Upper Level Math - Concepts and Formulas

slope between two points

(y₂-y₁)/(x₂-x₁)=rise/run

slope intercept form

y=mx+b

m=slope=Rise/run

b=y intercept=where the line crosses y axis

x intercept

where the line crosses the x axis

plug in 0 for y and solve for x

y intercept

where the line crosses y axis

plug in 0 for x and solve for y

parallel lines

same slope

perpendicular lines

negative reciprocal

1/2 and -2/1

distance between two points

d=√[(x₂-x₁)²+(y₂-y₁)²] (pythagorean theorem)

midpoint formula

((x₁+x₂)/2,(y₁+y₂)/2)

sum of all interior angles in a polygon

(n-2)180

n=number of sides

regular polygon

all angles and sides are the same

exterior angles

sum of all exterior angles is 360

interior+exterior=180

area of a trapezoid

1/2(b₁+b₂)h

rhombus

parallelogram with all sides congruent

circle

area:πr²

circumference:2πr

volume of a prism

base area * height

volume of a pyramid

1/3 * base area * height

surface area of a pyramid

base area + 1/2 * p * l

p = perimeter of base

l = slant height

whole numbers

numbers 0 or greater that don't have a fraction or decimal

0, 1, 2, 3...

integers

numbers that don't have fraction or decimal

-2, -3, 0, 2, 3

rational numbers

a number that can be written as a fraction, repeating decimal, or terminating decimal

irrational number

radical that can't be simplified, or a non-repeating, non-terminating decimal

real number

anything that can be put on a number line

imaginary number

square root of a negative number

i= √-1

exponent rules: x²*x³

x⁵ (add the exponents)

exponent rules: x⁵/x²

x³ (subtract the exponents)

exponent rules: (xy²)³

x³y⁶ (multiply the exponents)

x⁰

1 (anything to the 0th power is 1)

part, percent, whole formula

part=percent/100× whole

percent change

difference/original× 100%

5/0 vs. 0/5

5/0 is undefined and 0/5 is 0

pythagorean triples

3, 4, 5/ 5, 12, 13/ 7, 24, 25 (plus every multiple of them)

converse of the Pythagorean theorem

a²+b²=c²

what to do for each question

1. Annotate/read carefully

2. Write the formula

3. show your work

4. make sure you answered the question

venn diagram questions

add up all the number--difference between the sum and total is both--then calculate how many only have one or have both

Adding and subtracting radicals

Need to have like terms. Can only add and subtract radicals with the same radical.

√x *(√y)

√xy

average speed

distance/time

π = what fraction and what decimal

3.14 or 22/7

Area of Circle

A = πr²

Area of Trapezoid

Area of Parallelogram

A = b * h

Area of Triangle

A = ½ b h

What is a 3-4-5 Right Triangle?

A right triangle with legs of lengths 3 and 4; hypotenuse of length 5

45-45-90 Right Triangle (or half of a square)

Hypotenuse is equal to length of the side times √2.

A number is divisible by 3 if

the sum of the digits is divisible by 3

A number is divisible by 4 if

the last 2 digits are divisible by 4

Circumference of a Circle

C = π d or

C = 2 π r

1 pound (lb) = ? ounces

16 ounces (oz)

10 % equals what fraction and decimal

20% equals what fraction and decimal

70% equals what fraction and decimal

1/10 and 0.1

2/10 and 0.2

7/10 and 0.7

1/3 equals what percent and decimal

33 1/3 % and 0.33333..

25 % equals what fraction and decimal

75 % equals what fraction and decimal

0.25 and 1/4

0.75 and 3/4

Sides of a triangle

Longest side must be shorter than the sum of the other two sides

2/3 equals what percent and decimal

66 2/3 % and 0.666667

Slope-intercept form of a line

y = m x + b

Parallel lines have ___________ slope

the same

The slopes of perpendicular lines are _______________

inverse reciprocals

(Examples: 3 and -1/3; 1/2 and -2;

3/4 and -4/3; -2/5 and 5/2)

1/8 equals what percent and decimal

12.5 % and 0.125

How many odd integers are there between 1 and 9

3

What is the ten-digit in the number 182.09?

8

What is the hundredth digit in the number 182.09?

9

420/ (5 X 12 + 10) =

Remember PEMDAS!

