABC has sides AB=4(sqrt2) and AC=7 and Angle A = 45'. What is the length of BC?
Don't assume that because an angle looks like 90' that it is, unless it is marked. Create at right angle by dropping a perpendicular line (BD) between side AC and angle B. Mini triangle ABD is a right 45:45:90 with hypotenuse 4(sqrt2). Without calculation, 1:1:sqrt2, therefore sides AD and BD are 4. Side DC = ADC-AD=7-4=3 Mini triangle BCD has sides 3:4:BC, therefore BC=5.
If n=14x22x39, which is not an integer? A) n/21 B) n/24 C) n/26 D) n/42 E) n/77
Don't multiply out. Find the factors 14: 2x7 22: 2x11 39: 3x13 Determine if the denominators have a common factor: A) n/21: n/7x3 OK B) n/24: n/2x12 NOPE! C) n/26: 2x13 OK D) n/42: 6x7 = 2x3x7 OK E) n/77: 7x11 OK
If it takes three days for 10 workers to build one house, how long will it take 15 workers to build four houses? 15, 10, 8, 6, 4 days?
Don't try to set up ratio or equations. Think logically. For one house, 10 workers need 3 days; 15 workers should be faster: 3x10/15=2 days per house. There are 4 houses, so 2x4=8 days.
Inscribed angles holding chords/arcs of equal length in a circle are
Equal
Is 0 even or odd?
Even
Whats the difference between factors and multiples?
Factors are few, multiples are many.
If 3.2 = -n + 0.8m and 0.4m = -0.3n + 3.2, then what is the value of -n? a) 2 b) 1 c) 0 d) -1 e) -2
Find -n, NOT n. -n = -2. E, not A.
If given two points and asked to find the equation of a line that passes through them
First, find the slope using the formula for slope. Then plug one of the points into y = mx + b and slove for b.
x>2 A) 3^(x+2) B) 4^x Is A or B bigger, C equal, D undetermined?
For small values of x, A > B. If x=3, then A) 3^5 =243, and B) 4^3 = 64. However as two integers are raised to increasing integer powers, the larger base number will eventually outgrow the smaller base number. If x=10, then A) 3^12 = 531,441, and B) 4^10 = 1,048,576.
Which quadrant is the upper right hand?
I
Which quadrant is the upper left hand?
II
Which quadrant is the lower left hand?
III
Which quandrant is the lower right hand?
IV
X is a non-negative integer. A) 2X B) 3X Is A or B bigger, C equal, D undetermined?
If X>0, then 3X>2X always. But 0 is a non-negative integer! If X=0, 2X=3X=0. Answer D.
A) The area of a rectangle with perimeter 24 B) 20 Is A or B bigger, C equal, D undetermined?
If sides = 11+11+1+1=24, Area = 11. If sides = 10+10+2+2=24, Area = 20. If sides = 8+8+4+4=24, Area = 32 If sides = 6+6+6+6=24, Area = 36. A Square is also a rectangle. Area can be stretched thin, or blown up.
P and r are factors of 100. What is greater, pr or 100?
Indeterminable.
How many multiples does a given number have?
Infinite.
To find the degree measure of an arc of a circle
Look to the central angle
k is an integer for which 1/(2^(1-k)) < 1/8. Option A) k Option B) -2 Is A or B bigger, C equal, D undetermined?
NOT 2^(1-k) < 8. Be careful with reciprocals. For 1/(2^(1-k)) to be less than 1/8, the denominator of 2^(1-k) must be greater than 8. Thus: 2^(1-k) > 8 = 2^3 1-k >3 -k > 2 k < -2 therefore B!
3.7, 4.1, a, 8.5, 9.2, 2a The six numbers above are listed in increasing order. Which of the following values could be the range of the six numbers? Indicate all values. a) 4.0 b) 5.2 c) 7.3 d) 11.6 e) 12.9 f) 14.1
NOT which could be "in" the range, but what IS the range? Range = biggest - smallest = 2a - 3.7 The biggest value for 2a is when a<8.5. 2a<17. Range = 17-3.7 = 13.3. Therefore F) 14.1 is not possible. Anything less than E may be possible. The smallest values for a<4.1, but 2a cannot be less than 9.2. The smallest range possible is when 2a is a little bigger than 9.2. 9.2-3.7 =5.5. Therefore the possible ranges are 5.5 < Range < 13.3, which is C D E.
