Grade 9 Math – Course Review PDF

Grade 9 Math – Course Review Section A: Numeracy, Polynomials and Equations 1. The set of real numbers and its subsets are often represented using the symbols shown on the right. Describe how each of the following pairs of sets are related. a) N and W b) Z...

Grade 9 Math – Course Review Section A: Numeracy, Polynomials and Equations 1. The set of real numbers and its subsets are often represented using the symbols shown on the right. Describe how each of the following pairs of sets are related. a) N and W b) Z and Q 2. Express each improper fraction as a mixed number. Reduce to lowest terms. 7 9 20 28 36 113 80 a) b) 2 c) 13 d) e) − f) g) − 11 4 5 7 25 3. Express each mixed number as an improper fraction. Reduce to lowest terms. 3 5 6 1 4 a) 2 4 b) 4 8 c) −1 7 d) −2 3 e) 10 5 4. Express each of the following as a percentage. 73 42 7 18 90 38 67 a) 100 b) 50 c) 10 d) 23 e) 200 f) 60 g) 50 5. Evaluate. 6. Evaluate. 7. Find the unit rate for each of the following: a) $65 for 5 boxes b) 24 cans for $10.80 c) 9 boxes for $38.25 d) 175 𝑘𝑚 in 2.5 hours e) $25.50 for 204 boxes f) $6.75 for 750 mL 8. A box of Huggers diapers contains 198 diapers and costs $43.99. A box of Snuggies diapers contains 136 diapers and costs $24.89. Determine which brand is the better buy per diaper and by how much. 9. Determine the value of each of the following. a) 50% of 40 b) 25% of 40 c) 7% of 200 d) 35.4% of 430 10. A computer that has a regular price of $895.99 is on sale for 20% off. a) Determine the price of the computer after b) Determine the cost of the discounted the discount is applied (excluding tax). computer after tax (13% HST). 11. A magic carpet is made with three colors of yarn. The ratio of each color in the carpet is: 10 parts gold yarn, 7 parts bronze yarn, 3 parts silver yarn. The magic carpet is made with a total of 150 meters of yarn. How much silver yarn is in the magic carpet? 12. Express as a single power a) 𝑥12 ÷ 𝑥 5 = _________ b) (−3)12 ÷ (−3)3 = _________ c) 610 ÷ 6 = _________ d) 29 ÷ 28 = _________ 2 9 2 9 −24𝑡 4 e) (5) ÷ (5) = _________ f) = ____________ −3𝑡 4 13. Express as a single power a) 54 ∙ 55 = _________ b) (−7)4 (−7)8 = _________ c) 108 ∙ 10 = _________ d) (𝑥 7 )(𝑥 2 )(𝑥 8 ) = _________ 3 3 e) (−𝑎)4 (−𝑎)0= ____________ f) (4) (4) = _________ 14. Express as a single power a) (510 )5 = _________ b) [(−4)2 ]8 = _________ c) (𝑦 3 )5 = _________ d) (𝑥 8 )5 = _________ e) (0. 79 )4 = _________ f) [(−2)3 ]7= ____________ g) (470 )5 = ____________ h) (2𝑥 )3 = _________ 15. Evaluate or simplify as required a) 104 b) −34 c) (−2)3 1 5 2 3 d) −(−5)2 e) (2) f) (3 𝑥) 3 4 g) − (𝑦 2) h) (−6𝑤)0 i) (5𝑥 2 𝑦 4 )3 16. Simplify a) 4𝑥 + 9 + 6𝑥 + 5 b) 4𝑥 2 − 7𝑥 + 8 − 3𝑥 + 2𝑥 2 − 9 c) −3𝑎2 − 7𝑎 − 8𝑎 − 4𝑎2 d) 𝑥 2 + 3𝑥 − 𝑥 2 − 𝑥 17. Evaluate if 𝑥 = −4 and 𝑦 = 6 3𝑥𝑦 a) 2𝑥 + 𝑦 b) 4𝑥 2 c) d) (2𝑥)2 𝑥 18. For the trinomial, −3𝑥 3 + 2𝑥 + 5𝑥𝑦 2 , list the coefficients _________ 19. Simplify. −45𝑥 3 𝑦 5 a) b) (5𝑥 3 )(−2𝑥 2 ) c) (−3𝑥 2 𝑦)(−4𝑥 5 ) −5𝑥𝑦 4 20. Expand and simplify. a) 5𝑥𝑦(2𝑥 − 3𝑦) b) −4𝑥 3 (5𝑥 + 3) c) 7 + 2( x − 1) d) −3( x + 10) − 8 21. Expand and simplify. a) (8𝑥 2 + 2𝑥 − 3) + (−6𝑥 2 + 4𝑥 + 7) b) (3𝑥 2 + 5𝑥 + 7) − (2𝑥 2 + 4𝑥 − 9) c) 5𝑥(3𝑥 2 − 2𝑥 − 7) + 2𝑥(𝑥 + 7) d) 3𝑦 2 (2 − 3𝑦 − 𝑦 2 ) − 𝑦(4𝑦 − 𝑦 2 ) 22. Solve each of the following. Do a proper LS/RS check for parts (c), (d) and (f) a) 3x − 8 = 4 b) − x + 6 = 2 c) 2 x − 8 = 10 − x 2x −1 x x d) 6( x − 2) = 3x + 2( x − 1) e) =3 f) = +7 5 3 2 5𝑥 1 𝑥 1 g) 4(2 x + 1) = 9 − 3(1 − 4 x) h) +8=4−3 6 23. Check using a LS/RS check whether 𝑥 = −2 is a solution to the following equation. Do NOT solve. 