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MATHEMATICS STUDY GUIDE Practice Materials for the Developmental Mathematics Parts of the Mathematics Placement Test at California University of Pennsylvania Dear Prospective Student: When you come to New Student Registration at California University of Pennsylvania, you will be required to take a mathematics placement test. The attached practice materials are intended to help you review for the Basic Mathematics and Introductory Algebra tests. If your score on the first test (Part A) is below 11 of 17, you are required to work with the tutors in the Math Lab on Part I of this study guide and take Part A test again. The Math Lab is located in 115 Noss Hall – phone (724) 938-5893 for an appointment. If your score on the second part (Part B) is below 12 of 17, you will be scheduled for Introductory Algebra (DMA 092) if you are in a math-intensive major. You may study Part II of this study guide and retake Part B of the math placement test just once. Please contact Claire Pizer at (724) 938-5779 or [email protected] to set up an appointment to retake the test. It is important for you to know that the credit for Introductory Algebra will not count toward graduation. However, the grade earned in the course does count toward your grade point average. Also, included are study topics for Parts C and D covering College Algebra and Trigonometry if you want to test for Trigonometry, Pre-Calculus or Calculus. We hope you will spend some time reviewing the attached materials because they will be helpful to you in preparing for the placement test. (Calculators may not be used during the test.) Best wishes to you. We expect you will have enjoyable and productive experiences here at California. If you have any questions, please call Dr. Paul Williams at (724) 938-5894. Sincerely, The Mathematics Curriculum Committee Test Date: Test Time: Location: Noss Hall 215 2 Dear Student: In preparation for the developmental mathematics parts of the math placement test (Part A and/or Part B), construct at least a dozen practice tests. Do so by; • marking 17 of the Practice Problems with a pencil, • taking the phone off the hook, • timing yourself for 15 minutes, • taking the test, • checking your answers, and • referring to the worked out problem type of each question you missed. Construct the next practice test by erasing the marks you placed by the 17 practice problems and by marking another 17 questions, whether or not you marked them on the last test. A score of 11 is passing on Part A and 12 is passing on Part B, but you should try to get scores of 13 on the practice tests you take on your own. Call the Math Lab in 115 Noss Hall at (724) 938-5893 for free help when you are having difficulty with particular problems. Sincerely, Dr. Paul Williams Math Lab Director 3 PART I – Basic Mathematics Type of Question Example Solution 1. Add whole numbers Find 126 + 5,123 + 57. 126 5123 + 57 5306 2. Subtract whole numbers Compute 753 – 68. 