Square Root of Negative
No real solution
√3 + √5?
No, only like radicals can be added.
Germany's Gasoline tax = 20.4%, Income tax = 28.6%. The amount of Germany's gasoline tax revenue was approximately what percent less than the amount of its income tax revenue? a) 10% b) 20% c) 30% d) 40% e) 50%
Not 28.6-20.4 = 8.2% (change in values)/starting value = % 8.2/28.6 = 30% C.
From a box of 10 lightbulbs, you remove 4. How many different sets of 4 lightbulbs could you remove?
Number if all were ordered = 10! Number if first four were picked in order = 10!/(10-4)! = 10!/6! = 10 x 9 x 8 x 7 = 5040 But order doesn't matter = 10!/(10-6)!4! = 5040/(4x3x2x1) = 210 different sets
Real Number
Numbers that appear on the number line including pi, the square root of 2, etc.
Let A, B, C, D be events for which P(A or B)=0.6, P(A)=0.2, P(C or D)=0.6, and P(C)=0.5. The events A and B are mutually exclusive, and the events C and D are independent. What is P(D)?
P(A)=0.2 P(B)=P(A or B) - P(A)=0.6-0.2=0.4 P(C)=0.5 P(CuD)=P(C)+P(D)-P(CnD)=0.6 0.5+P(D)-[P(C)xP(D)]=0.6 P(D)[1-0.5]=0.1 P(D)=0.1/0.5=0.2
Probability of an Event
P(E) = number of favorable outcomes/total number of possible outcomes
45 is what percent of 9?
Part = Percent x Whole 45 = Percent x 9 45/9 = Percent 5 = percent 500% = percent
What is 12 percent of 25?
Part = Percent x Whole Part = .12 x 25 Part = 3
15 is 3/5 percent of what number?
Part = Percent x Whole 15 = 3/5 x Whole 15 = .60 x Whole 15 = .006 x Whole 15/.006 = Whole 2,500 = Whole
The price goes up from $80 to $100. What is the percent increase?
Percent Change = (Amount of Change/Original Whole) x 100 Increase = (20/80) x 100 Increase .25 x 100 Increase = 25%
2x/3 = 2y/5 = 2z/7 z is positive. A) x+y B) z Is A or B bigger, C equal, D undetermined?
Pick numbers. If z = 7, then 2z/7 = 2. Then 2x/3=2, thus x=3. Likewise y=5. A) x+y = 3+5 = 8 B) z = 7 Thus A is bigger.
a is positive, aⁿ is
Positive
Distance
Rate x Time
Output
Rate x Time
Which is greater? 200x²⁹⁵ or 10x²⁹⁴?
Relationship cannot be determined (what if x is negative?)
A circle of area of 9Π cm2 has a square inscribed inside of it, so that each of the four corners of the square lie on the circle. What is the area of the square?
Remember Diameter = 2 radius = 6 Ans 18cm^2
Adding Roots
Roots can be added the same way you would variables
S and T are positive and follow S=k/T, where k is a constant. If S increases by 50%, then the value of T decreases by what percent?
ST=k E.g. S=10, T=1, k=10 If S=15, k=10, then T=k/S=10/15=2/3 NOT 66.7%! By what PERCENT has T decreased? T'=1, T''=2/3. % decrease = (T'-T")/T'=1/3 33.3%!
Formula for the area of a sector of a circle?
Sector area = (n/360) X πr²
Relationship between lengths of the sides of a triangle and the measure of the angles of the triangle
Sides with the same lengths are opposite angles with the same measure.
Pat invested a total of $3000. Part of it was invested in an account that had 10% simple annual interest, and the remainder was in another account that paid 8% simple annual interest. If the interest earned at the end of the first year from these investments was $256, how much did Pat invest at 10% and how much at 8%?
Simultaneous equations: A+B=3000 0.1A + 0.08B = 256 --> A+0.8B=2560 B-0.8B = 0.2B = 3000-2560 = 440 --> B = 2200 A = 800
x<y<0 A) x/y B) y/x Is A or B bigger, C equal, D undetermined?
