Transcript Regents Review #5

Regents Review #5
Properties
Ratios, Rates, Proportions
Direct Variation
Percents
Roslyn Middle School
Research Honors
Integrated Algebra
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
0 is the identity
element of addition
1 is the identity element
of multiplication
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
Properties
Binary Operations: An operation performed
using 2 elements from a set
Does this set display commutativity?
Yes
b * c = d and c * b = d
Does this set display associativity?
b*c=d
a * d =d
c*c=a
Yes
(b * c) * a = b * (c * a)
d*a = b*c
d =
d
What is the identity element of this set?
a
because a * a = a
a*c=c
a*b=b
a*d=d
Ratios, Rates, Proportions
A bicyclist travels 6 miles in 20 minutes. Find
the rate of speed of the bicyclist in miles per
minute and miles per hour.
D
R
T
6 miles
 0.3
20 min
0.3 miles per minute
0.3 miles /min
0.3 miles
x miles 18 miles per hour

1 min
60 min 18 mph
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
Rain is falling at a rate of 2 inches per hour. At
this rate, how many inches of rain will fall in x
minutes?
Let’s first find out how many inches per minute it rains.
2 inches 2 inches
1


1 hour
60 min 30
If it rains 1/30 of an inch per
minute and it rains for x
minutes, multiply to find the
number of inches of rainfall.
It rains 1/30 of an inch per minute
1
x
x
inches of rain
30
30
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
Find the rate that Mike reads in words per minute
3 min
x

312 words 884 words
312x = 2652
x = 8.5
312 words
 104
3 min
8 ½ minutes
8 minutes and 30 seconds
Mike reads 104
words per minute
884/104 = 8.5
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
• When converting from a bigger unit to a smaller unit,
multiply
• When converting from a smaller unit to a bigger unit,
divide
Ratios, Rates, Proportions
Convert 20 feet to yards
1 yard = 3 feet
20 feet = ? yards
Small Unit

Big
Unit
20/3 = 6.6666…..
6 2/3 yards
1 yd
x yd

3 ft
20 ft
3x = 20
x = 6.66666….
Ratios, Rates, Proportions
Convert 25 yards to inches
1 yard = 3 feet
1 foot = 12 inches
25 yards = ? feet
Big Unit 
Small Unit
(25)(3) = 75 feet
75 feet = ? Inches
Big Unit 
Small Unit
(75)(12) = 900 inches
900 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
Direct Variation
y = kx is read as “y varies directly as x”
y
k
x
k is the constant of variation
y = kx has a y-int of 0 and a slope of k
Direct Variation
How do we recognize a direct variation relationship
from a table?
x
y
-5
-1
y
Does
k ?
x
-1/-5 = .2
-10
-2
-15
-3
-20
-4
-2/-10 = .2
-3/-15 = .2
-4/-20 = .2
The constant of variation is 0.2
This table does show a direct
variation relationship
Direct Variation
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?
w = kh
w: wages earned
h: hours worked
w = kh
29.80 = k(4)
7.45 = k
w = 7.45h
w = 7.45(30)
w = 223.50
We can also solve this problem using a proportion
w = 7.45h
(direct variation equation)
$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
Regents Review #5
Now it’s time to study!
Using the information from this
power point and your review packet,
complete the practice problems.
There is one more slide!!!!!
THIS IS THE END OF REGENTS
REVIEWS
The Regents Reviews included all material covered
primarily from Quarters 1 – 3
We will not be reviewing Quarter 4 material
(Trigonometry, Geometry, Probability and Statistics)
Make sure you review those topics on your own using
your notes and the exams that were administered on
those topics
THE REGENTS IS THURSDAY, JUNE 14th
STUDY!!!!