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%.

(12sqrt15) / (2sqrt5) =

(12/2) x (sqrt15 / sqrt5) = 6sqrt3

2³×7³

(2x7)³

(6sqrt3) x (2sqrt5) =

(6 x 2)(sqrt3 x sqrt5) = 12sqrt15

Factor a^2 + 2ab + b^2

(a + b)^2

a^2 + 2ab + b^2

(a + b)^2

a^2 - b^2

(a - b)(a + b)

a^2 - b^2 =

(a - b)(a + b)

a^2 - 2ab + b^2

(a - b)^2

How to determine percent decrease?

(amount of decrease/original price) x 100%

Simplify the expression [(b^2 - c^2) / (b - c)]

(b + c)

Area of a triangle?

(base*height) / 2

Time

(distance)/(rate) d/r

The negative exponent x⁻ⁿ is equivalent to what?

1/xⁿ i.e. 5^-3 = 1/(5^3) = 1/ 125 = .008

Legs 6, 8. Hypotenuse?

10

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

What does scientific notation mean?

Expressing a number as the product of a decimal between 1 and 10, and a power of 10.

Whats the difference between factors and multiples?

Factors are few, multiples are many.

Probability of an Event

P(E) = number of favorable outcomes/total number of possible outcomes

If Event is impossible

P(E) = ø

Perimeter of a rectangle

P= 2L + 2w

The Perimeter of a rectangle

P=2(l+w)

The Perimeter of a Square

P=4s (s=side)

What is the order of operations?

PEMDAS (Parentheses Exponents Multiplication/Division Addition/Subtraction)

What is the set of elements found in both A and B?

The interesection of A and B.

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)

A number is divisible by 6 if...

its divisible by 2 and by 3.

A number is divisible by 4 is...

its last two digits are divisible by 4.

If the two sides of a triangle are unequal then the longer side.................

lies opposite the greater angle

the slope of a line in y=mx+b

m

Slope given 2 points

m= (Y1-Y2)/(X1-X2)

The objects in a set are called two names:

members or elements

To multiply a number by 10^x

move the decimal point to the right x places

To increase a number by x%

multiply by 1+x% i.e. 100 x (1+50%)=100x1.5=150

To decrease a number by x%

multiply by 1-x% i.e. 100 x (1-50%)=100x.5=50

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)!

Circumference of a circle

pi(diameter)

1 is the

smallest positive integer

If an inequality is multiplied or divided by a negative number....

the direction of the inequality is reversed.

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.

A number is divisible by 9 if...

the sum of digits is divisible by 9.

A number is divisible by 3 if ...

the sum of its digits is divisible by 3.

0^0

undefined

Solve the quadratic equation ax^2 + bx + c= 0

x = [(-b)+/- (sqrt b^2 - 4ac)]/2a

Factor x^2 - xy + x.

x(x - y + 1)

30 60 90

x, x(SR3), 2x

Consecutive integers

x, x+1, x+2

(x^2)^4

x^(2(4)) =x^8 = (x^4)^2

x^4 + x^7 =

x^(4+7) = x^11

x^6 / x^3

x^(6-3) = x^3

(x+y)²

x²+2xy+y²

factored binomial product of (x+y)²

x²+2xy+y²

(x-y)²

x²-2xy+y²

factored binomial product of (x-y)²

x²-2xy+y²

(x-y)(x+y)

x²-y²

binomial product of (x+y)(x-y)

x²-y²

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=2x^2 down 3 units of spaces. What is the new equation?

y = 2x^2 - 3

If y is directly proportional to x, what does it equal?

y/x is a constant

Slope

y₂-y₁/x₂-x₁

Any Horizontal line slope

zero

#3 What is the 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.

#2 What is an important property of a 30-60-90 triangle?

• The hypotenuse is twice the length of the shorter leg.

#1 What is the relationship between lengths of the sides of a triangle and the measure of the angles of the triangle?

• The longest side is opposite the largest (biggest) angle.

#3 What is an important property of a 30-60-90 triangle?

• The ratio of the length of the three sides is x:x√3:2x

#3 What are the important properties of a 45-45-90 triangle?

• The ratio of the lengths of the three sides is x:x:x√2.

#2 What is the relationship between lengths of the sides of a triangle and the measure of the angles of the triangle?

• The shortest side is opposite the smallest angle.

#1 What are the important properties of a 45-45-90 triangle?

• The triangle is a right triangle.

#1 What is an important property of a 30-60-90 triangle?

• The triangle is a right triangle.

#2 What are the important properties of a 45-45-90 triangle?

• The triangle is isosceles (AC=BC).

If a product of two numbers is ∅, one number must be

∅

Product of any number and ∅ is

∅

∅ divided by 7

∅

∅²

∅

The only number that is equal to its opposite

∅ ∅=∅

Probability of Event all cases

∅≤P(E)≤1

Is 0 even or odd?