Since both x and y are negative, their ratios will be positive. Since x is more negative (i.e. bigger) than y, then A) x/y will always >1, whereas B) y/x will be a fraction <1. Check with negative fractions! Answer A
Dividing numbers with exponents (same base)
Subtract them i.e 5⁷/5³= 5⁴
What is the sum of the integers from 10-50 inclusive?
Sum = Average x Number of Terms Sum = ((10+50)/2) x (50 - 10 + 1) Sum = 30 x 41 Sum = 1,230
17.5 is the average (arithmetic mean) of 24 numbers. What is the sum of the 24 numbers?
Sum = Average x Number of Terms Sum = 17.5 x 24 Sum = 420
Absolute Value
The absolute value of a number is its distance from a number line.
Describe the relationship between 3x² and 3(x - 1)²
The graph of 3(x - 1)² is a translation (shift) of the graph one unit or space to the right.
What is the "range" of a series of numbers?
The greatest value minus the smallest.
The side opposite the right angle in a right triangle is
The hypotenuse and is always the longest side
The larger the degree of the angle of a triangle
The larger the length of the opposite side
Two sides of a triangle T are 3 and 4. A) The third size B) 8 Is A or B bigger, C equal, D undetermined?
The largest side cannot be more than the sum of the two smaller sides, otherwise it would be incomplete. The largest that Side X can be < 3+4 = 7, which is less than 8. Answer B.
What is a major arc?
The longest arc between points A and B on a circle's diameter.
If 6 builders can build a house in 45 days, how long will it take 15 builders to construct the same house?
The number of days needed to complete building a house varies inversely as the number of builders employed. If t is time in days and n is the number of men, then t and n are related by t = (k/n), where k is the work that each man can perform. When t = 45 and n = 6, then k = 45×6 = 270, so it would take 270 days for 1 man to build a house. t = 270/15 = 18.
What are the members or elements of a set?
The objects within a set.
What is the "solution" for a set of inequalities.
The overlapping sections.
What is the "solution" for a system of linear equations?
The point of intersection of the systems.
Describe the relationship between the graphs of x² and (1/2)x²
The second graph is less steep.
What is the intersection of A and B?
The set of elements found in both A and B.
What is the "union" of A and B?
The set of elements which can be found in either A or B.
What is the "domain" of a function?
The set of input values for a function.
What is the "range" of a function?
The set of output values for a function.
What is a minor arc?
The shortest arc between points A and B on a circle's diameter.
Number is a multiple of 3
The sum of the digits is a multiple of 3 (i.e. 45 ... 4 + 5 = 9 so the whole thing is a multiple of 3)
Number is multiple of 12
The sum of the digits is a multiple of 3; the last two digits are multiples of 4. Ex: 144 - 1+4+4=9 - a multiple of 3 - 44 - multiple of 4
What are the roots of the quadrinomial x² + 2x + 1?
The two Xs after factoring.
If two inscribed angles hold the same chord in a circle
The two inscribed angles are equal
After 5 adults leave a party, there are three time as many children as adults. After a further 25 children leave the party, there are twice as many adults as children. A) The original number of adults B) 14 Is A or B bigger, C equal, D undetermined?
The wordy question is difficult to set up as an equation. You only need to compare the original number of adults to 14. 14-5=9 adults, so children=9x3=27. After 27-25=2 children left, so then 2x2=4 adults left, which does not equal 9. Pick either a smaller or bigger than 14. 13-5=8 adults, so children=8x3=24. After 24-25=-1 children left. Doesn't make sense, therefore adults > 14. (A)
Of the 180 people, 30% were women, and 25% were from minority groups. If 1/9 of the women are from minority groups, how many were neither women nor from minority groups? a) 75 b) 81 c) 87 d) 93 e) 99
Total minority = 0.25 x 180 = 45 Minority = Minor women + Minor men Minor women = 0.3 x 1/9 x 180 = 6 Minor men = Minority - Minor women = 45 - 6 = 39 Men = (180x0.7) - 39 = 126 - 39 = 87 C.
What are congruent triangles?
Triangles with same measure and same side lengths.
True or false? 4.809 X 10⁷ = .0004809 X 10¹¹
True
What are "supplementary angles?"
Two angles whose sum is 180.
What are complementary angles?
