Evaluating Expressionspjk.scripts.mit.edu/lab/mdm/Prerequisite_Skills_ALL.pdfEvaluating Expressions McGraw-Hill Ryerson Mathematics of Data Management 1. ... h m. where . m. is the

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Evaluating Expressions McGraw-Hill Ryerson Mathematics of Data Management

1. Write an algebraic expression for each of the following.

a) the product of x and 13 b) the quotient of r divided by d c) the total of b, w, and 19 d) 7 less k

2. Evaluate each expression for the given value of the variable.

a) 5g2, g = 3.2 b) 3m, m = 2.7 c) 11 – 7t3, t = –2 d) x3 – y3, x = 3, y = –2

3. Evaluate each expression, to the nearest tenth, given x = 2.1, y = –3.3, z = 4.7.

a) x2 – 2y + z2 b) x3 – 4xy – y3 c) y2 – x2 – 3z

4. Evaluate each expression, given x = –3, y = 5, z = –2.

a) x2 – y2 + z2 b) y + z2 – 3x2 c) 4y – 3x2 + z d) (x – 2y)(z + 2y)

5. Expand and simplify. Then, evaluate for r = 4, s = 5, and t = –3.

a) 6(3r – 4s – 9) b) –5(2r2 + 3rs – s2) c) 3t(t – 4) – t(t + 5) – 2t(t – 1) d) 3s(2s2 + st – t2) – t(4s2 – 5r + 3t2)

6. The formula for the height of a model rocket is h = –t2 + 20t, where h is the height, in metres, and t is the time, in seconds, after liftoff.

a) What is the height of the rocket after 9 s? 15 s? b) Find the rocket's height after 20 s. Explain your answer.

7. Use the formula ,7

)90(6

hm where m is the approximate mass of a young adult, in kilograms,

and h is the person's height, in centimetres, to find the mass of a young adult whose height is

a) 175 cm b) 220 cm

8. Evaluate each expression, given a = 7, b = –4, c = 2.5.

a) b2 – c2 b) a2 – 3abc + b3 c) a3 – b3 – c d) (a + b)3 – (c – b)2

9. Complete the columns for s and A, using the information in parts a) and b).

a b c s A

5 cm 7 cm 8cm

11 cm 19 cm 20 cm

8.4 m 3.6 m 6 m

1.5 m 1.5m 2.1m

a) Given the side lengths a, b, and c of a triangle, calculate half the perimeter, s, using the formula

2

cbas

.

b) Find the area, A, of the triangle, using the formula ))()(( csbsassA .

Copyright 2003 McGraw-Hill Ryerson Limited Review of Prerequisite Skills MHR 101

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Exponent Laws McGraw-Hill Ryerson Mathematics of Data Management

1. Simplify.

a) 22 × 23 b) 35 × 33 c) 44 × 42 d) 103 × 10 e) 94 × 93

f) 8 × 84 g) x2 × x5 h) y3 × y3 i) z3 × z2

2. Find the missing exponent.

a) 32 × 3□ = 34 b) 5□ × 54 = 57 c) 83 × 8□ = 85 d) 7□ × 73 = 74

e) j5 × j□ = j8 f) b□ × b5 = b9 g) k × k9 = k□ h) s6 × s□ = s7

3. Simplify.

a) 54 ÷ 52 b) 46 ÷ 43 c) 33 ÷32 d) 95 ÷ 92

e) 74 ÷ 73 f) 26 ÷ 24 g) m7 ÷ m5 h) p8 ÷ p6


a) 25 ÷ 2□ = 23 b) 34 ÷ 3□ = 32 c) 4□ ÷ 42 = 44 d) 5□ ÷ 53 = 5 e) n4 ÷ n□ = n2 f) c□ ÷ c4 = c3 g) y□ ÷ y2 = y2 h) z9 ÷ z□ = z

5. Simplify.

a) (32)3 b) (24)2 c) (73)4 d) (62)4 e) (53)2 f) (45)3 g) (x3)3 h) (r5)2


a) (33)□ = 39 b) (25)□ = 210 c) (5□)2 = 58 d) (4□)3 = 412

e) (g2)□ = g6 f) (m3)□ = m9 g) (s□)5 = s20 h) (t□)2 = t6

7. