AVSAB Mathematics Knowledge
14 x 8
112
13 x 9
117
How much greater is 3x + 5 than the value of 3x − 7?
12
15 x 8
120
11 x 11
121
14 x 9
126
11 x 12
132
15 x 9
135
13 x 11
143
12 x 12
144
14 x 11
154
13 x 12
156
If the tire of a car rotates at a constant speed of 552 times in one minute, how many times will the tire rotate in half-an-hour?
16,560 (552 x 30 = 16,560)
15 x 11
165
14 x 12
168
13 x 13 = ?
169
3 X 6
18
Karen is 6 years older than her sister Michelle, and Michelle is 2 years younger than their brother David. If the sum of their ages is 62, how old is Michelle?
18
Volume of a cylinder can be found by?
Volume for a cylinder can be found by multiplying the two dimensional area (πr²) by the height to get πr²h.
What does volume measure?
Volume measures the space inside a 3 dimensional figure
If Jill needed to buy 9 bottles of soda for a party in which 12 people attended, how many bottles of soda will she need to buy for a party in which 8 people are attending?
We can set up a proportion to solve: 9 bottles / 12 people = x bottles / 8 people. Cross-multiply to solve a proportion: (9)(8) = (12)(x) 72 = 12x 6 = x
b.)x > 3
What inequality is represented by the number line seen in the figure? a.)x b.)x > 3 c.)x = 3 d.)x ≤ 3
multiplicative inverse
When the product of two numbers is 1, one is called the reciprocal or multiplicative inverse of the other. 1/4 and 4 are multiplicative inverses.
septagon
a polygon with 7 sides
octagon
a polygon with 8 sides
quadratic
a polynomial equation of the second degree like x² + x + 2
trapezoid
a quadrilateral with exactly one pair of parallel sides
circles have?
a radius which measures the length from the center of the circle to any point on its curve
rate
a ratio that compares two quantities measured in different units such as miles/hour
monomial
a single term made up of numbers and variables
hexagon
a six-sided figure
constant
a term that has no variable and does not change
acute triangle
a triangle with 3 acute angles
scalene triangle
a triangle with no equal sides
obtuse triangle
a triangle with one obtuse angle
equilateral triangle
a triangle with three equal sides
4 x 8
32
3 x 12
36
4 x 9
36
The first four terms in a sequence are shown below. What is the sixth term in the sequence? {3, 6, 11, 18, ...}
38 Begin by examining the sequence for a pattern. In order to go from 3 to 6, 3 must be added; moving from 6 to 11 requires 5 to be added; moving from 11 to 18 requires 7 to be added. The pattern emerges here — adding by consecutive odd integers. The 5th term is equal to 18 + 9 = 27, and the 6th term is equal to 27 + 11 = 38.
13 x 3
39
3^5 x 3^3 = ?
3^8
4^5 % 4^2 = ?
4 ^ (5-2) 4 ^3
14 x 3
42
6 x 7
42
7 x 6
42
15 x 3
45
5 x 9
45
4 x 12
48
6 x 8
48
(4x + 2) (x + 1) =
4x^2 + 6x + 2
13 x 4
52
6 x 9
54
14 x 4
56
Lisa and Robert have taken the same number of photos on their school trip. Lisa has taken 3 times as many photos as Claire and Robert has taken 12 more photos than Claire. How many photos has Claire taken?
6
polygon
A closed plane figure consisting of a certain number of sides.
cylinder
A figure that has two circular bases that are the same size and are in parallel planes. It has one curved face.
altitude of a triangle
A line segment drawn from a vertex point perpendicular to the opposite side (base); it is referred to as the height of the triangle. In a right triangle, the altitude is one of the legs.
whole number
A natural number or zero; one of the numbers {0, 1, 2, 3, ...}
How do exponents work?
A number in superscript next to a base (for example, 2³). They represent the number of times (3) that the base (2) will be multiplied by itself. Another way of expressing a number with an exponent of two is squared and with an exponent of three, cubed so 2³ is two cubed and 4² is four squared.
rational number
A number that can be written as a/b where a and b are integers, but b is not equal to 0.
coordinate plane
A plane that is divided into four regions by a horizontal line called the x-axis and a vertical line called the y-axis.
triangle
A polygon with three sides
trinomial
A polynomial of three terms
Composite Number
A positive, whole number that can be divided evenly by numbers other than 1 or itself. Composite numbers have more than 2 factors.
rectangle
A quadrilateral whose angles are right angles.
parallelogram
A quadrilateral with two pairs of parallel sides
dividend
A quantity being divided in a division problem; e.g., 30 ÷ 5 = 6 (30 is the dividend)
Trick to see if the number is divisible by 3
Add up the digits: if the sum is divisible by three, then the number is as well
average
Also called the mean. The sum of the values in the set divided by the number of items.
linear equation
An equation of the first degree. The graph of a linear equation in two variables is a straight line. example: y = 3x + 4
real number
Any number that is a rational number or an irrational number.
The reciprocal of 1/6 is a.)1 b.)3 c.)6 d.)1/3
a.)1 *multiply 1/6 by 6= 1
Solve for x: 2x-3=x+7 a.)10 b.)6 c.)21 d.)-10
a.)10
The surface area of a rectangular prism with sides 3 x 5 x 5 inches is: a.)110 square inches b.)140 square inches c.)72 square inches d.)85 square inches
a.)110 square inches *The area of each of the three unique sides are 15, 15, and 25. We need to double (by 5) this area because each side is mirrored in a rectangular solid. This gives us a total surface area of 110 square inches.