753 - 68 685 3. Multiply whole numbers Jake worked 8 hours during each of 6 days last week. He received time and one-half for weekends. How many hours did he work last week? 64 4. Divide whole numbers 8 x6 48 Hours 608 54 32,832 Divide: 54 32,832 - 324 432 - 432 0 5. Add fractions or mixed numbers Find: 5 7 5 + + 6 8 12 LCD = 24 = = = 5 7 5 + + 6 8 12 5x4 7x3 6x4 8x3 20 21 10 + + 24 24 24 51 24 2 24 51 - 48 3 = = 2 3/24 2 1/8 5x2 12 x 2 4 Type of Question 6. Subtract fractions or mixed numbers Example Sue missed 5/16 of a shift. For how much of the work shift should she be paid? Solution 1– 5 16 16 5 16 16 11 = shift 16 = 7. Multiply fractions or mixed numbers Find: 4 8. Divide fractions or mixed numbers Divide: 16 ÷ 9. Add decimals Find 0.27 + 6 + 0.3 10. Subtract decimals Find 8.2 - .67 2 x 6. 3 7 8 2 x6 3 = 14 x 6 2 31 1 28 = 1 = 28 7 16 ÷ 8 16 8 = x 1 7 128 = 7 2 = 18 7 4 0.27 6.00 + 0.30 6.57 7 11 8.20 - .67 7.53 11. Multiply decimals Find 4.63 x .015 4.63 2 places x.015 3 places 2315 463 .06945 5 places in answer 5 12. Divide decimals Find 16.03 ÷ .014 1145. .014 16.030 Move decimal point three places to the right. 13. Using decimals If 6 cans of soda costs $1.20, how much will 4 cans cost? .20 6 1.20 14 20 14 63 56 70 70 0 per can .20 x 4 = .80 for 4 cans 14. Renaming percents Name 24% as a fraction and a decimal. 24 out of 100 = 24 or reduce to 6 100 25 or 0.24 15. Finding percents of a number Find 62% of 120. 62% = 62 = 0.62 100 120 x 0.62 = 120 x 0.62 240 720 74.40 62% of 120 is 74.4 16. Using percents A television priced at $360 is on sale for 20% off. Find the sale price. Find 20% of 360 360 x .20 72.00 Regular price = 360.00 - 72.00 $288.00 6 PART I – Basic Math Practice Problems Problem Type 1. Find: 423 + 5,469 + 32 (1) 2. Subtract: 5,023 - 3,697 (2) 3. Multiply: 786 x 709 (3) 4. Divide: 63 44,667 5. Find: 7 1 1 - + 15 5 25 6. Calculate: 5 13 – 24 16 7. Multiply: 4 8. Compute: 1 7 x 5 8 13 7 5 ÷ 6 12 (4) (5 & 6) (6) (7) (8) 9. Find: 32 + 0.58 + 0.064 (9) 10. Find: 108.6 – 65.93 (10) 11. Multiply: 0.675 x 3.82 (11) 12. Find: 10.224 ÷ 3.6 (12) 13. Sarah can ride her bicycle 1.35 miles in 45 minutes. At this rate, how far can Sarah ride in one minute? (13) 14. Write 48% as a decimal (14) 15. Find: 54% of 208 16. Everything in Marty’s hardware store is on sale for a 25% discount. What would a hammer that was originally priced at $16.90 cost? (15) 17. Joe ran 5 miles, walked 2 miles, ran 3 miles, and drove 75 miles. How far did Joe run? (1) (16) 7 18. The Civic Club added 8,227 new members last year. They now have 30,419 members. How many members did the Civic Club have before last year? (2) 19. Compute: 694 x 780 (3) 20. Calculate: 9,318 ÷ 49 (4) 21. Compute: 42 1 5 + 35 6 8 (5) 22. Subtract: 34 1 7 - 18 12 8 (6) 2 1 pounds of grapes and sold of them. 3 3 How many pounds of these grapes does John have left? (7) 1 3 sections of a fence in 8 hours. If 7 2 she continues to paint at this rate, how many more sections can she paint in one more hour? (8) 23. John bought 7 24. Joan can paint 5 25. Peter ran 0.16 of a mile, walked 0.5 of a mile and then jogged back the same distance. How far did Peter travel? (9) 26. Joyce lost 16.8 kg on her new diet. If Joyce originally weighed 160 kg, how much does she weigh now? (10) 27. Mrs. Reynolds bought 6.5 yards of material costing $1.98 per yard. How much did she pay for the material? (11) 28. Compute: 66 - .0033 (12) 29. Mr. King charges $36.95 for a new automobile tire. How much would four tires cost? (13) 30. What fraction represents 65%? (14) 31. What is 6% of 72? (15) 32. How much sales tax would you pay on a refrigerator that costs $478.00 at a rate of 6% tax? (16) 8 Answers to Practice Problems 1. 