Two angles whose sum is 90.
For all integers x, the function f is defined as: f(x)=(x-1) when x is even. f(x)=(x+1) when x is odd. If a and b are integers, and f(a)+f(b)=a+b, which of the following statements must be true? A) a=b B) a=-b C) a+b is odd D) Both a and b are even. E) Both a and b are odd.
Use all pieces of the question to help: f(a)+f(b)=a+b A) If a=b, whether or not a/b is even or odd, f(a)+f(a)=2a-2 OR 2a+2, which is not a+b=2a B) If a=-b and even, f(a)+f(-a)=(a-1)+(-a-1)=-2, which is not 2a. If a=-b and odd, it would be the same result -2. Rules out D and E. C) If a+b is odd, one must be even and the other must be odd. Then f(a=even)+f(b=odd)=(a-1)+(b+1)=a+b. The 1 + -1 = 0.
I expect to earn at least $1000 in interest after an initial investment of $20K. If I invest for one year when interest is compounded quarterly, what is the least annual interest rate to achieve the goal?
Value of investment = (Principal investment)(1+r/100n)^nt r = annual interest rate. n = compounded n times per year. t = years 21000 < 20000 (1+r/400)^4 21/20 < (1+r/400)^2 (sqrt(sqrt(1.05))) < 1+r/400 1.0122 - 1 < r/400 r > 4.91
Golden rule of solving equations
What you do to one side, you must do to the other
When does a function automatically have a restricted domain (2)?
When we need to avoid having a zero in the denominator or avoid taking the square root of a number.
Exponent Laws
X^A x X^B = X^(A + B) X^A / X^B = X^(A-B) (X^A)^B = X^(A x B)
5x² - 35x -55 = 0
[(7+ √93) /2], [(7 - √93) / 2]
If an integer is randomly selected from all positive 2-digit integers, what is the probability that there is: 1) a 4 in the tens place? 2) at least one 4 in the tens place or the units place?
_ _ Tens place possibilities: 1-9 Units place possibilities: 1-10 1) P(4 in the tens place) = 1/9 2) P(at least one four) = P(A) + P(B) - P(AnB) = 1/9 + 1/10 - (1/9x1/10) = (10 + 9 - 1)/90 = 18/90 = 1/5
How many 3-digit positive integers are odd and do not contain the digit 5?
_ _ _ First digit possibilities: 1-9 Second digit possibilites: 1-10 Third digit possibilities: 1-10 Third digit odd, not 5 possibilities: 1, 3, 7, 9 Probability (first digit is not 5) = 8/9 Probability (second digit is not 5) = 9/10 Probability (third digit is odd, not 5) = 4/10 Probability = 8/9 x 9/10 x 4/102 = 288/900 Possibilities = 288
Commutative Property
a + b = b + a A x B = B x A
What is a subset?
a grouping of the members within a set based on a shared characteristic.
An equation like y = 4 is
a horizontal line at y = 4
An equation like x = 3 is
a vertical line at x = 3
The Distributive Property
a x (b+c) = ab + ac a x (b - c) = ab - ac
Describe the domain, x and y intecepts of: a) y=2lx-1l b) y=100-900x c) y=5-(x+20)^2 d) y=(x+2)^1/2 e) y=x+lxl
a) y=2lx-1l is lxl shifted to the right by 1, and stretched vertically by a factor of 2. Domain: all real numbers. b) y=100-900x is a line with slope -900, y-cept=100, and x-cept=1/9. Domain: all real numbers c) y=5-(x+20)^2 is a parabola pointing down with vertex (-20,5), y-cept=-395, x-cept=-20+/-sqrt(5). Domain: all real numbers d) y=(x+2)^1/2 is a half parabola opening to the right with vertex (-2,0), x-cept=-2, y-cept=sqrt(2). Domain: numbers >= -2. e) y=x+lxl. For x>=0, y=2x. For x<0, y=0. Two lines, one is y=2x, and second is along x-axis for x<0. Domain: all real numbers.
If a<b, then
a+c<b+c
The length of the shortest side of a triangle
can never be less than the positive different of the other two sides
What is the common monomial factor in the expression 4c³d - c²d² + 2cd?
cd
If cube root(x) = 3, and x=square root (y), what is y?
cube root(x) = 3 --> x=3^3 = 27. 27=square root (y) --> y = 27^2 = 729.