What is the value of YZ If XY is 4 and XZ is 5? a.)3 b.)2 c.)6 d.)5
a.)3 *a^2+b^2=c^2 4^2+b^2=5^2 16+b^2=25 b^2=9 b=_/9\ b=3
If a rectangle has a perimeter of 36 feet and is 4 feet wide, what's its area? a.)56 sq ft b.)128 sq ft c.)112 sq ft d.)16 sq ft
a.)56 sq ft *A=lw P=2l+2w =36-2(4)-36-8=28 Divide the answer by the 2 to determine the length of one side 28/2=14 Multiply the area by the with A=14x4= 56
A circle has a radius of 5 inches. What's its approximate area? a.)78.5 in b.)70 in c.)314 in d.)25 in
a.)78.5 in *A=pi(r)^2 =3.14(5)^2 =3.14(25) 78.5
Find the area and the perimeter of the triangle if the sides are 10in, 8in, base is 13in and the height is 6in high. a.)A=39in^2;P=31in b.)A=45in^2;P=56in c.)A=23in^2;P=44in d.)A=88in^2;P=36in
a.)A=39in^2;P=31in *A=1/2bh A=1/2(13)(6) A=3(13) A=39in^2 P=s1+s2+b P=10+8+13 P=31in
Solve by factoring x^4-16=0 a.)x=_+2 b.)x=_+4 c.)x=_+5 d.)x=_+8
a.)x=_+2
x^2(x^4) a.)x^6 b.)x^8 c.)2x^6 d.)2x^8
a.)x^6
What are mixed fractions?
combination of whole and fractional numbers. Examples- 1 1/4 and 3 3/5
Solve: 7 1/7 + 3 3/14 a.)64/52 b.)1 5/8 c.)9 d.)2 2/9
d.)2 2/9 *7 1/7 + 3 3/14 =7*7+1/7 + 3*14+3/14 =50/7 +45/14(flip it into a proper faction) =50/7* 14/45 =50/5/7+7 * 14/7/45/5 =10*2/1*9 =20/9 2 2/9
4!= a.)16 b.)40 c.)0 d.)24
d.)24 *4x3x2x1=24
Which of the following fractions is the largest? a.)2/3 b.)5/8 c.)11/16 d.)3/4
d.)3/4 *3/4=0.75
product
The answer to a multiplication problem. example: the product of 8 and 5 is 40.
To convert temperatures in degrees Celsius to Fahrenheit...
multiply by 1.8 (or 9/5) and add 32.
Fraction to Percent without using calculator
multiply the top by 100 first then divide by the bottom number
coefficient
number in front of a variable
consecutive integers
numbers that follow each other in order example: 1,2,3
What are improper fractions?
numerator equal to or greater than the denominator. Examples- 6/3 and 2/2
What are proper fractions?
numerator smaller than the denominator. Examples- 1/2 and 7/8
volume
the amount of 3-dimensional space occupied by an object
diameter
the distance across a circle through its center (is also twice the radius)
natural number
One of the numbers {1, 2, 3, ...}
semi-circle
One-half of a circle; the two areas in a circle formed by drawing a diameter.
Prime Number
Only can divide by itself and 1 (i.e. 2, 3, 5, 7..etc) (ONE IS NOT A PRIME NUMBER) *the smallest prime number is 2*
18 x^3 - 57 x^2 - 85 x + 100
(x - 4) (3 x + 5) (6 x - 5)
x^2 + 2 x - 24
(x - 4) (x + 6)
(x^2-y^2)/ (x-y) = ?
(x+y) (x-y)/ (x-y) (x + y)
P.E.M.D.A.S. stands for what?
Operations inside parentheses and brackets Exponents and roots Multiplication Division Addition Subtraction
(1/2) % (5/6) = ?
(1/2) x (6/5) (1x6)/(5x2) 6/10 %2 to each # (3/5)
One in every 9 people in a town vote for party A. All others vote for party B. How many people vote for party B in a town of 810?
(8x810)/(9x1) --- 6480 % 9 = 720
area of a trapezoid
(b₁ + b₂) h/2
calculating simple interest
(interest = principal x interest rate)
x^2 + 3 x - 10
(x - 2) (x + 5)
x^2 - 9 x + 18
(x - 3) (x - 6)
2 x^2 - 5 x - 12
(x - 4) (2 x + 3)
4^-2 = ?
--> 1/4^2 1/16
If y + 3y + 5y = −18, then what is the value of y?
-2
0^4 and 0^6
0
x^0 = ?
1
Area of a triangle
1/2 x h x b
Area of a triangle =
1/2(base)(height)
13 x 8
104
15 x 7
105
9 x 12
108
15 x 12
180
all the angles of a right triangle add up to?
180 degrees
sum of the interior angles of a triangle
180 degrees
The sum of 7 numbers is greater than 140 and less than 210. Which of the following could be the average (arithmetic mean) of the numbers?
20, 26, 30, 34 ? 26
3 X 7
21
3 X 8
24
3 x 9
27
8 x 9
27
4 x 7
28
diameter of a circle
2r
Firs 6 digits of Pi
3.141592
12 x 5
60
15 x 4
60
5 x 12
60
9 x 7
63
13 x 5
65
14 x 5
70
6 x 12
72
15 x 5
75
A barista averages making 16 coffees per hour. At this rate, how many hours will it take until she's made 1,200 coffees?
75
How many 1/3 pound paperback books together weigh 25 pounds?
75 25/1 x 3/1 25 x 3
13 x 6
78
12 x 7
84
14 x 6
84
15 x 6
90
Degrees of a right angle
90 degrees
13 x 7
91
8 x 12
96
14 x 7
98
If a square box has a volume of 125 in3in3 what's the perimeter of each side of the box?