5,924 17. 8 miles 2. 1,326 18. 22,192 members 3. 557,274 19. 541,320 4. 709 20. 190 R8 5. 23 75 21. 77 19 24 6. 11 48 22. 15 5 24 7. 24 8. 1 9. 32.644 3 4 3 7 23. 2 24. 4 pounds 9 76 sections 119 25. 1.32 miles 10. 42.67 26. 143.2 kg 11. 2.5785 27. $12.87 12. 2.84 28. 65.9967 13. 0.03 miles 29. $147.80 14. 0.48 30. 65 13 or 100 20 15. 112.32 31. 4.32 16. $12.68 32. $28.68 9 PART II – Introductory Algebra Type 1. Problem Adding integers with like signs -14 + (-19) Method: To add integers with the same sign – (1) add the absolute value of the addends, (2) attach to the result the common sign of the addends. (1) |−14| = 14 19 |−19| = 19 + 14 33 (2) Since both addends are negative, the result is negative. -14 + (-19) = -33 2. Add integers with unlike signs 23 + (-35) Method: To add integers with different signs: (1) subtract these absolute values, (2) give the result the same sign as the addend with the greater absolute value 23 + (-35) positive negative |23| (1) = 23 35 |−35| = 35 - 23 12 (2) Since |−35| > |23|, the result is the same sign as -35; negative 23 + (-35) = -12 3. Finding the sum of more than two integers Method: 4. -6 + (-8) + 13 + (-4) -6 + (-8) + 13 + (-4) = -14 + 13 + (-4) = -1 + (-4) = -5 Subtracting Integers -17 – (-20) Method: To subtract two integers, (1) change the subtraction sign to addition, (2) find the opposite of the second number, and (3) add: -17 – (-20) = -17 + 20 = 3 10 Type 5. Problem Subtraction occurring several times Method: 18 – 24 – 7 18 – 24 – 7 = 18 + (-24) – 7 = -6 – 7 = -6 + (-7) = 6. -13 Multiplying or dividing integers with unlike signs. (1) 6 (-13) (2) 120 ÷ (-8) Method: If the signs are different, the result is negative. (1) 6 (-13) = -78 (2) 120 ÷ (-8) = -15 7. Multiplying or dividing integers with like signs. (1) -8 (-7) (2) -162 ÷ (-9) Method: If the signs are the same, the result is positive. (1) -8 (-7) = 56 (2) -162 ÷ (-9) = 18 8. Multiplying more than two factors. (1) -5 (6) (-7) (-2) Method: (1) -5 (6) (-7) (-2) = -30 (-7) (-2) = = 9. 210 (-2) -420 Solving two-step equations. Method: Solve: 7x – 8 = -29 7x – 8 + 8 = -29 + 8 7x = -21 7x − 21 = 7 7 x = -3 Solve: 7x – 8 = -29 Check: 7 (-3) – 8 = -29 -21 - 8 = -29 -29 = -29 Therefore; -3 is the solution. 11 10. Solving equations of the form ax + b = cx + d Method: 11. Solving equations containing parentheses Method: 12. Solve: 6a – (3a – 4) = 5 (a + 2) 6a – 3a + 4 = 5a + 10 3a + 4= 5a + 10 3a – 5a + 4= 5a -5a + 10 -2a + 4= 10 -2a + 4 – 4 = 10 – 4 -2a = 6 -2a = 6 -2 -2 a = -3 Adding polynomials Method: 13. Solve: 4x - 5 = 6x + 11 4x – 6x – 5 = 6x – 6x +11 -2x – 5 = 11 -2x – 5 + 5 = 11 + 5 -2x = 16 -2x = 16 -2 -2 x = -8 Subtracting polynomials = = = = Solve: 6a – (3a - 4) = 5 (a + 2) Check; the solution is -3 (12x2 – 17x + 4) + (9x2 + 13x – 5) (3x2 + 2x – 2) – (5x2 – 8x – 6) (3x2 + 2x – 2) – (5x2 – 8x – 6) (3x2 + 2x – 2) + - (5x2 – 8x – 6) 3x2 + 2x – 2 – 5x2 + 8x + 6 3x2 – 5x + 2x + 8x - 2 + 6 -2x2 + 10x + 4 Multiplying monomials Method: Check; the solution is -8 (12x2 – 17x + 4) + (9x2 + 13x – 5) = 12x2 – 17x + 4 + 9x2 + 13x – 5 = 12x2 + 