Transcript Document

Chapter 8
Basic Algebra
But First… Let’s Review!
Sets
The Intersection of two sets X and Y is the set of elements common to X and Y.
An element has to be in both sets to be in the intersection.
Intersection is written X ∩ Y.
The Union of two sets X and Y is the set of all elements in either set X or set Y,
with no element repeated twice.
An element that is in one set or both sets is in the union.
Union is written X U Y.
Which of the following number sets has the property
that the sum of any two numbers in the set is also in the
set?
I. Even integers
II. Odd integers
III. Composite numbers
A.
B.
C.
D.
E.
I
II
III
I and II
I and III
Answer: A
The sum of two even numbers is always an even number.
Sequences
Arithmetic Sequence – a sequence in which each term is a constant
difference d from the previous term.
The formula
an  a1  d  (n  1) can be used to find the nth term of an arithmetic
sequence.
Example: 3, 6, 9…
Fourth term:
a1  3 and d = 3
a4  3  3(4  1)  3  3  3  3  9  12
Geometric Sequence – a sequence such that each term is given by a constant
multiple r of the previous one.
Find the next three terms in the sequence: 3, 6, 12,… In this sequence r = 2.
Therefore, the next three terms in the sequence are 24, 48, 96
The formula
Sixth term:
an  a1  r ( n 1)
can also be used to find the nth term of the
Sequence. In this problem a1 = 3 and r = 2.
 61
5
a6  3  (2)
 3  2  3  32  96
Arithmetic Sequence –
an  a1  d  (n  1)
Sequences
( n 1)
a

a

r
Geometric Sequence –
n
1
The first term in a geometric sequence is 2, and the
common ratio is 3. The first term in an arithmetic
sequence is 3, and the common difference is 3. Let
set X be the set containing the first six terms of the
geometric sequence and set Y be the set containing
the first six terms of the arithmetic sequence. What is
the sum of the elements in X ∩ Y?
Answer: 24
Geometric sequence
Arithmetic sequence
X = {2, 6, 18, 54, 162, 456}
Y = {3, 6, 9, 12, 12, 15, 18}
X ∩ Y = {6, 18} => 6 + 18 = 24
GCF and LCM
The GREATEST COMMON FACTOR (GCF) of two numbers is
the largest factor the two numbers have in common.
The LEAST COMMON MULTIPLE (LCM) of two numbers is the
smallest multiple two numbers have in common.
In the repeating decimal 0.714285714285…,
what is the 50th digit to the right of the
decimal point?
Answer: 1
In the repeating decimal 0.714285714285…, 5 is the 6th,
12th, 18th, 24th, 30th, 36th, 42nd, and 48th digit.
7 is the 49th digit and 1 is the 50th digit.
You might also realize that there are 6 digits that repeat,
divide 50 by 6, and get a remainder of 2. Therefore the
2nd of the 6 digits that repeat, which is 1, will be the 50th
digit.
Ratio and Proportion
If Greg lost 20 pounds, then the ratio of Ted’s weight to
Greg’s weight would be 4:3. If Ted weighs 180 pounds,
what was Greg’s initial weight?
A.
B.
C.
D.
E.
115 pounds
125 pounds
135 pounds
145 pounds
155 pounds
4
T
4
180

 
Answer: E
3 G  20
Write a proportion and solve: 3 G  20
4(G  20)  540
G  20  135
G  155 pounds
Percent Increase and Percent Decrease
Percent Increase and Percent Decrease
60% of the students at the high school play sports. 14% of the
students who play sports play baseball. What percent of the
students in the school play baseball?
A.
B.
C.
D.
E.
4.6%
4.8%
6.4%
8.4%
10.6%
Answer: D
14% of the 60% play baseball so
0.14 x 0.60 = 0.084 = 8.4% of the
students in the school play
baseball.
Chapter 6: Mean, Median, Mode
The Arithmetic Mean (average) is the sum of the items
divided by the number of items.
The Median is the middle number when the list is placed
in order if there is an odd number of items. If
there is an even number of items, the median
is the mean of the two middle numbers.
The Mode is the item that occurs most frequently. If
every item appears the same number of times,
then there is no mode in the set.
a, b, and c are all positive integers such that a
+ b + c = 150, and none of these values are
equal to each other. What is the smallest
possible value for the median of a, b, and c?
A.
B.
C.
D.
E.
5
4
3
2
1
Answer: D
First, find the mean of a+b+c.
a + b + c = 150 => 150/3 = 50
Because a, b, and c are all positive
integers, the set of numbers that
would create the smallest median
is {1, 2, 147}. The median is 2.
Basic rules for exponents:
x  4a bc
2
3
What does x equal?
 x  (4a bc ) 
2
x  16a b c
4 2 6
3 2
Chapter 8
Basic Algebra
Practice SAT Problems:
Homework:
Chapter 8: 24 Practice Problems