Transcript Regents Review #5 - Hacked By Shadows Team Hackers

Regents Review #5
Set Notation
Universal Set = {10,11,12,13,14,15,16,17,18,19,20,21}
A = {14, 21}
B = {12,15,18,21}
A and B are subsets of the Universal Set
A
A intersection B “In Common”
B 
{21}
A
A union B “Combination of both Sets”
B
{12,14,15,18,21}
B’ or Bc
The Complement of B “Not B”

{10,11,13,14,16,17,19,20}
Interval Notation
[ means to include  or 
( means do not include < or >
For Example: [12,16) means….
All Real Numbers 12 through 16, including 12 but not 16
Ex:
Which interval notation represents the following inequality?
-4 < x  6
(1) [-4,6]
(4) (-4,6]
(2) (-4,6)
(3) [-4,6)
(4) (-4,6]
Properties
A+B=B+A
1) Commutative (addition)
2) Commutative (multiplication)
(A)(B) = (B)(A)
3) Associative (addition)
(A + B) + C = A + (B + C)
4) Associative (multiplication)
(A x B) x C = A x (B x C)
5) Distributive
A(B + C) = AB + AC
or
A(B – C) = AB – AC
Properties
6) Identity (addition)
A+0=A
7) Identity (multiplication) (A)(1) = A
8) Inverse (addition)
(additive inverse)
A + (-A) = 0
9) Inverse (multiplication)
(multiplicative inverse)
A x (1/A) = 1
Properties
10) Closure
Given a set of numbers, if an operation is performed and the result is part of
that set, then the set of numbers is closed under that operation.
Example: Is the set of whole numbers closed under addition?
Yes, any whole number added to another whole number will always result
in a whole number
Example: Is the set of whole numbers closed under subtraction?
No, if the difference is taken between 3 and 4, the result is an integer
3 – 4 = -1, therefore the set is not closed under subtraction
Properties
Name The Property
1) x 
1
 1 Inverse Property of Multiplication
x
2) (q)(y + z) = (y + z)(q)
3) a x 1 = a
Commutative Property of Multiplication
Identity Property of Multiplication
4) (9 + 8) + 11 = 9 + (8 + 11)
5) r + 0 = r
Identity Property of Addition
6) m(p + q) = mp + mq
7) c + (-c) = 0
Associative Property of Addition
Distributive Property
Inverse Property of Addition
8) For which operations are the set of integers closed?
Add, Sub, Mult
Ratios, Rates, Proportions
A paper copy machine takes 4 minutes and 20
seconds to duplicate 520 pages. Find the rate at
which the machine duplicates in pages per second.
4 minutes = 4(60) = 240 seconds
240 seconds + 20 seconds = 260 seconds
520 pages
2
260 sec
2 pages per second
2 pgs/sec
Ratios, Rates, Proportions
Two inches of rain fell in an hour and a half. At
this rate, how many inches of rain, to the
nearest hundredth, will fall in 10 minutes?
2 inches
2 inches
is the sam e as
1.5 hour
90 min
Set up a proportion to find
the number of inches it will
rain in 10 minutes.
2 in
x

90 min 10 min
90x  20
0.22 inches of rain will
fall in ten minutes
x  .222222....
Ratios, Rates, Proportions
Mike takes 3 minutes to read an article of 312
words. How many minutes will it take him to
read an article of 884 words at the same rate?
Set up a Proportion
3 min
x

312 words 884 words
312x = 2652
x = 8.5
8 ½ minutes
8 minutes and 30 seconds
Ratios, Rates, Proportions
The measures of two supplementary angles are
in the ratio of 4:11. What is the measure of the
larger angle?
Let 4x = the measure of the smaller angle
The smaller angle = 48 degrees
Let 11x = the measure of the larger angle
The larger angle = 132 degrees
4x + 11x = 180
15x = 180
x = 12
The larger angle measures 132○
Ratios, Rates, Proportions
Converting Units of Measurement
You can convert units of measurement by setting up a
proportion
EX: Convert 20 feet to yards
1 yard = 3 feet
1 yd
x yd

