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Basic Algebra: Sets, Sequences, Ratios, and Percentages

Review of basic algebra concepts including sets, sequences, arithmetic and geometric progressions, ratios, proportions, and percent increase/decrease.

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Basic Algebra: Sets, Sequences, Ratios, and Percentages

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  1. Chapter 8 Basic Algebra

  2. 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.

  3. 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. 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.

  4. Sequences Arithmetic Sequence – a sequence in which each term is a constant difference d from the previous term. The formula can be used to find the nth term of an arithmetic sequence. Example: 3, 6, 9… and d = 3 Fourth term: 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 the Sequence. In this problem a1 = 3 and r = 2. Sixth term:

  5. Sequences Arithmetic Sequence – Geometric Sequence – 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

  6. 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.

  7. 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.

  8. Ratio and Proportion

  9. 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. 115 pounds B. 125 pounds C. 135 pounds D. 145 pounds E. 155 pounds Answer: E Write a proportion and solve:

  10. Percent Increase and Percent Decrease

  11. 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. 4.6% B. 4.8% C. 6.4% D. 8.4% E. 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.

  12. 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.

  13. 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. 5 B. 4 C. 3 D. 2 E. 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.

  14. Basic rules for exponents: What does x equal?

  15. Chapter 8 Basic Algebra

  16. Practice SAT Problems:

  17. Homework: Chapter 8: 24 Practice Problems

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