Math Review

Question 1

If an integer b is not a divisor of a, then the result can be viewed in 3 diff ways…

Answer 1

If an integer b is not a divisor of a​, then the result can be viewed in 3 diff ways…

1) A fraction
2) A decimal
3) A quotient with remainder

Question 2

The product of two positive integers is…

Answer 2

The product of two positive integers is a positive integer

Question 3

The product of two negative integers is…

Answer 3

The product of two positive integers is a positive integer

Question 4

The product of a positive integer and negative integer is…

Answer 4

The product of a positive integer and negative integer is a negative integer

Question 5

Each of the multiplied integers that result in a product

Answer 5

Factor/Divisor

Question 6

When an integer *a * is divided by an integer b, where *b * is a divisor of *a, *the result is always

Answer 6

When an integer a is divided by an integer b, where b is a divisor of a, the result is always

A divisor of a

EX: When 60 is divided by 6(one of its divisors), the result is 10, which is another divisor of 60

Answer 7

When a positive integer a is divided by a positive integer *b, you first find the first greatest multiple of b that is less than or equal to a. That multiple of b can be expressed as the product qb(where q is quotient). Then the remainder is equal a *minus that multiple of b, or r = a - qb, where *r *is the remainder.

The remainder is always greater than or equal 0 and less than b.

Question 8

When a positive odd integer is divided by 2…

Answer 8

When a positive odd integer is divided by 2…

The remainder is always 1

Question 9

To divide one fraction by another…

Answer 9

To divide one fraction by another…

First invert the second fraction(i.e. find it’s reciprocal), then multiply the first fraction by the inverted fraction.

17/8 ÷3/4 = (17/8)(4/3) = (4/8)(17/3) = (1/2)(17/3) = 17/6

Answer 10

Numbers of the form a/b, where either *a *or b is not an integer and b NE 0, are fractional expressions that can be manipulated just like fractions.

EX: Numbers π/2 and π/3 can be added as:

π/2 + π/3 = (π/2)(3/3) + (π/3)(2/2) = (3π/6)(2π/6) = 5π/6

Question 11

Exponents are..

Answer 11

Exponents are..

Used to denote the repeated multiplication of a number by itself.

-Remember the base **and exponent

Question 12

A negative number raised to an odd power…

Answer 12

A negative number raised to an odd power..

is always negative

Answer 13

NOTE: The exponent is applied before the negative sign when you have an expression with exponents

Without parentheses the expression -3² means “the negative of ‘3 squared.’” So (-3)²= 9,

but -3² = -9

Answer 14

All positive numbers have two square roots, one positive and one negative

Answer 15

The symbol √n is used to denote the nonnegative square root of the nonnegative number n.

Therefore, √100 = 10 and –√100 = –10

Question 16

(– √a)² =

Answer 16

(– √a)² = a

(– √3)²= 3

Question 17

√a²=

Answer 17

√a² = a

√π² = π

Question 18

√a√b =

Answer 18

√a√b = √ab

√3√10 = √30

Question 19

√a/√b =

Answer 19

√a/√b = √a/b

√18/√2 = √18/2 = √9 = 3

Question 20

For odd-order roots

Answer 20

For odd-order roots, there is exactly one **root for every number n, even when n is negative.

EX: 8 has exactly one cube root, ³√8 = 2

and –8 has exactly one cube root, ³√–8 = –2

Question 21

For even-order roots

Answer 21

For even-order roots, there are exactly two roots for every positive number n and **no roots **for any **negative **number n.

EX: 8 has two fourth roots, ⁴√8 and ^–4√8,

but -8 has no fourth root, since it is negative

Question 22

All rational numbers and irrational numbers, including all integers, fractions and decimals.