𝑥−3 = 2(3 − 4𝑥) − 21 5 24. Solve the following inequalities. For (a) and (b) graph the solution on the number line. a) 𝑥 – 3 > 5 b) 12 ≤ −𝑥 + 5 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 –10–9 –8 –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 8 9 10 𝑥−6 5(𝑥 – 4) c) 2𝑥 – 8 > 4(𝑥 − 5) d) ≤ 2 3 25. If 𝑥 = 5 is a solution to the equation 2(𝑥 − 3) + 𝑘(1 + 2𝑥) = 𝑘 − 𝑥 − 1 determine the value of 𝑘. 26. David earned four times as much as Mitchell. Together they earned a total of $120. How much did David earn? Make sure to include let statements and a concluding statement. 27. Three houses are numbered with three consecutive EVEN numbers. If their sum is 186, what are the house numbers? Make sure to include let statements and a concluding statement. 28. The ages of Jon and Samantha total 27 years. In 4 years, Samantha’s age plus twice Jon’s age will be 43. What are Jon and Samantha’s ages? Make sure to include let statements and a concluding statement. Section B: Data & Linear Relations Scores on a Science Test in Mr. A’s Class 29. Given the box plot to the right, answer the following questions: a) What is the median of the data? b) What is the range of scores on the test? c) What interval contains the middle 50% of the grades? d) Between which quartiles is the data most concentrated? e) What percentage of students got above 70 on the test? 30. The mass, in kilograms, of ten people randomly chosen from a crowd is listed below. 𝟕𝟒, 𝟕𝟏, 𝟔𝟖, 𝟖𝟒, 𝟓𝟒, 𝟔𝟎, 𝟔𝟒, 𝟔𝟖, 𝟔𝟓, 𝟔𝟒 a) Use the table below to calculate the mean, median and mode. Mean Median Mode/s 𝑆𝑢𝑚 = 𝐷𝑎𝑡𝑎 𝑝𝑙𝑎𝑐𝑒𝑑 𝑖𝑛 𝑜𝑟𝑑𝑒𝑟: 𝑀𝑒𝑎𝑛 = 𝑀𝑒𝑑𝑖𝑎𝑛 = b) Graph a Box and Whisker Plot for the data above. Make sure to include all features of a graph. 𝑄0 = _____ 𝑄1 = _____ 𝑄2 = _____ 𝑄3 = _____ 𝑄4 = _____ 31. The data for 𝟏𝟐 professional basketball players is given in the chart. Most Points in Disqualifications a) Draw a scatter plot. One Game in One Season 𝟓𝟓 𝟎 b) If there is a relationship, draw a line of 𝟑𝟑 𝟏 best fit. 𝟐𝟕 𝟎 𝟏𝟗 𝟎 𝟏𝟗 𝟑 𝟐𝟕 𝟏 𝟏𝟒 𝟓 𝟏𝟔 𝟎 𝟏𝟏 𝟎 𝟏𝟒 𝟏 𝟏𝟒 𝟎 𝟏𝟔 𝟎 32. Use the data in the table to answer each question. Time (s) 𝟎 𝟏 𝟐 𝟑 𝟒 𝟓 𝟔 Distance (m) 𝟎 𝟓 𝟕 𝟏𝟐 𝟏𝟓 𝟏𝟗 𝟐𝟓 a) Which quantity is the independent variable? The dependent variable? b) Graph the data using the grid to the right. c) Is this relationship linear or non-linear? d) Draw a line or curve of best fit. e) Estimate the distance traveled in 2.5 𝑠, 6.25 𝑠, and 8 𝑠. f) Estimate the time taken to travel 3 𝑚, 21 𝑚, and 36 𝑚. g) Which of the following is the best match for correlation coefficient of the above data. Circle the correct answer then explain. −0.97 −0.6 0.01 0.54 0.99 33. Determine the first differences for each table of values. State which are linear and which are non- linear relations. For each linear relation, state an equation which represents the relation. a) 𝑥 𝑦 b) 𝑥 𝑦 c) 𝑥 𝑦 d) 𝑥 𝑦 1 3 0 0 −1 2 1 6 2 6 1 1 0 0 2 5 3 9 2 4 1 −2 3 4 4 12 3 9 2 −4 4 3 34. For each of the following graphs: a) determine the equation of each