Chapter 8Chapter 8

Basic Algebra

SetsThe 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.

But First… Let’s Review!

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 integersIII. Composite numbers

A. I B. II C. III D. I and II E. I and III

Answer: A The sum of two even numbers is always an even number.

Arithmetic Sequence – a sequence in which each term is a constant difference d from the previous term.

Sequences

The formula can be used to find the nth term of an arithmetic sequence.

1 ( 1)na a d n

Example: 3, 6, 9… 1 3a and d = 3

Fourth term: 4 3 3(4 1) 3 3 3 3 9 12a

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 can also be used to find the nth term of theSequence. In this problem a1 = 3 and r = 2.

( 1)1

nna a r

Sixth term: 6 1 5

6 3 (2) 3 2 3 32 96a

Arithmetic Sequence – Sequences1 ( 1)na a d n Geometric Sequence –

( 1)1

nna a r

The first term in a geometric sequence is 2, and the common ratio is 3. The first term in an arithmeticsequence is 3, and the common difference is 3. Letset X be the set containing the first six terms of thegeometric sequence and set Y be the set containingthe first six terms of the arithmetic sequence. What isthe sum of the elements in X ∩ Y?

Answer: 24Geometric sequence Arithmetic sequenceX = {2, 6, 18, 54, 162, 456} Y = {3, 6, 9, 12, 12, 15, 18}X ∩ Y = {6, 18} => 6 + 18 = 24

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.

GCF and LCM

In the repeating decimal 0.714285714285…, what is the 50th digit to the right of the decimal point?

Answer: 1In 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 toGreg’s weight would be 4:3. If Ted weighs 180 pounds,what was Greg’s initial weight?

A. 115 poundsB. 125 poundsC. 135 poundsD. 145 poundsE. 155 pounds

Answer: EWrite a proportion and solve:

4 4 180

3 20 3 204( 20) 540

20 135

155 pounds

T

G GG

G

G

Percent Increase and Percent Decrease

Percent Increase and Percent Decrease

60% of the students at the high school play sports. 14% of thestudents who play sports play baseball. What percent of the students in the school play baseball?

A. 4.6%B. 4.8%C. 6.4%D. 8.4%E. 10.6%

Answer: D14% 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. 5B. 4C. 3D. 2E. 1

Answer: DFirst, 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:

2 34x a bcWhat does x equal?

2 3 2

4 2 6

(4 )

16

x a bc

x a b c

Chapter 8Chapter 8

Basic Algebra

Practice SAT Problems:

Homework:

Chapter 8: 24 Practice Problems