Answer 22

Real number

Question 23

If a + b = b + a, then

Answer 23

If a + b = b + a, then ab = ba

EX: 8 + 2 = 2 + 8 = 10

and (-3)(17) = (17)(-3) = -51

Question 24

If (a + *b) + c = a + (b* + c), then

Answer 24

If (a + b) + c = a + (b + c), then (ab)c = *a(bc*)

EX: (7 + 3) + 8 = 7 + (3 + 8) = 18,

and (7√2) √2 = 7(√2√2) = (7)(2) = 14

Question 25

If *a*(*b* +*c*) =

Answer 25

If *a*(*b* + *c*) = *ab* + *ac* EX: 5(3 + 16) = (5)(3) + (5)(16) = 15 + 80 = 95

Answer 26

*a *+ 0 = *a*, (*a*)(0) = 0, and (*a*)(1) = *a*

Question 27

If *ab* = 0, then

Answer 27

If *ab*​ = 0, then *a* = 0 or *b* = 0 or both EX: -2b = 0, then b = 0

Question 28

|*a* + *b*| \<= l*a*| + l*b*|

Answer 28

The **triangle inequality** EX: If *a* = 5 and *b *= –2, then |5 + (–2)| = |5 - 2| = |3| = 3 and |5| + |–2| = 5 + 2 = 7 Therefore, |5 + (–2)| \<= |5| + |–2|

Answer 29

If* a *\> 1, then *a*² \> a. If 0 \< *b* \< 1, then *b*² \< *b* EX: 5² = 25 \> 5, but (1/5)² = 1/25 \< 1/5

Question 30

Ratio is

Answer 30

Ratio is a way to express relative size, often in the form of a fraction.

Question 31

Given the *percent *and *whole, *compute the percent

Answer 31

Given the percent and whole, ​compute the percent by dividing the *part * by the whole. Result will be decimal equivalent, so multiply result by 100

Question 32

To find the *part* that is a certain * percent* of the *whole*,

Answer 32

To find the part that is a certain percent of the whole, you can: 1) either **multiply the *percent* by the decimal equivalent of the percent.** EX: Find 30% of 350 by *x* = (350)(.3) = 105 2) or set up a proportion to find the part. EX: Find the numer of parts of 350 that yields the same ratio as 30 out of 100 parts. *x*/350 = 30/100 and solve for *x*

Question 33

Given the *percent *and the *part*, calculate the *whole*

Answer 33

Given the *percent* and the *part*, calculate the *whole*. 1) Either **use the decimal equivalent* *of the percent** EX: 15 is 60% of what number? 0. 6*z* = 15 and slove for *z* 2) or set up a proportion and solve * part*/*whole* = 60/100 --\> 15/*z = 60/100 *and cross multiply

Question 34

The whole is _____ of the percent

Answer 34

base

Question 35

The amount of change as a percent of the initial amount

Answer 35

percent change

Question 36

When computing a percent *increase* the base..

Answer 36

When computing a percent *increase* the base is the *smaller number.* NOTE: Base is initial number, before the change

Question 37

When computing a percent *decrease*, the base is...

Answer 37

When computing a percent *decrease*, the base is the *larger* number. NOTE: Base is initial number, before the change

Question 38

Roots and exponents

Answer 38

**Roots ** are closely related to **exponents**. **Exponents** and **roots** can also undo each other √5² = 5 and (³√7³) = 7

Question 39

What to do? 2/x-1

Answer 39

Always get variables out of denominators.

Question 40

Whenever you see an equation with a squared variable, you need to:

Answer 40

1. Recognize that the equation may have two solutions 2. Know how to find both solutions x² = 4 √x² = √4 x = 2 or -2

Question 41

Before factoring a quadratic expression...

Answer 41

you MUST make sure that the other side of the equation equals 0

Question 42

x² - y² =

Answer 42

(x + y) (x - y)

Question 43

x² + 2xy + y² =

Answer 43

(x + y) (x + y) = (x + y)²

Question 44

x² - 2xy +y² =

Answer 44

(x - y) (x - y) = (x - y)²

Question 45

Variables in absolute value |y| = 3

Answer 45

When there is a variable inside an absolute value, you should look for the variable to have two possible values. +(y) = 3 or -(y) = 3 y = 3 or -3 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ |2x+4| = - (2x+4) OR (2x+4)

Question 46

When performing operations on compound inequalities

Answer 46

Perform operations on **every term** in the inequality: x + 3 \< y \< x + 5 x \< y - 3 \< x + 2 \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ c/2 \< b - 3 \< d/2 c \< 2b - 6 \< d

Question 47

When dealing with optimization problems:

Answer 47

Problems involving optimization are related to extreme values: specifically, minimizing or maximizing. **Focus on the largest and smallest possible values for each of the variables**.