9x2 – 17x + 13x + 4 – 5 = 21x2 – 4x – 1 Method: 14. Solve: 4x – 5 = 6x +11 (-3x6y2z) (-5x2y3z) = (-3) (-5) (x6 x2) (y2 y3) (z z) = 15x6+2y2+3z1+1 = 15x8y5z2 (-3x6y2z) (-5x2y3z) 12 15. Method: 16. (-2x4y2z)3 = (-2)3 (x4)3 (y2)3 (z1)3 = -8x43 y23 z13 = -8 x12 y6 z3 Multiplying a monomial by a polynomial. Method: 17. (-2x4y2z)3 Simplifying powers of monomials 4x3 (3x2 – 2x + 6) 4x3 (3x2 – 2x + 6) = 4x3 (3x2) + 4x3 (-2x) + (4x3) (6) = 12x5 – 8x4 + 24x3 (2x – 3) (x2 – 3x – 5) Multiplying two polynomials x2 – 3x – 5 Method: 2x – 3 -3x2 + 9x + 15 2x3 – 6x2 – 10x 2x3 – 9x2 – x + 15 18. Multiplying two binomials Method: (5a – 2b) (3a + 4b) To find the product of two binomials use the “FOIL” method: (5a – 2b) (3a + 4b) product of the first terms (5a) (3a) + product of the outer terms (5a) (4b) + product of the inner terms (-2b) (3a) + = 15a2 + 20ab – 6ab – 8b2 = 15a2 + 14ab – 8b2 19. Finding the square of the binomial Method: (1) (2) (3a – 5)2 (3a + 5)2 (1) (3a – 5)2 = = = (3a – 5) (3a – 5) 9a2 – 15a – 15a + 25 9a2 – 30a + 25 (2) (3a + 5)2 = = = (3a + 5) (3a + 5) 9a2 + 15a + 15a +25 9a2 + 30a + 25 product of the last terms (-2b) (4b) 13 20. Dividing a polynomial by a monomial Method: = 12b7 3b3 + 36b5 3b3 12b7 + 36b5 – 3b3 3b3 -3b3 3b3 + = 4b4 + 12b2 - 1 21. Factoring difference of 2 squares Method: Factor: 4x2 – 9 Find the square roots of both terms. Since 9 is negative, one parenthesis will have a positive (+) in the middle and the other a negative (-). √4𝑥 2 = 2x (2x + 3) (2x – 3) √9 = 3 The result can be checked by multiplying (2x + 3) (2x – 3) 4x2 – 6x + 6x – 9 = 4x2 – 9 22. Factor: 6ax2 – 3a2 x3 – 9a2 x2 Common factoring Method: Find the greatest common factor of all the terms in the polynomial. Divide each term by the greatest common factor. 3ax2 divides into all 3 terms 6ax2 - 3a2 x3 3ax2 3ax2 2 - ax - 3a 3ax2 (2 – ax – 3a) - 9a2 x2 = 3ax2 The result can be checked by multiplying. 23. Common factoring Method: Factor: 3x (a + b) – 2 (a + b) The parenthesis in the original problem groups together a + b so that they act like a monomial. Divide the terms by the common binomial. 3x (a + b) - 2 (a + b) a + b a + b (a + b) (3x – 2) 24. Factoring by grouping Method: Factor: 2x2 – 2xy + 3x – 3y 1. Group the first 2 terms and the last 2 terms together (2x2 – 2xy) + (3x – y) 2. Common factor each group of terms 2x (x – y) + 3 (x – y) 3. Common factor the common binomial (x – y) (2x + 3). (Refer to problem 23.) 14 25. Factoring a perfect square trinomial Method: 26. 27. A perfect square trinomial is a trinomial where (1) first and last terms are squares (2) middle term – 2 times square root of first term times square root of last term. To factor – find the square roots of the first and last terms. The sign inside the parenthesis is the same as the sign in front of the middle term. √9𝑥 2 = 3x √4 = 2 2 (3x + 2) Factoring a trinomial Method: Factor: 6x2 – x – 12 This trinomial can be factored using trial and error. (1) find two factors of the first term 6x2 = 3x 2x (2) find two factors of the last term -12 = -6 2 (3x – 6) (2x + 2) Check your guess through multiplication (3x – 6) (2x + 2) 6x2 + 6x – 12x – 12 = 6x2 – 6x – 12 Incorrect. If your first guess is unsuccessful, try again. Second try: 6x2 = 3x 2x (3x ) (2x ) -12 = 4 -3 (3x + 4) (2x – 3) Check: (3x + 4) (2x – 3) = 6x2 – 9x + 8x – 12 = 6x2 – x – 12 Correct. Reducing a rational algebraic expression Method: Factor: 9x2 + 12x + 4 Reduce: 2x2 – 7x + 3 3x2 – 7x – 6 This fraction is reduced by factoring the numerator and denominator. 