Even

∅ is

Even

The reciprocal of any non-zero number is

1/x

2 is the only

Even prime number

∅ is a multiple of

Every number

1 is a divisor of

every number

(a^-1)/a^5

1/a^6

The reciprocal of any non-zero #x is

1/x

Volume of a rectangular solid

(length)(width)(height)

The sum of the measures of the n angles in a polygon with n sides

(n-2) x 180

Simplify the expression (p^2 - q^2)/ -5(q - p)

(p + q)/5

Area of a circle

(pi)r²

Distance

(rate)(time) d=rt

binomial product of (x+y)²

(x+y)(x+y)

binomial product of (x-y)²

(x+y)(x-y)

What is the maximum value for the function g(x) = (-2x^2) -1?

-1

-3³

-27

3 is the opposite of

-3

6w^2 - w - 15 = 0

-3/2 , 5/3

For what values should the domain be restricted for the function f(x) = sqrt(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

What is the slope of a horizontal line?

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

The product of any number x and its reciprocal

1

a^0 =

1

b¹

1

Evaluate 4/11 + 11/12

1 & 37/132

Probability of E not occurring:

1 - P(E)

Perfect Squares 1-15

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Write 10,843 X 10^7 in scientific notation

1.0843 X 10^11

8.84 / 5.2

1.7

1/2 divided by 3/7 is the same as

1/2 times 7/3

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

1/6 in percent?

16.6666%

Which is greater? 64^5 or 16^8

16^8 64^5 = (4^3)^5 = 4^15 16^8=(4^2)^8 = 4^16

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

Number of degrees in a triangle

180

If a pair of parallel lines is cut by a transversal that's not perpendicular, the sum of any acute angle and any obtuse angle is

180 Acute Angle an angle that is less than 90° Obtuse Angle:angle that is greater than 90° but less than 180°

What is the sum of the angles of a triangle?

180 degrees

The consecutive angles in a parallelogram equal

180°

the measure of a straight angle

180°

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 x^2 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

Circumference of a circle

2(pi)r

First 10 prime #s

2, 3, 5, 7, 11, 13, 17, 19, 23, 29 A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself

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 (4(a^3)(b^2) - 12(a^2)(b^5)) / (16(a^3)(b^2))?

20.5

20<all primes<30

23, 29

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

Simplify 4sqrt21 X 5sqrt2 / 10sqrt7

2sqrt6

2⁵/2³

2²

(2²)³

2⁶

2⁵+2³

2⁸

200 <_ x <_ 300. How many values of x are divisible by 5 & 8?

3

Simplify 9^(1/2) X 4^3 X 2^(-6)?

3

x^2 = 9. What is the value of x?

3, -3

30 60 90

3, 4, 5

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 all angles around a point

360°

The sum of the angles in a quadrilateral is

360°

3/8 in percent?

37.5%

30 60 90

3x, 4x, 5x

What is the name for a grouping of the members within a set based on a shared characteristic?

A subset.

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 (4^3)^2

4096

40 < all primes<50

41, 43, 47

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 (a^2 + b)^2 - (a^2 - b)^2

4a^2(b)

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 sqrt. 25 =

5 OR -5

30 60 90

5, 12, 13

5 bakeries sell an average of 300 muffins per bakery per day. If 2 stop making muffins but the total muffins sold stays the same, what is the average of muffins per bakery sold among the remaining?

500

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

53, 59

What is the area of a regular hexagon with side 6?

54sqrt3. (divide the hexagon into 6 congruent equilateral triangles.

What percent of 40 is 22?

55%

10^6 has how many zeroes?

6

How many sides does a hexagon have?

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

Evaluate 3& 2/7 / 1/3

9 & 6/7

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

In a Rectangle, each angles measures

90°

The percent decrease of a quantity

= (actual decrease/Original amount) x 100%

If a lamp decreases to $80, from $100, what is the decrease in price?

= (actual decrease/Original amount) x100% = 20/100x100% = 20%

If a lamp increases from $80 to $100, what is the percent increase?

= 25%. = (actual increase/original amount) x 100% = 20/80 x 100% = 1/4 x 100% = 25%

1:sqrt3:2 is the ratio of the sides of what kind of triangle?

A 30-60-90 triangle.

Area of a rectangle

A = length x width

Formula for the area of a circle?

A = pi(r^2)

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.

∅ is

A multiple of every integer

What is a percent?

A percent is a fraction whose denominator is 100.

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 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, 4x^2 and 2a/c)

Area of a triangle

A= (1/2) b*h

Area of a Parallelogram:

A=(base)(height)

Area of a circle

A=pi*(r^2)

formula for area of a triangle

A=½bh

When multiplying exponential #s with the same base, you do this to the exponents...

Add them. i.e. (5^7) * (5^3) = 5^10

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)

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, x^2 + 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, 2a^2)

What is the name of set with a number of elements which cannot be counted?

An infinite set.

1:1:sqrt2 is the ratio of the sides of what kind of triangle?

An isosceles right triangle.

Formula to calculate arc length?

Arc length = (n/360) x pi(2r) where n is the number of degrees.