Question 48

1/100

Answer 48

0.01 or 1%

Question 49

1/50

Answer 49

0.02 or 2%

Question 50

1/25

Answer 50

0.04 or 4%

Question 51

1/20

Answer 51

0.05 or 5%

Question 52

1/10

Answer 52

0.10 or 10%

Question 53

1/9

Answer 53

0.11 ≈ 0.111 ≈ 11.1%

Question 54

1/8

Answer 54

0.125 or 12.5%

Question 55

1/6

Answer 55

0.16 ≈ 0.167 ≈ 16.7%

Question 56

1/5

Answer 56

0.2 or 20%

Question 57

1/4

Answer 57

0.25 or 25%

Question 58

3/10

Answer 58

0.3 or 30%

Question 59

1/3

Answer 59

0.3 ≈ 0.333 ≈ 33.3%

Question 60

3/8

Answer 60

0.375 or 37.5%

Question 61

2/5

Answer 61

0.4 or 40%

Question 62

3/5

Answer 62

0.6 or 60%

Question 63

5/8

Answer 63

0.625 or 62.5%

Question 64

2/3

Answer 64

0.6 ≈ 0.667 ≈ 66.7%

Question 65

4/5

Answer 65

0.8 or 80%

Question 66

5/6

Answer 66

0.83 ≈ 0.833 ≈ 83.3%

Question 67

7/8

Answer 67

0.875 or 87.5%

Question 68

5/4

Answer 68

1.25 or 125%

Question 69

4/3

Answer 69

1.3 ≈ 1.33 ≈ 133%

Question 70

7/4

Answer 70

1.75 or 175%

Question 71

X percent

Answer 71

X/100

Question 72

of

Answer 72

Multiply (usually)

Question 73

of Z

Answer 73

Z is the whole

Question 74

Y is X percent of Z

Answer 74

Y is the Part, Z is the Whole Y = (X/100)Z Part = (Percent/100)xWhole \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ Alternative Y/Z = X/100 Part/Whole = Percent/100

Question 75

A is 1/6 of B

Answer 75

A = (1/6)B

Question 76

C is 20% of D

Answer 76

C = (0.20)D

Question 77

E is 10% greater than F

Answer 77

E = (1+ 10/100)F = (1.1)F

Question 78

G is 30% less than H

Answer 78

G = (1 - 30/100)H = (0.70)H

Question 79

The dress cost $J. Then it was marked up 25% and sold. What is the profit?

Answer 79

Profit = Revenue - Cost Profit = (1.25)J - J Profit = (0.25)J

Question 80

Pythagorean Theorem

Answer 80

For ***any right triangle***, the relationship that a² + b² = c² Where **a** and **b** are the legs of a triangle and **c** is the hypotenuse.

Question 81

Circumference

Answer 81

C = 2πr where **r** is the radius

Question 82

Diameter of a circle

Answer 82

D = 2r where **r** is the radius

Question 83

Area of a circle

Answer 83

A = πr² where **r** is the radius

Question 84

Ignore the numbers: Rectangle is inscribed in a circle. If the circumference of the circle is 5π and the diameter is 4, what is the area of the rectangle?

Answer 84

**Remember: Any line that passes through the center is a diameter, choose the best angle to solve.** Where **b** = 4 and **d** = diameter (hypotenuse) C = 2πr ; C = 5π, so 5π = 2πr now find r. 5 = 2r; 2 = 5/2 = 2.5 **d** = 2r = 2(2.5) = 5 a² + b² = d² -\> a² + 4² = 5² a² + 16 = 25 a² = 9 -\> a = 3 **a** = 3 and **b** = 4 A(rectangle) = length\*width = (4)(3) = 12

Question 85

The sum of any two side lengths of a triangle will always be __________ than the third side length.

Answer 85

The sum of any two side lengths of a triangle will always be **greater** than the third side length. This is because the shortest distance between two points is a straight line.

Question 86

The third side length will always be ________ than the difference of the other two side lengths.

Answer 86

The third side length will always be **greater** than the difference of the other two side lengths

Question 87

isosceles triangle

Answer 87

A triangle that has two equal angles and two equal sides (opposite the equal angles)

Question 88

equilateral triangle

Answer 88

A triangle that has three equal sides (all 60) and three equal sides

Question 89

Sides of a triangle correspond to their opposite angles

Answer 89

The longest side is opposite the largest angle, and the smallest side is opposite the smallest angle.