A square box is a cube which means that length = width = height. Volume for a cubic box is the length of one side of the box cubed or V = s3s3 where s is the length of one side of the box. The cube root of 125 is 5 so the size of each side of the box is 5. The perimeter of a box is 2w + 2h which, in our case, is: (2 inches x 5) + (2 inches x 5) = 10 inches + 10 inches = 20 inches
radical
A symbol (√) indicating the positive square root of a number
isosceles trapezoid
A trapezoid with two equal sides (legs) and two equal base angles.
right triangle
A triangle that has a 90 degree angle.
isosceles triangle
A triangle with 2 equal sides
prime number
A whole number that has exactly two factors, 1 and itself.
Evaluate 3x + 7 when x = -3. A. -2 B. 10 C. 16 D. 30
A. -2 A. Substitute -3 for x. Then 3(-3) + 7 = -9 + 7 = -2.
If 2b+3=1/8, b= A. -6 B. -3 C. 0 D. 2
A. -6 A. 1/8=1/23=2-3 so 2b+3=2-3 and b + 3 = -3. Therefore, b + 3 - 3 = -3 - 3 = -6.
How many minutes are there in 1 week? A. 10,080 B. 1,440 C. 420 D. 168
A. 10,080 A. There are 60 minutes in 1 hour, 24 hours in 1 day, and 7 days in 1 week. So 1 week = ? = 7 × 24 × 60 = 10,080 minutes.
If x is a positive integer, solve x2 + 6x = 16. A. 2 B. 4 C. 8 D. 10
A. 2 A. Set the equation equal to 0 and factor. x2 + 6x - 16 = 0 and (x + 8)(x - 2) = 0. Then, either x + 8 = 0 or x - 2 = 0, so x = -8 or x = 2. Since x is positive, x = 2 only.
If the area of a square is 400, what is the length of its side? A. 20 B. 40 C. 100 D. 200
A. 20 A. The area of a square is s2 where s is a side of the square. If s2 = 400, then s = √400 = 20.
Subtract (2x3-3x+1)-(x2-3x-2) A. 2x3-x2+3 B. 2x3-x2-6x-1 C. x3-6x-1 D. x2+3
A. 2x3-x2+3 A. Subtraction can be changed to addition by changing the signs in the entire term being subtracted. (2x3 -3x +1) - (x2-3x-2)=(2x3-3x+1) + (-x2+3x+2).. Combine like terms: 2x3-x2-3x+3x+1+2=2x3-x2+3.
Trick to see if the number is divisible by 9
Add the digits. If that sum is divisible by nine, then the original number is as well.
Find the length of the radius in the following figure. A. 3 B. 4 C. 5 D. 10
C. 5 C. The hypotenuse of the triangle is the diameter of the circle. By the Pythagorean theorem, d2 = 62 + 82= 36 + 64 = 100. So d = √100 = 10 and the radius is 10/2=5.
Mean, Median, and Mode
Average, middle, most common
If 7p + 5q = -3, find q when p = 1. A. -1 B. -2 C. -1.142857143 D. -0.285714286
B. -2 B. Substitute 1 for p and solve for q. 7(1) + 5q = -3 and 7 + 5q = -3. 7 + 5q - 7 = -3 - 7 and 5q = -10. Dividing both sides by 5 results in q = -2.
The value of x if the angles are 140 degrees, 40 degrees and 70 degrees. A. 70° B. 110° C. 140° D. 210°
B. 110° B. The angle adjacent to the 140° angle is 40° since supplementary angles add to 180°. The angles of a triangle add to 180° so the angle adjacent to angle x is 180° - 70° - 40° = 70°. Angle x and 70° are supplementary, so x = 180° - 70° = 110°.
The greatest common factor of 24 and 36 is A. 6 B. 12 C. 36 D. 72
B. 12 B. Factors of 24 are 2 × 2 × 2 × 3. Factors of 36 are 2 × 2 × 3 × 3. The greatest common factor is 2 × 2 × 3 = 12.
What is the area of the figure shown? A. 130 ft2 B. 145 ft2 C. 160 ft2 D. 175 ft2
B. 145 ft2 B. Divide the figure into a rectangle and triangle as shown. The area of the figure equals the area of the rectangle plus the area of the triangle. The rectangle = length × width or 10 × 13 = 130 ft2; the triangle = ? base × height or ? ?. Together, the area is 130 + 15 = 145 ft2.
How many distinct prime factors are there in 120? A. 2 B. 3 C. 4 D. 5
B. 3 B. Prime factors of 120 are 2 × 2 × 2 × 3 × 5. Distinct factors are 2, 3, and 5. Therefore, there are three distinct prime factors.
Find the diagonal of a square whose area is 36. A. 6 B. 6√2 C. 9 D. 9 √2
B. 6√2 B. The area of a square is s2 where s is a side of the square. If s2 = 36, then s = 6. The diagonal of a square forms two right triangles; d is the hypotenuse and the two legs are 6 units long. Using the Pythagorean theorem, d2 = 62 + 62 = 36 + 36 = 72. Therefore, d = √72 = 6√2.
Which expression represents the volume of a cylinder whose height is equivalent to the length of the radius? A. pr2 B. pr3 C. (pr)2 D. (pr)3
B. pr3 B. The volume of a cylinder is given by the formula V = pr2h, where r is the radius of the circular base and h is the height. Since h = r, V = pr2r = pr3.
Circumference of Circle
C=2πr
In a classroom of 32 students, 14 are male. What percentage of the class is female?
First, solve how many females are in the class: Number of females = 32 - 14 = 18 To compute for the percentage of females in the class: Percentage = (18/32)(100%) = 56.25% = 56%
In a standard deck of playing cards, a king of hearts is drawn and not replaced. What is the probability of drawing another king from the deck? A. 1/4 B. 1/13 C. 1/17 D. 3/52
C. 1/17 C Probability is ?. Since one king was drawn and not replaced, three kings remain in the deck of 51 cards. So the probability of drawing another king is ?.