THE DENOMINATOR CAN NEVER

BE ZERO! 1/∅=null

Formula to find a circle's circumference from its diameter?

C = (pi)d

Formula to find a circle's circumference from its radius?

C = 2(pi)r

For any number x

Can be negative, zero, or positive

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 do you solve proportions? a/b=c/d

Cross multiplication a/b=c/d 4/6=10/15 4(15)=6(10) 60=60

Circumference of a circle?

Diameter(Pi)

Vertical lines

Do not have slopes!

∅ Is

EVEN

P and r are factors of 100. What is greater, pr or 100?

Indeterminable.

How many multiples does a given number have?

Infinite.

One is (a prime or not?)

NOT A PRIME

Slope of any line that goes down as you move from left to right is

Negative

The product of odd number of negative numbers

Negative

Can you add sqrt 3 and sqrt 5?

No, only like radicals can be added.

7 divided by ∅

Null

a/∅

Null

1 is an

ODD number

If E is certain

P(E) = 1/1 = 1

A quadrilateral where two diagonals bisect each other

Parallelogram

Pi is a ratio of what to what?

Pi is the ratio of a circle's circumference to its diameter.

If a is positive, aⁿ is

Positive

Slope of any line that goes up from left to right

Positive

∅ Is neither

Positive or Negative

Positive integers that have exactly 2 positive divisors are

Prime numbers (2, 3, 5, 7, 11, 13, 17, 19, 23)

Dividing by a number is the same as multiplying it by its

Reciprocal

Which is greater? 200x^295 or 10x^294?

Relationship cannot be determined (what if x is negative?)

Describe the relationship between 3x^2 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.

In a rectangle, all angles are

Right

Formula for the area of a sector of a circle?

Sector area = (n/360) X (pi)r^2

An Angle that's 180°

Straight Angle

When dividing exponential #s with the same base, you do this to the exponents...

Subtract them. i.e (5^7)/(5^3)= 5^4

How to recognize a multiple of 6

Sum of digits is a multiple of 3 and the last digit is even.

How to recognize a # as a multiple of 4

The last 2 digits are a multiple of 4. (i.e 144 .... 44 is a multiple of 4, so 144 must also be a multiple of 4.)

What is a major arc?

The longest arc between points A and B on a circle's diameter.

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 x^2 and (1/2)x^2

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.

How to recognize a # as 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)

How to recognize a # as a multiple of 9

The sum of the digits is a multiple of 9.

How to recognize if a # is a multiple of 12

The sum of the digits it a multiple of 3 and the last two digits is a multiple of 4. (i.e 144: 1+4+4=9 which is a multiple of 3, and 44 is a multiple of 4, so 144 is a multiple of 12.)

What are the roots of the quadrinomial x^2 + 2x + 1?

The two xes after factoring.

What is the set of elements which can be found in either A or B?

The union of A and B.

What are congruent triangles?

Triangles with same measure and same side lengths.

T or F? Given d,e &f =/ 0, [(d^3)e(f^5)] / 2d(e^3) / [3(d^2)(e^3)(f^7)] / [6(e^5)(f^2)]?

True

True or false? 4.809 X 10^7 = .0004809 X 10^11

True

∅ is a multiple of

Two (∅×2=∅)

What are "supplementary angles?"

Two angles whose sum is 180.

What are complementary angles?

Two angles whose sum is 90.

What is an isoceles triangle?

Two equal sides and two equal angles.

What is the slope of a vertical line?

Undefined, because we can't divide by 0.

Volume of a rectangular box

V=L*w*h

formula for volume of a rectangular solid

V=l×w×h

formula for the volume of a cube

V=side³

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.

Find distance when given time and rate

d=rt so r= d/t and t=d/r

formula for distance problems

distance=rate×time or d=rt

Can you simplify sqrt72?

Yes, because you can factor out a perfect square (36). Sqrt(36 x 2) = sqrt36 X sqrt2 = 6sqrt2.

Can you subtract 3sqrt4 from sqrt4?

Yes, like radicals can be added/subtracted.

5x^2 - 35x -55 = 0

[(7+ sqrt93) /2], [(7 - sqrt93) / 2]

What is a subset?

a grouping of the members within a set based on a shared characteristic.

If a<b, then

a+c<b+c

a<b then a - b is positive or negative?

a-b is negative

a>b then a - b is positive or negative?

a-b is positive

a(b+c)

ab+ac

a(b-c)

ab-ac

If a is inversely porportional to b, what does it equal?

ab=k (k is a constant)

Pythagorean theorem

a²+b²=c²

If a is negative and n is even then aⁿ is (positive or negative?)

aⁿ is positive

bⁿ

b∧b∧b (where b is used as a factor n times)

Circumference of a Circle

c=2 x pi x r OR pi x D

What is the common monomial factor in the expression 4(c^3)d - (c^2)(d^2) + 2cd?

cd

Rate

d/t (distance)/(time)

Volume of a cube

edge³