Given: time and rate Find: distance
distance = rate*time rate = distance/time time = distance/rate
formula for distance problems
distance=rate×time or d=rt
What is the graph of f(x) shifted left c units or spaces?
f(x + c)
What is the graph of f(x) shifted upward c units or spaces?
f(x) + c
What is the graph of f(x) shifted downward c units or spaces?
f(x) - c
What is the graph of f(x) shifted right c units or spaces?
f(x-c)
An inscribed angle holding the diameter of a circle
is a right angle
Slope given 2 points
m= (Y2-Y1)/(X2-X1) ex: (1,3); (5,6) m = (6-3) / 5-1) = 3/4
The objects in a set are called two names:
members or elements
To multiply a number by 10ⁿ
move the decimal point to the right n places
Suppose you have a set of n objects, and you want to select k of them, but the order doesn't matter. What formula do you use to determine the number of combinations of n objects taken k at a time?
n! / (k!)(n-k)!
If you have a set of n objects, but you only want to order k of them, what formula do you use to determine the number of permutations?
n! / (n-k)!
Martha invited 4 friends to go to the movies. How many ways can they sit in a row of 5 seats, one person per seat? How many of those ways is Martha sitting in the middle?
n=5, pick and order all 5. 5! = 5x4x3x2x1 = 120 (Number of ways to select without order) x (Number of ways to order) = (Number of ways to select with order) Number of ways with Martha in the middle = (Number of ways to select with order) / (Number of ways to order) = 120/5 = 24
a is negative & n is even > aⁿ is
positive
1 is the
smallest positive integer
The slope of a line can be found by
subtracting the y values of a pair of coordinates and dividing it by the difference in the x values.
Number is a multiple of 9
sum of the digits is a multiple of 9.
What is a set with no members called?
the empty set, denoted by a circle with a diagonal through it.
The larger the absolute value of the slope...
the steeper the slope.
0⁰
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Factor x² - xy + x.
x(x - y + 1)
Consecutive integers
x, x+1, x+2
Square Roots
x^2 = 49 Square root of 49 = 7 X = 7 and X = -7
(x+y)²
x²+2xy+y²
(x-y)²
x²-2xy+y²
(x+y)(x-y)
x²-y²
(x-y)(x+y)
x²-y²
x⁶ / x³
x⁶⁻³ = x³
Suppose that the graph of f(x) is the result of stretching y=x + 5 away from the x-axis by a factor of 2. What is the new equation for the graph f(x)?
y = (x + 5)/2
Suppose that the graph of f(x) is the result of sliding the graph of y=2x² down 3 units of spaces. What is the new equation?
y = 2x² - 3
If x is an integer and if |x-2| > 1, then what are all of the possible values of x? A. x>3 B. x<-3 or x>1 C. x<1 or x>3 D. 1<x<3 E. x>1
|x-2| > 1 Solve: +(x-2)>1 x>1+2 Solve: -(x-2)<1 Don't forget to switch inequality sign! -x+2<1 -x<-1 (if multiplying both sides to remove -1, leave sign intact. If multiply by -1 to move negative to other sign, switch sign!) x<1 So x<1 and x>3, which is C. Also draw a graph!
Area of a circle
πr²
A 30-60-90 triangle
• is a right triangle. • The hypotenuse is twice the length of the shorter leg. • 1, √3, 2 OR x, x√3, 2x
A 45-45-90 triangle
• is a right triangle. • is isosceles (AC=BC). • ratio of the lengths of the three sides is x:x:x√2.
A parallelogram
• opposite angles and sides = same • same side angles add to 180° • area = (base)(height)
Can you simplify √72?
√(36 x 2) = 6√2
A contest has 8 competitors. Judges award prizes for gold, silver and bronze medals, with no ties. 1) How many ways can the judges award prizes? 2) How many different groups of 3 can get prizes?
#ways to order all 8 contestants = n! = 8! #ways to order first 3 prizes = n!/(n-k)! = 8!/(8-3)! = 8!/5! = 8x7x6 = 336 #ways that a group of 3 get prizes (order doesn't matter) = n!/(n-k)!k! = 336/3! = 336/6 = 56