What percent of 3/4 is 1/8? A. 9.38% B. 12% C. 16.67% D. 25%
C. 16.67% C. Let p represent the unknown percent. p×3/4=1/8. Solve for p by multiplying both sides by the reciprocal of 3/4. p×3/4×4/3=1/8×4/3=4/24=1/6. As a percent, 1/6 is 16 2/3%.
If a + b = 6, what is the value of 3a + 3b? A. 9 B. 12 C. 18 D. 24
C. 18 C. 3a + 3b = 3(a + b). Since a + b = 6, 3a + 3b = 3(6) = 18.
Solve for m: 3m - 12 = -6 A. -6 B. 0 C. 2 D. 6
C. 2 C. 3m - 12 + 12 = -6 + 12; 3m = 6; Dividing both sides by 3 results in m = 2.
The slope of the line shown is A. -2/5 B. -5/2 C. 2/5 D. 5/2
C. 2/5 C. Slope is found by identifying two points on the line and finding the (change in y)/(change in x). The points (0, 0) and (5, 2) form the slope (2 - 0)/(5 - 0) = 2/5.
If a = 5/2 then 1/a = A. 2 B. 5 C. 2/5 D. 5/2
C. 2/5 C. Substitute 5/2 for a, giving you 1/a = 1/(5/2) = 1 x 2/5 = 2/5.
Seven more than 3 times a number is equal to 70. Find the number. A. 10 B. 17 C. 21 D. 30
C. 21 C. Translate to a mathematical expression and solve. 3x + 7 = 70 so 3x + 7 - 7 = 70 - 7 and 3x = 63. Divide both sides by 3. Therefore, x = 21.
Simplify (9x2y3z-12xy2z2)/3yz A. 3xy2z2 - 4xyz B. 3xy2z - 12xyz C. 3x2y2 - 4xyz D. 3y2 - 4xy2z2
C. 3x2y2 - 4xyz C. (9x2y3z - 12xy2z2)/3yz = 9x2y3z/3yz - 12xy2z2/3yz = 3x2y2 - 4xyz
The angles of a triangle are in the ratio 3:4:5. What is the measure of the smallest angle? A. 15° B. 30° C. 45° D. 75°
C. 45° C. Angles in a triangle add to 180°. So 3x + 4x + 5x = 180° and 12x = 180°. Dividing both sides by 12 results in x = 15°. The smallest angle is represented by 3x = 3(15°) = 45°.
If (3.2 + 3.3 + 3.5)w = w, then what is the value of w?
First add up the three numbers that are in parentheses: 10w = w 10w = w will only be true when w = 0.
Trick to see if the number is divisible by 6
Check 3 and 2. If the number is divisible by both 3 and 2, it is divisible by 6 as well.
Trick to see if the number is divisible by 8
Check the last three digits.if the last three digits of a number are divisible by 8, then so is the whole number.
(3 - 1)×7 - 12 ÷ 2 = A. 1 B. -2 C. 4 D. 8
D. 8 D. Following the correct order of operations produces: (3 - 1) × 7 - 12÷2 = 2 × 7 - (12÷2) = 14 - 6 = 8.
12 is 15% of what number? A. 0.0125 B. 1.8 C. 18 D. 80
D. 80 D. Let n represent the number. If 12 is 15% of n, then 12 = 0.15n. Divide both sides by 0.15. Therefore, n = 80.
What is a factorial?
Denoted by a number followed by an exclamation point. Also is the product of all whole numbers from 1 through that number. Example- 4! is 4 x 3 x 2 x 1 = 24.
Speed =
Distance/time
Cory's average score after 8 tests is 91%. If he gets a 78% on his 9th test, what will his new average be?
For the 8 tests, the average is presented as 91, so you can assign the value of 91 to each of the 8 previous scores. This gives us 8 tests with a score of 91 each. 91⋅8=728 91⋅8=728 To get new average after the 9th test, use the same formula: 728+789=8069=89.5 728+789=8069=89.5 Since there are 2 answer choices that are very close to this, and you don't know which way the .5 will be rounded, extend the division another place by adding a 00 to the dividend, then divide. You find that the answer is now 89.5589.55, which rounds up to 89.689.6. Therefore, you now know that 89.589.5 would not be correct. Always extend the decimal places when 2 answer choices are as close as these and you don't know which one is correct. 89.6%
pythagorean theorem
Given sides a and b of a right triangle with hypotenuse c, then a² + b² = c²
associative law of addition
Grouping of terms when adding doesn't affect answer. a + (b + c) = (a + b) + c
associative law of multiplication
Grouping of terms when multiplying doesn't change answer a x (b x c) = (a x b) x c
Steve bought a total of 6 packages of pens, and each package contained either 3 or 7 pens. If exactly 4 of the packages Steve bought contained 7 pens, how many pens did Steve buy?
If Steve bought 4 packages of 7 pens and 6 packages total, then he must have purchased 2 packages of 3 pens. 4(7) + 2(3) = 28 + 6 = 34
Trick to see if the number is divisible by 5
If the last digit is a five or a zero, then the number is divisible by 5.
b.)8
In the circle seen below, what is the value of x. Where one section of the circle is 8? a.)4 b.)8 c.)10 d.)12
Formula for simple interest is what?
Interest = principal x (interest x rate)
The volume of the brick is
L x W x H
Degrees in types of angles?