Question 90

Area of a triangle

Answer 90

A(tri) = 1/2 (base)(height) - **base** and **height** MUST be perpendicular to each other, so that it forms a 90° angle - Any side of the triangle could act as a base, only thing that matters is that base and height are perpendicular to each other.

Question 91

Pythagorean triplet

Answer 91

A relationship of **right triangles** (the hypotenuse, or largest side, corresponds to the largest number in the triplet) where all three sides have lengths that are integer values.

Question 92

45-45-90 tri

Answer 92

Isosceles right triangle has a 90° angle (opposite the hyp) and two 45° angles (opposite two equal legs) and has a specific ratio: 45° --\> 45° --\> 90° leg --\> leg --\> hyp 1 : 1 : √2 x : x : x√2

Question 93

30-60-90 tri

Answer 93

An equilateral triangle in which all three sides (and all three angles) are equal. Each angle is 60°. A close relative of the equilateral triangle is the 30-60-90 triangle, when put together, form an equilateral triangle 30°--\>60°--\>90° short--\>long--\>hyp 1 : √3 : 2 x : x√3 : 2x

Question 94

diagonal of square/face of cube

Answer 94

* d* = s√2, where s is a side of a square - recall, any squre can be divided into two 45-45-90 triangles and can use the ration 1 : 1 : √2

Question 95

main (space) diagonal of a cube

Answer 95

*d* = s√3, where s is the edge of the cube

Question 96

Trapezoid

Answer 96

Quadrilateral, where one pair of opposite sides is parallel.

Question 97

Parallelogram

Answer 97

Quadrilateral, where opposite sides and opposite angles are equal.

Question 98

Rhombus

Answer 98

Quadrilateral, where opposite sides and opposite angles are equal. All sides are equal.

Question 99

The sum of interior angles of a polygon

Answer 99

(n-2) x 180 where n = number of sides Because the polygon can be cut into (n-2) triangles, each of which contains 180°

Question 100

Area of a trapezoid

Answer 100

(Base₁ + Base₂)/2 x Height i.e. Average of the 2 bases and multiply it by the height

Question 101

Surface Area

Answer 101

the **SUM** of the areas of **ALL** of the faces 2(LW + LH + WH)

Question 102

Volume

Answer 102

Length x Width x Height

Question 103

Of all quadrilaterals with a given perimeter, the _______ has the largest area

Answer 103

Of all quadrilaterals with a given perimeter, the **SQUARE** has the largest area

Question 104

Of all quadrilaterals with a given area, the ______ has the minimum perimeter.

Answer 104

Of all quadrilaterals with a given area, the **SQUARE** has the minimum perimeter.

Question 105

Circumference

Answer 105

The distance around a circle π x d 2πr

Question 106

Area of a circle

Answer 106

πr²

Question 107

sector

Answer 107

fractional portion of circle

Question 108

Arc length

Answer 108

portion of circumference remaining in a sector central angle/360\*Circumference of full circle (*n*°/360°) \* 2πr **central angle** - An angle whose vertex lies at the center point of a circle (an arc and sector of a circle)

Question 109

Sector area

Answer 109

(*n​*°/360°) \* Area of whole circle (n​°/360°) \* πr²

Question 110

Inscribed angle

Answer 110

has its vertex *on the circle itself* (rather than on the *center* of the circle).

Question 111

An integer is divisible by 3 if....

Answer 111

An integer is divisible by 3 if the SUM of the interger's DIGITS is divisible by 3. 72 divisble by 3 b/c sum of its digits is 9, which is divisible by 3

Question 112

An integer is divisble by 4 if...

Answer 112

An integer is divisble by 4 if the integer is divisible by 2 TWICE, or if the TWO-DIGIT number at the end is divisble by 4. 23, 456 is divisble by 4 because 56 is divisible by 4.

Question 113

An integer is divisible by 6 if...

Answer 113

An integer is divisible by 6 if the integer is divisible by BOTH 2 and 3. 48 is divisible by 6 since divisible by 2 AND by 3 (4+8=12, which is divisible by 3)

Question 114

An integer is divisible by 8 if

Answer 114

An integer is divisible by 8 if the integer is divisible by 2 THREE TIMES in succession, or if the THREE-DIGIT number at the end is divisible by 8. 23, 456 is divisible by 8 because 456 is divisible by 8.

Question 115

An integer is divisible by 9 if...