3 ft
20 ft
3x = 20
x = 6.66666….
6 2/3 yards
Ratios, Rates, Proportions
Convert 25 yards to inches
1 yard = 3 feet
1 foot = 12 inches
1 yd
25 yd

3 ft
x ft
75 = x
75 feet
1 ft
75 ft

12 in
x in
900 = x
900 inches
Proportions
Anita’s wages vary directly as the number of hours she
works. If she can earn $29.80 for working 4 hours, how
much will she earn when she works 30 hours?
This problem can be solved using a proportion
$29 .80
x

4 hours
30 hours
4x = 894
x = 223.5
Anita will earn $223.50
Percents
Ronald bought a DVD that cost $18.99. His total
payment including sales tax was $20.51. Find
the sales tax rate to the whole percent?
Tax = % (Tax Rate) x Original Price
1.52 = N% (18.99)
N% = 1.52/18.99
N% = 0.080042….
8% tax rate
Tax = 20.51 – 18.99
Tax = $1.52
Percents
The accompanying circle graph shows expenditures for state and
local governments.
1. If the total spending is $50,000, how
much money was spent on
highways?
2. Approximately how many times the
amount of spending on highways is
spent on education?
1.
N = .07(50,000)
N = $3,500
2. N = .34(50,000)
N = $17,000
17,000/3500 = 4.857….
About 5 times as much
3. Approximately what fraction of the
total expenditures are spent on
highways and public welfare
3. 7% + 14% = 21%
combined?
21/100 is about 20/100
About 1 is spent on highways and pw
5
Trigonometry
Make sure calculator is in degrees!
Go to Mode and highlight degree (ENTER)
Finding a Side
Use a Proportion
Sine, Cosine or Tangent
cos 16 
20 units
x ( adj)
21 ( hyp)
Finding an Angle
Use the inverse Trig Functions (2nd Key on Calc)
Sin-1 ,Cos-1 or Tan-1
A ladder leans against a building. The foot of
the ladder is 6 feet from the building. The
ladder reaches a height of 14 feet on the
building. To the nearest foot, find the
measure of the angle that the ladder makes
with the ground.
14(opp)
6(adj)
cos 16 x

1
21
x  21(cos16)
tan x 
x  20.186...
x  66.8014...
tan1 (14 / 6)  x
67◦
Statistics
Quantitative Data is data that is numerical, meaning it can be counted or
measured
Ex: height of a flagpole, weight of a backpack
Qualitative Data is data that is not numerical, meaning it can be observed
Ex: type of toppings on a pizza, favorite ice cream flavor
Univariate refers to single variable data
Ex: the number of pets each student owns
Bivariate refers to two variable data
Ex: A person’s shoe size compared to their height
Statistics
Correlation vs. Causation
A correlation is a relationship between two variables
Ex:
Number of
potted plants
purchased
Number of
winter coats
sold
Money Spent on potted plants
Number of swimsuits sold
Not every correlation shows causation (one variable causes the other)
Which graph above shows causation?
The first graph because the more money you spend on
plotted plants, the more plants you will have
The second graph shows a correlation
but the number of swimsuits sold does
not cause the number of winter coats
sold. The temperature outside and
seasons cause the sales of the items.
Statistics
A population is a group that you want information about.
A sample is part of a population that is used to make estimates about the
population
In a random sample, each member has an equal chance of being selected,
and the sample is representative of the entire population
A biased sample favors one or more parts of the population over others
Ex:
You want to conduct a survey to find out what type of music people listen to
Determine which of these scenarios is biased
A. You ask every fifth person leaving a Taylor Swift concert about the type of
music they listen to
B. You ask every fifth person leaving the local mall about the type of music
they listen to
END OF REGENTS REVIEWS
Now it’s your turn to review on your own!
Using the information presented today and your review
packet, complete the practice problems in the packet.
THE REGENTS IS WEDNESDAY, JUNE 12th
STUDY! STUDY! STUDY!