Line- 180 Right-90 Acute- <90 Obtuse- >90 but less than 180 Reflex- > 180 but less than 360
Trick to see if the number is divisible by 4
Look at the last two digits. If the number formed by its last two digits is divisible by 4, the original number is as well.
When solving equations with inequalities when do you switch the signs around?
Make sure to reverse the direction of the inequality when you multiply or divide by both sides by a negative
In algebra constants are?
Numbers
P =
P = W ÷ time
order of operations
PEMDAS parentheses or brackets Exponents and roots Multiplication Division Addition Subtraction
integers
Positive and Negative Whole Numbers and 0. Examples: .... -3, -2, -1, 0, 1, 2, 3.....(Does not include fractions or decimals)
2 − 8 ÷ (24 ÷ 2) =
Remember your order of operations: Complete operations within parentheses first. Calculate exponents. Then multiply and divide in order from left to right. Then add and subtract in order from left to right. 2 − 8 ÷ (24 ÷ 2) = 2 − 8 ÷ (16 ÷ 2) = 2 − 8 ÷ 8 = 2 − 1 = 1
To convert temperatures in degrees Fahrenheit to Celsius...
Subtract 32 and multiply by .5556 (or 5/9).
Blair's average (arithmetic mean) on two physics tests is 7. What should Blair's score be on the next test to have an overall average of 8 for all the tests?
The arithmetic mean is defined as the sum of the terms divided by the number of terms. 7 = sum / 2 14 = sum Adding the next test's score will change the average to an 8, and the total terms to 3. 8 = (14 + new score) / 3 24 = 14 + new score 10 = new score
There are two pizza ovens in a restaurant. Oven 1 burns three times as many pizzas as oven 2. If the restaurant had a total of 12 burnt pizzas on Saturday, how many pizzas did oven 2 burn?
The correct answer is (A). Begin by setting up a system of equations to represent the situation, letting x represent oven 1 and y represent oven 2. x = 3y and x + y = 12 Substituting the first equation into the second: 3y + y = 12 so y = 3.
The hour hand of a watch rotates 30 degrees every hour. How many complete rotations does the hour hand make in 6 days?
The correct answer is (C). There are 360 degrees in a complete circle, so 360/30 = 12 hours to make one full circle. In 6 days there are 24 hours x 6 = 144 hours total. The total number of rotations will be 144/12 = 12.
If a store adds 50 chairs to its current inventory, the total number of chairs will be the same as three-halves the current inventory of chairs. If the manager wants to increase the current inventory by 40%, what will the new inventory of chairs be?
The correct answer is (D). This word problem requires careful translation. Let's say t = total current inventory of chairs. The first sentence states that 50 + t = (3/2)t. First solve for the current inventory: 50 + t = (3/2)t 50 = (3/2)t − t 50 = (1/2)t 100 = t The manager wants to increase this by 40%. 40% of 100 is 40, so the new inventory will be 140.
Which of the following expressions is equal to 38 when p = 3^5?
The correct answer is 27p 38 = 3 x 3 x 3 x 3 x 3 x 3 x 3 x 3 or 3^5 x 3^3 = p x 3^3 p x 3^3 = 27p
circumference
The distance around a circle.
abolute value
The distance from zero on a number line
4!
The factorial function (symbol: !) means to multiply a series of descending natural numbers 4 × 3 × 2 × 1 = 24
greatest common factor
The largest factor that two or more numbers have in common.
lowest common denominator
The least common multiple of the denominators of two or more fractions.
probability
The likelihood of something happening
commutatuve law of multiplication
The order of the numbers does not affect the product. a x b x c = b x c x a
commutative law of addition
The order of the numbers does not affect the sum. a + b + c = b + c + a
How can you find the perimeter of a triangle?
The perimeter of a triangle can be found by adding the lengths of each side
Julie gives 7 pieces of candy to each of her friends. If Julie gives all of her candy away, which amount of candy could have been the amount she distributed?
The problem states that all of Julie's friends received 7 pieces each. There is no mention about the number of friends Julie gave the candies to. But no matter how many friends Julie gave the candies to, the total number of candies she has given away must be divisible by 7. There is no other way to solve the problem but try each given choice and pick one that is divisible by 7 and results in a whole number. 1617=23 1617=23 1287=18.28 1287=18.28 1457=20.71 1457=20.71 1627=23.14 1627=23.14 Therefore, the answer is 161 candies (distributed to 23 friends with 7 candies each).
trisect
The process of separating into three equal parts
divisor
The quantity by which the dividend is being divided; e.g., 30 ÷ 5 = 6 (5 is the divisor).
quotient
The quantity resulting from the division of two numbers. example: 2 is the quotient of 6 divided by 3
radicand
The quantity under a radical sign
What is the inverse of an exponent?
The root which is indicated by a radical sign √. Example- in ³√8 the number 8 is the radicand and the number 3 is the root. The third root os 8 is 2 because 2x2x2x2 = 8
root of an equation
The solutions or zeroes; the value that makes the equation true; x-intercepts are also solutions
What is the sum of the interior angles of a triangle?
The sum of the interior angles of a triangle always equal 180°
perimeter
The sum of the lengths of the sides of a polygon
Trapezoid question
The trapezoid consists of two smaller shapes: a triangle and a rectangle. Draw a line from point A across this figure to form the two shapes. Find the area of each shape and then add them together to find the area of the trapezoid. Area of rectangle = length x width. Therefore, the area of the rectangle is 18x20 = 360 Area of a triangle = 1/2(base)(height). The base of the triangle is 20. The height of the triangle is unknown. Use the Pythagorean Theorem to find the triangle's height. c2 = a2 + b2 where c is the triangle's hypotenuse. 292 = 202 + (height)2 841 = 400 + (height)2 441 = (height)2 The square root of 441 is 21 height = 21. Therefore, area of the triangle = (1/2)(20)(21) = 210. The total area of the trapezoid = 360+210 = 570.