420/(70) = 6; [NOT 420/(5 ×22)]

6

List the first five multiples of 11

11, 22, 33, 44, 55

Is 2 a multiple of 8?

No, it is a factor

2/5 - 3/4 =

2/5 - 3/4 =

8/20 - 15/20 =

-7/20

180/ 0.03 =

6000

0.1/ 0.02 =

.1/.02 =

10/2=

5

.6 = what fraction

3/5

A student answered 75% of the questions on a test correctly and left 7% of the questions blank. What percentage of the questions did the student answer incorrectly?

18%

1/4 equals what decimal?

0.25

16% of 10% of 500 =

.16 × .1 × 500=

.16 × 50 =

8

Is 1 a prime number?

no

How many factors does 100 have?

9 and they are 1, 2, 4, 5, 10, 20, 25, 50, 100

If 3(x+y) = 21 Then what does x + y =?

Divide both sides by 3

7

12 + 2x > 24 - x What does x =

combine like terms:

3x > 12

x > 4

16% of what number is 25% of 80?

.16 × x = .25 × 80

.16x = 20

x = 20/.16 = 2000/16 =

125

60% of what percent of 500 is equal to 6?

.6 × x × 500 = 6

300x = 6; x= 6/300 = 2/100 = 0.2

2%

What Percent of 32 is 24?

24 = x × 32

x = 24/32 = 3/4 = .75 =

75%

What is the length of the diagonal of a square whose side is 5 inches?

5√2 inches

1 quart (qt) = ? pints

2 pints (pts)

1 gallon = ? quarts

4 quarts (qts)

What is the sum of the angles in a triangle?

180°

What is the sum of the angles in a rectangle

4 right angles = 4 × 90 =

360°

what is the sum of the angles in a pentagon?

add 180° every time you add a side (start with triangle)

180 + 180 + 180 OR

(n-2)×180 = (5-2)×180 = 3×180

540°

What is the sum of the angles in an octagon?

add 180° every time you add a side (start with triangle)

180 + 180 + 180 + 180 + 180 + 180 OR

(n-2)×180 = (8-2)×180 = 6×180 =

1080°

Which is greater: 5/9 or 6/11?

Use criss-cross method

5×11 = 55; 6×9 = 54; therefore

5/9 > 6/11

What are supplementary angles?

They add to 180°

What is 3² × 3⁵

Add exponents in multiplication

3⁷

What is 2¹⁰ ÷ 2⁶

Subtract exponents in division

2⁴ = 16

What is the slope of the line containing the points (4,2) and (2,1)?

Slope = change in y divided by change in x

= (2-1)/(4-2) = ½

OR

= (1-2)/(2-4) = -1/-2 = ½

How much money does Bob have after 2 years if he started with $500 and the interest rate is 6%?

Interest = 500 × .06 = $30/year

$30 × 2 years = $60

Total = $500 + $60 =

$560

If r+s+t= 40 and s−7=20, what is the value of r?

Not enough information!

3 variables, and only 2 equations

The most you know is that (r+t) = 13

If Sam lives 6 miles from school, and Peter lives 3 miles from school, how far apart do they live?

Not enough information!

If Sam and Peter live in the same direction, it is 3 miles; if they live on opposite sides of the school, it is 9 miles, or it could be any distance in between.

What is the largest of 7 consecutive numbers if the average is 17?

20

If there are the same number of tricycles as bicycles at school, and the total number of wheels is 30, how many bicycles are there?

3x + 2x = 30

5x = 30; x = 6

6 bicycles and 6 tricycles

A restaurant sells 100 hamburgers a day at $10 each. To increase business, they reduce the price to $8 before noon. If they sell 40 hamburgers before noon, how many will they have to sell after noon to keep their hamburger revenue the same?

100 × $10 = $1000 (daily burger revenue)

40 × $8 = $320 (revenue from $8 burgers)

$1000 - $320 = $680 = revenue from $10 burgers

$680/$10 = number of $10 burgers they have to sell

68

The average weight of 4 girls is 110. The average weight of 3 boys is 125. What is the average weight of all 7 children rounded to the nearest pound?

110 × 4 = 440; 125 × 3 = 375

440+375 = 815;

815 ÷ 7 ≅ 116