2x2 – 7x + 3 = (2x – 1) (x – 3) 3x2 – 7x – 6 (x – 3) (3x + 2) Common binomial factors are divided out (2x – 1) (x – 3) = 2x – 1 (x – 3) (3x +2) 3x + 2 28. Factoring a common factor with factoring a trinomial Method: Factor: 45x2 + 6x – 24 In any type of factoring, common factoring should be done first. 3(15x2 + 2x – 8). Then, use an appropriate factoring method to factor the result. 3(5x + 4) (3x – 2). Trial and error should be used. 15 PART 2 – Introductory Algebra Practice Problem Problem Type 1. 14 – 26 #4 2. -25 + (-59) #1 3. 13 + (-22) + 9 + (-6) #3 4. 13 – (-23) #4 5. #6 6. 135 ÷ (-9) #7 7. -42 ÷ (-3) -6 – (-19) – (-9) #5 8. -5 (23) #6 9. #7 10. -84 ÷ (-7) (-7) (8) (-2) (-3) #8 11. -12 – (-19) #4 12. Solve: -8x + 3 = -29 #9 13. Solve: -5a – 3 = 2a + 18 #10 14. Solve: 5 – (9 – 6x) = 2 (x – 1) #11 15. (8a4 b3) (-7ab5) #14 16. (-8y2 + y – 15) – (6y2 + 2y + 7) #13 17. 15b4 – 5b2 + 10b 5b #20 18. (4a3b4c)2 #15 19. -2x (4x2 + 7x – 9) #16 20. (4x – 3)2 #19 21. (3x3 – 2x2 – 4x) + (8x2 – 8x +7) #11 16 Practice Problem Problem Type 22. (4x + 3)2 #19 23. (3x2 + 2x – 5) (4x – 3) #17 24. (2x – 3) (4x +7) #18 25. Factor: 5x2y – 10x3y2 – 15x2y5 #22 26. Factor: 15x2 – 35x + 6x – 14 #24 27. Factor: 6x2 – 25x – 9 #26 28. Factor: 49x2 – 28x + 4 #25 29. Reduce: x2 – 25 2x + 7x – 15 #27 2 30. Factor: 6x2 – 2x – 9x + 3 #24 31. Factor: 3y(x – 2) + 5(x – 2) #23 32. Factor: 10x2 – 55x + 25 #28 33. Factor: 64y2 – 25 #21 34. Factor: 25x2 + 40x + 16 #25 35. Factor: 9 – y2 #21 36. Factor: 27x2 – 18x + 3 #28 37. Reduce: 38. Factor: 14b2c3 – 12b4c5d + 6b2c2d2 #22 39. Factor: 21x2 + 23xy + 6y2 #26 40. Factor: 5(y – x) – 3x (y – x) #23 8x2 – 26x + 15 12x2 – 5x – 3 #27 17 ANSWER KEY 1. -12 21. 3x3 + 6x2 – 12x + 7 2. -84 22. 16x2 + 24x + 9 3. -6 23. 12x3 – x2 – 26x + 15 4. 36 24. 8x2 + 2x – 21 5. -15 25. 5x2y (1 – 2xy – 3y4) 6. 14 26. (5x + 2) (3x – 7) 7. 22 27. (3x + 1) (2x – 9) 8. -115 28. (7x – 2)2 9. 12 29. x–5 2x - 3 10. -336 30. (2x – 3) (3x – 1) 11. 7 31. (x – 2) (3y + 5) 12. 4=x 32. 5 (2x – 1) (x – 5) 13. -3 = a 33. (8y + 5) (8y – 5) 14. ½=x 34. (5x + 4)2 15. -56 a5b8 35. (3 – y) (3 + y) 16. -14y2 – y – 22 36. 3 (3x – 1)2 17. 3b3 – b + 2 37. 18. 16a6b8c2 38. 2b2c2 (7c – 6b2c3d + 3d2) 19. -8x3 – 14x2 + 18x 39. (3x + 2y) (7x + 3y) 20. 16x2 – 24x + 9 40. (y – x) (5 – 3x) 2x – 5 3x + 1 18 Study Guide for California University of Pennsylvania Math Placement C and D Tests For Math Placement Test C, take any college algebra book and do the chapter review problems for chapters that include the following; • • • • • • Special products Factoring trinomials Simplification of radicals Positive, negative, and fractional exponents Complex numbers Solution of systems of linear equations For Math Placement Test D, take any college trigonometry book and do the chapter review problems for chapters that include the following; • • • • • • The trig functions Trig functions of an acute angle Trig identities Trig functions of any special angle w/o a calculator Properties of logarithms Inverse trig functions