Trick to see if the number is divisible by 7
To find out if a number is divisible by seven, take the last digit, double it, and subtract it from the rest of the number. If you get an answer divisible by 7 (including zero), then the original number is divisible by seven. If you don't know the new number's divisibility, you can apply the rule again.
Each year, a cyber café charges its customers a base rate of $25, with an additional $0.30 per visit for the first 50 visits, and $0.10 for every visit after that. How much does the cyber café charge a customer for a year in which 72 visits are made?
Translate the information into arithmetic. The café charges $25 + $0.30(first 50) + $0.10(additional after 50). For 72 visits there are 50 visits with an additional 22 visits. $25 + $0.30(50) + $0.10(22) =$25 +$15 + $2.20 =$42.20
What is a equilateral triangle?
Triangles are equilateral when all three sides have the same length
complementary angles
Two angles that add up to 90 degrees
What are supplementary angles?
Two angles that the sum equals 180 degrees.
What are complementary angles?
Two angles whose sum equals 90 degrees.
supplementary angles
Two angles whose sum is 180 degrees
distributive law
a (b + c) = ab + ac
ratio
a comparison of two numbers by division
In algebra coefficients are ?
a constant multiplied by a variable
discount
a decrease in the price of an item
square
a four-sided figure with four right angles and four equal side
variable
a letter used to represent one or more numbers
radius
a line segment from the center of a circle to any point on the circle (is also half the diameter)
exponent
a mathematical notation indicating the number of times a quantity is multiplied by itself. The exponent of the term 4x³ is 3
What is the Additive inverse?
a number or variable added to its inverse equals zero. E Example, 7y + (-7y) = 0.
What is the Additive identity?
a number or variable added to zero equals that number or variable. Example, 7y + 0 = 7y.
What is the Multiplicative inverse?
a number or variable multiplied by its reciprocal equals one. Example, (7y)(1/7y) = 1.
What is the Multiplicative identity?
a number or variable multiplied by one equals that number or variable. Example, 7y x 1 = 7y.
irrational number
a number that can not be written a/b. An example is π
perfect square
a number that has a whole number for a square root ex: 9, 16, 121
composite number
a number that has factors other than 1 and itself
prime factorization
a number written as the product of its prime factors
rhombus
a parallelogram with four equal sides
decagon
a polygon with 10 sides
quadrilateral
a polygon with 4 sides
pentagon
a polygon with 5 sides
Second Law of Motion
acceleration imparted upon an object is directly related to the strength of the force and inversely related to the object's mass and always occurs in the direction of the force.
supplementary angles
added = 180 degrees
complementary angles
added = 90 degrees
Whole numbers
all positive integers and zero 0, 1, 2, 3
What are whole numbers?
all positive integers including zero. Examples - 0,1,2,3
straight angle
an angle that measures 180 degrees
right angle
an angle that measures 90 degrees
acute angle
an angle with a measure greater than 0 and less than 90
proportion
an equation that states that two ratios are equal Ex: 1/2 = x/10
polynnomial
an expression consisting of variables and coefficients. Exponents must be positive integers. An example is x³ − 4x + 7.
What is a polynomial?
an expression with one or more terms
First Law of Motion
an object at rest tends to remain at rest unless acted upon by an outside force and that an object in motion tends to remain in motion at the same speed and at the same direction unless acted upon by an outside force
Rational numbers
any number that can be written as a fraction with integer numerator and denominator -2/5, -1/3, 0, 1, 4/3
Irrational numbers
any number that is not a rational number and that cannot be written as a fraction or as a non repeating decimal Pi (3.1415...), √3 (1.7320...)
In Geometry what are Parallel lines?
are equidistant lines that never cross
what are quadrilaterals?
are four-sided geometric shapes whose four angles sum 360°
In Geometry what are perpendicular lines?
are lines that intersect at a right angle (90°).
At 3:00pm, the angle between the hands of the clock is? a.) 90 degrees b.) 180 degrees c.) 120 degrees d.) 360 degrees
b.) 180 degrees
Solve the inequality |x|<3 a.)3<x<3 b.)-3<x<3 c.)-3>x>-3 d.)-3>x<3
b.)-3<x<3
Solve for x: -x^2-x+30=0 a.)4,-8 b.)-6,5 c.)-4,5 d.)6,-3
b.)-6,5 *-x^2-x+30=0 -(x^2+x-30)=0 -(x+6)(x-5)=0 x=-6,5
A tube has a radius of 3 inches and a height of 5 inches. What's its approximate volume? a.)34 cubic inches b.)141 cubic inches c.)565 cubic inches d.)45 cubic inches
b.)141 cubic inches V=pi(r)^3(h) =3.14(3)^3(5) =3.14(9)(5) =141
A square box has a volume of 64, cubic inches. What's the perimeter of one of its faces? a.)8 in b.)16 in c.)64 in d.)32 in
b.)16 in *V=64 find the cube root of 64: 4^3 Add the 4 sides of the square 4+4+4+=16
In any triangle, the sum of the interior angles add up to how many degrees? a.)90 b.)180 c.)270 d.)360
b.)180
Solve for x in the equation (x-7)^2-4=(x+1)^2 a.)2 1/2 b.)2 3/4 c.)4 1/2 d.)4 3/4
b.)2 3/4 *(x-7)(x-7)-4=(x+1)(x+1) x^2-14x+49-4=x^2+2x+1 x^2-14x+45=x^2+2x+1 -14x+45=2x+1 44=16x 2 3/4=x
What's the prime factorization of 90? a.)2*3*5 b.)2*3^2*5 c.)2^2*3^2 d.)2*3*5^2
b.)2*3^2*5
One complementary angle is 62 degrees. What is measure of the other angle? a.)158 degrees b.)28 degrees c.)62 degrees d.)26 degrees
b.)28 degrees *complementary angle are always 90 degrees Subtract 90-62=28
What's the product of _/36\ and _/49\? a.)1,764 b.)42 c.)13 d.)6
b.)42
What angle is complementary to 47 degrees? a.)33 degrees b.)43 degrees c.)93 degrees d.)133 degrees
b.)43 degrees *Two angles are coplementary if they add up to 90 degrees. This means that the complementary angle to 47 degrees is equal to 90-47 = 43 degrees.