line by finding the slope and intercept. b) state the co-ordinates of the point of intersection of the two lines. c) verify that this point satisfies the two equations. i) ii) 35. Determine the slopes of the line segments joining the following pairs of points: a) 𝐴(1, −1) and 𝐵(−3,3) b) 𝐶(4, −1) and 𝐷(4, −6) 36. High school theatre companies earn their income through start-up grants and ticket sales. The graph shows the relationship between income, I, in dollars & number of tickets sold, n. a) If 30 tickets are sold, which company will make more money? b) What does the point (20, 200) represent? c) Under which conditions will Company A make more money? Company B make more money? 37. Complete the following table: equation slope y-intercept 2 𝑦 = − 𝑥−4 3 5 3 2 2 −4 0 0 2 𝑦 = −𝑥 + 3 38. Sketch all 4 lines below on the same graph (shown above to the right), using the method indicated. Use space provided to algebraically calculate the 𝑥- and 𝑦-intercepts for part c). a) slope y-intercept b) slope y-intercept c) 𝒙 & 𝒚 intercepts d) Any method 2 𝑦 = 3𝑥 − 5 𝑦 = −3𝑥 + 1 3𝑥 − 2𝑦 + 6 = 0 𝑥=5 39. Complete the chart below. Inequality a) 𝑥 ≥ 1 b) 𝑦 8 b) 𝑥 ≤ −7 c) 𝑥 < 6 d) 𝑥 ≥ 7 25. 𝑘 = −1 26. Mitchell $24, 27. 60, 62, 64 28. Samantha 23, David $96 Jon 4 Section B: Data & Linear Relations 29. a) median = 75 b) 100 − 55 = 45 c) 70 to 85 d) 𝑄1-𝑄2 e) 75% 30. a) mean = 67.2 ; median = 66.5 ; mode = 64,68 b)𝑄0 = 54, 𝑄1 = 64, 𝑄2 = 66.5, 𝑄3 = 71, 𝑄4 = 84 31. b) The data shows no correlation, so drawing a line of best fit is not appropriate. 32. a) time is independent b) linear d) 10𝑚, 25𝑚, 32𝑚 e) 0.75𝑠, 5.25𝑠, 9𝑠 f) 𝑟 = 0.99 since the data has a very strong positive correlation. 33. a) 1st diff's: 3,3,3 ; linear ; 𝑦 = 3𝑥 b) 1st diff's: 1,3,5 ; non-linear c) 1st diff's: –2,–2,–2 ; linear ; 𝑦 = −2𝑥 d) 1st diff's: –1,–1,–1 ; linear ; 𝑦 = −𝑥 + 7 34. a) i) 𝑦 = 2𝑥 + 4, 𝑦 = −𝑥 + 1 ii) 𝑦 = −𝑥 − 1, 𝑦 = −2𝑥 + 1 b) i) (−1,2) ii) (2, −3) 35. a) −1 b) Does not exist 36. a) Company A b) Point of intersection. Both companies will earn $200 when selling 20 tickets. c) Company A when more than 20 tickets. Company B when less than 20 tickets are sold. 37. 𝑎) 𝒆𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝒔𝒍𝒐𝒑𝒆 𝒚 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕 𝑏) 2 2 𝑦 =− 𝑥−4 − –4 3 3 5 3 5 3 𝑦= 𝑥+ 2 2 2 2 𝑦 = −4𝑥 –4 0 𝑦=2 0 2 𝑐) 𝑦 = −𝑥 + 3 –1 3 𝑑) 38. See the graph above to the right. For c) x-int: (−3,0), y-int: (0, 2) 44. b) 39. a) b) 𝑦 < 1 c) 1 5 40. a) 𝑦 = −𝑥 + 6 b) 𝑦 = − 3 𝑥 + 3 c) 𝑦 = −𝑥 + 5 d) 𝑦 = −𝑥 + 5 41. 𝑘 = 2 42. a) line would rotate, and slope would be less steep b) Line would translate 8 units up. 43. a) 𝑦 = 27.75𝑥 + 40 b) $789.25 44. a) For both scenarios: 𝑃 = $2000, 𝑟 = 0.07, 𝑡 = 15. Scenario #1: Simple Interest which is linear; Scenario #2: Compound Interest which is non-linear b) See above to the right c) Compound interest will earn more interest is earned on previous interest. Section C: Measurement and Geometry 45. 40 𝑘𝑚 46. a) ∼ 9.64 𝑓𝑡 b) ∼ 2.56 𝐿 47. 𝑃 ≐ 22.7 𝑚𝑚, 𝐴 ≐ 37.0 𝑚𝑚2 48. a) 𝑥 = 27° b) Using Thales Theorem: 𝑎 = 45°, 𝑏 = 67° 49. a) 𝑉 = 297 𝑐𝑚 , 𝑆𝐴 ≐ 295.2 𝑐𝑚2 3 b) 𝑉 ≐ 1588.6 𝑚3 , 𝑆𝐴 ≐ 831.3 𝑚2 50. The surface area of the cube will be 102 = 100 larger. 51. The volume of the prism will be 3 times larger than the pyramid. 52. ℎ = 5 𝑐𝑚, 𝑆𝐴 = 527.52 𝑐𝑚2