Answer 115

An integer is divisible by 9 if the SUM of the integer's DIGITS is divisible by 9. 4, 185 is divisible by 9 since the sum of its digits is 18.

Question 116

If ***a*** is divisble by ***b***, and ***b*** is divisible by ***c***, then...

Answer 116

If ***a ***is divisble by ***b***, and ***b*** is divisible by ***c***, then ***a*** is divisible by ***c ***as well. 100 is divisible by 20, and 20 is divisible by 4, so 100 is divisible by 4 as well.

Question 117

If ***d*** and ***e ***has*** ******f*** as prime factors, then

Answer 117

If ***d*** and ***e*** has ***f ***as prime factors, ***d*** is also divisible by ***e*** x ***f***. 90 is divisible by 5 and 3, so 90 is also divisible by 5x3=15. You can let ***e*** and ***f*** be the same prime, as long as there are at least two copies of that prime in ***d***'s prime factors. E.g., 30 = 2 x 3 x 5. Its factors are 1, 2, 3, 4, 5, 6(2x3), 10(2x5), 15(3x5), and 30 (2x3x5).

Question 118

If you add or subtract multiples of **N**, the result..

Answer 118

If you add or subtract multiples of **N**, the result is a multiple of **N.** Multiples of 7: 35+21=56 (5x7) + (3x7) = (5x3) +7 = 8x7 35-21=14 (5x7) - (3x7) = (5-3) x7 = 2x7

Question 119

Odd ± Even =

Answer 119

Odd ± Even = ODD

Question 120

Odd ± Odd =

Answer 120

Odd ± Odd = EVEN

Question 121

Even ± Even =

Answer 121

Even ± Even = EVEN

Question 122

Odd x Odd =

Answer 122

Odd x Odd = ODD

Question 123

Even x Even =

Answer 123

Even x Even = EVEN

Question 124

Odd x Even =

Answer 124

Odd x Even = EVEN

Question 125

All prime numbers are odd....

Answer 125

All prime numbers are odd, except number 2

Question 126

When the base of an exponential expression is a positive proper fraction (fraction between 0 and 1), what happens when the exponent increases?

Answer 126

When the base of an exponential expression is a positive proper fraction (fraction between 0 and 1), and the exponent increases then **the value of the expression decreases** (3/4)² = (3/4)(3/4) = 9/16 (0.5)⁴ = (0.0625) **EXCEPT: ***When something greater than 1 is raised to a power, it gets bigger.* **(10/7)² \> (10/7) **

Question 127

COMPOUND BASE: When the base of an exponential expression is a product...

Answer 127

COMPOUND BASE: When the base of an exponential expression is a product, you can multiply the base together and then raise it to the exponent, OR you can distribute the exponent to each number in the base. (2x5)³ = (10)³ = 1,000 **OR** (2x5)³ = 2³ x 5³ = 8x125 = 1000

Question 128

COMPOUND BASE: When the base of an exponential expression is a **SUM** or **DIFFERENCE**

Answer 128

COMPOUND BASE: When the base of an exponential expression is a **SUM** or **DIFFERENCE​**, you must add or subtract the numbers inside the parentheses first. (2 + 5)³ = (7)³ = 343 (5 - 2)⁴ = (3)⁴ = 81

Question 129

When you see a negative exponent...

Answer 129

When you see a negative exponent, **think reciprocal!** (3/4)^-3 = (4/3)³ = 64/2744

Question 130

When multiplying exponential terms that share a common base...

Answer 130

When multiplying exponential terms that share a common base, **add the exponents.** a⁵ x a = a⁶

Question 131

When dividing exponential terms with a common base...

Answer 131

When dividing exponential terms with a common base, **subtract the exponents**. a⁵ / a² = a³

Question 132

The powers of 2:

Answer 132

The powers of 2: 2, 4, , 8, 16, 32, 64, 128

Question 133

The powers of 3:

Answer 133

The powers of 3: 3, 9, 27, 81, 243

Question 134

13²

Answer 134

13² = 169

Question 135

14²

Answer 135

14² = 196

Question 136

15²

Answer 136

15² = 125

Question 137

4³

Answer 137

4³ = 64

Question 138

5³

Answer 138

5³ = 125

Question 139

a³ + a³ + a³ =

Answer 139

a³ + a³ + a³ = 3a³

Question 140

x^-a =

Answer 140

x^-a = 1/x^a