What's the sum of the integers from 1 to 300? a.)38,243 b.)45,150 c.)49,923 d.)52,024
b.)45,150 *S=n/2(a+b) =300/2(1+300) =150(301) 45,150
If a^3+b^3=a^3+x^3, then b= a.)b^3-a^3 b.)x c.)a^3-b^3 d.)a
b.)x *a^3+b^3=a^3+x^3 b^3=x^3 3_/b^3\=3_/x^3\ b=x
(y^2)^3+y^2= a.)y^7 b.)y^6+y^2 c.)y^8+y^2 d.)3y^2
b.)y^6+y^2
How to simplify a fraction?
by determining what number will divide evenly into both the numerator and the denominator. Example, with the fraction 4/8 four divides evenly into both 4 and 8 giving 4/8 ÷ 4/4 = 1/2.
A circle has a radius of 15 feet. What's most nearly its circumference? a.)30 ft b.) 225 ft c.) 94 ft d.) 150 ft
c.) 94 ft add the radius twice 15+15=30 Multiply 30 by 3.14 3.14x30=94
If I=Prt, and P=$1,000, r=7% and t=1. What does I equal? a.)$35 b.)$1,000 c.)$700 d.)$70
c.)$700 *I=Prt =1,000(0.07)(1) 700
Simplify 8x^2-3x+4xy-9x^2-5x-20xy a.)5x^2+9xy b.)8x-9x^2 c.)-x^2-8x-16xy d.)8x+9x^2
c.)-x^2-8x-16xy
What is the area of the trapezoid? If AB is 7cm, AE is 10cm and DEC is 15 a.)32 cm^2 b.)157 cm^2 c.)110 cm^2 d.) 220 cm^2
c.)110 cm^2 *A=h(base1+base2/2) A=10(7+15/2) A=10(22/2) A=10(11) A=110cm^2
Solve (4*3)*3-6*(9/3) a.)54 b.)14 c.)18 d.)28
c.)18
A cube has a volume of 64 cubic inches. What's its surface area? a.)16 sq in b.)64 sq in c.)96 sq in d.)32 sq in
c.)96 sq in *V=64 find the cube root of 64: 4^3 To find the surface area add 4+4=16 Multiply by 16 by 4=96
A pizza with an 18-inch diameter is cut into 12 equal slices. What is the area of each slice? How many linear inches of crust does each one have? What is the total perimeter of each slice? a.)A=37/5in^2;C=1 1/2piin;P=6+2piin b.)A=89/3in^2;C=1 3/4piin;P=4+9.7piin c.)A=27/4piin^2;C=1 1/2piin;P=18+1.5piin d.)A=65/2piin^2;C=1 2/9piin;P=19+2.7piin
c.)A=27/4piin^2;C=1 1/2piin;P=18+1.5piin *A=pir^2 half of 18 is 9. 9 is the radius A=pi(9)^2 =81piin^2 A=81piin^2/12= 27/4piin^2 C=pid =pi(18) =18piin =18piin/12= 6/4 = 1 1/2piin P=s1+s2+b =1.5+9+9 =18+1.5piin
Solve the inequality 5-3x-3x+2/4<12 a.)x<2 b.)x>2 c.)x>-2 d.)x<-2
c.)x>-2 *5-3x-3x+2/4<12 20-12x-(3x+2)<48 20-12x-3x-2<48 -15x+18<48 -15x<30 x>-2
(x^3)^3 a.)3x^3 b.)x^6 c.)x^9 d.)2x^6
c.)x^9
compound interest
calculate simple interest for each year making sure to use the new principal + accumulated interest for each year's new interest calculation
Find the area and circumference of the following circle in terms of pi and round to the nearest tenth. The diameter is 22cm. a.)A=290.5cm^2;C=55.2cm b.)A=477.6cm^2;C=89.7cm c.)A=150.4cm^2;C=45.2cm d.)380.1cm^2;C=69.1cm
d.)380.1cm^2;C=69.1cm *r=1/2d r=1/2(22) r=11 A=pi*r^2 =pi(11)^2 =121picm^2 Multiply 121 by 3.14 and round it to the nearest tenth A=380.1cm^2 C=pid =pi(22cm) =22picm C=69.1cm
Find the volume of a cube that has only 4in. a.)56in^3 b.)89in^3 c.)24in^3 d.)64in^3
d.)64in^3 *V=s^3 =(4)^3 =64in^3
The angle shown here is a(n) a.)complementary angle b.)supplementary angle c.)acute angle d.)obtuse angle
d.)obtuse angle
The following figure is what type of quadrilateral? a.)square b.)rhombus c.)trapezoid d.)parallelogram
d.)parallelogram
Solve the following inequality 2/3(6x-9)+4>5x+1 a.)x>6 b.)x<6 c.)x>-3 d.)x<-3
d.)x<-3 *2/3(6x-9)+4>5x+1 4x-6+4>5x+1 4x-2>5x+1 -x>3 x<-3
How do you convert an improper fraction to a mixed fraction?
divide the numerator by the denominator to get a quotient of one and then write the remainder over the original denominator. Example, 15/7 = 2 remainder 1 which is 2 1/7.
odd consecutive integers
example: 1,3,5,...
even consecutive integers
example: 2,4,6,...
similar figures
figures that have the same shape but different sizes; their angles are congruent and their sides are proportional
How to find volume of a cube?
for a cube with equal side lengths, can be simplified to side³
torque =
force x distance Torque is a measure of moments of rotational force around a specific point
Reflex angle
greater than 180, less than 360
Obtuse
greater than 90 degrees, less than 180
What is the Property of inequality?
if two numbers are equal you can add or multiply those numbers by the same value to get the same result. Example, if 4 = 2y then 4(7) = 2y(7).
terms
in an expression are separated by addition and subtraction signs
What are Arithmetic operations?
include finding the sum (adding), finding the difference (subtracting), finding the product (multiplying), and finding the quotient (dividing).
In Geometry what is a line?
is a connection of points that go in a straight path in only two directions.
In Geometry what is a ray?
is a line that has one defined endpoint but other end goes infinitely in a straight path
area of a circle?
is pi times radius squared (πr²).
range of data
is the difference of the greatest value and the least value in a set of data.
median
is the middle value in a data set where the numbers are ordered least to greatest
base of a term
is the number that is repeatedly multiplied in a power. In the expression x³ , x is the base, 3 is the exponent.
mode
is the value in a data set that occurs most often. If all values occur the same amount of times there is no mode for that set.
To find volume of a rectangular prism?
length x width x height
Area of rectangle =
length x width.
Acute angle
less than 90 degrees
In algebra variables are ?
letters that represent unknown numbers
Salt helps to melt ice because it
lowers the temperature at which water freezes.
area of a parallelagram
lxh
If 5^11 = 5^2 x 5^m, what is the value of m?
m = 9
radius of a circle
measures the length from the center of the circle to any point on its curve
tripled
multiplied by 3
quadrupled
multiplied by 4
What is the least common multiple?
of two numbers is the smallest number that both numbers factor into evenly. Example- The first five multiples of of 4 are {4, 8, 12, 16, 20} making the least common multiple of 3 and 4 equal to 12.
The least common multiple
of two numbers is the smallest number that both numbers factor into evenly. The first five multiples of of 4 are {4, 8, 12, 16, 20} making the least common multiple of 3 and 4 equal to 12.
What is the greatest common factor?
of two numbers, it is the largest number that can divide equally into both numbers. Example, the factors of 18 are {1, 2, 3, 6, 9, 18} and the factors of 15 are {1, 3, 5} so the greatest common factor of 18 and 15 is 3.
reciprocal
one of a pair of numbers whose product is 1: the reciprocal of 2/3 is 3/2
quadrant
one of four sections into which the coordinate plane is divided
Real numbers
set of all rational and irrational numbers -√7, -1/3, 0, 1, 3/4
What are real numbers?
set of all rational and irrational numbers. Examples- -√7, -1/3, 0, 1, 3/4
If b does not equal zero, and ab = b/4, what is the value of a?
solve for a, divide both sides of the equation by b ab = b/4 (ab)/b = (b/4)/b a = (b/4)*1/b a = 1/4
Third Law of Motion
states that for every action there is an equal and opposite reaction.
like terms
terms that have identical variable parts raised to the same power
vertex of an angle
the intersection of two rays or line segments
area
the number of square units needed to cover a flat surface. Important formulas are area of a circle, square, rectangle, trapezoid, triangle, parallelogram.
scale factor
the number used to enlarge or reduce similar figures
x-intercept
the point where a graph crosses the x-axis
y-intercept
the point where a graph crosses the y-axis
net
the profit after all expenses have been paid
domain
the set of all the input (x-values) for a function
range of a function
the set of all the output (y-values) for a function
What are multiples of numbers?
the set of numbers formed by multiplying a number by other numbers. Example, the first five multiples of 3 are {3, 6, 9, 12, 15}.
multiples
the set of numbers formed by multiplying a number by other numbers. For example, the first five multiples of 3 are {3, 6, 9, 12, 15}.
base of a figure
the side of the figure to which the altitude is drawn
least common multiple
the smallest multiple that two or more numbers have in common
slope
the steepness of a line on a graph, rise over run
mean or average
the sum of the values in a data set divided by the number of values in the set
additive inverse
the value with the sign changed. The additive inverse of 3 is -3.
bisect
to divide in half
What is the Associative property?
when adding or multiplying you can group numbers in any order. Example, (3+5) + 7 = 3 + (5+7)
What are equivalent Fractions?
when two fractions identify the same quantity and simplify equally. Examples 1/2 and 2/4
What is a isosceles triangle?
when two sides have equal length and two angles are equal
What is the Distributive property?
when you have an equation that contains both multiplication and addition you can distribute the multipliers. Example, (2+4)(3+5) = 2(3+5) + 4(3+5).
Integer
whole number (-3, 0, 3)
slope intercept form
y = mx + b
What is the Commutative property?
you can add or multiply numbers in any order. Example, 2 x 4 x 7 = 7 x 2 x 4.
When adding and subtracting fractions the denominator must be the same for both numbers. If you have different denominators what do you do?
you need to find the least common denominator by finding a number that both denominators will divide into evenly.
Volume for a cylinder
πr2h
Area of Circle
πr^2
√50 = ?
√ (25 x 2) 5√2
√(27/3) = ?
√27/√3 √27 --> √ (3 x 9) = √3 x √9 = 3√3 (3√3)/√3 3 √(3/3) 3 √ (1) 3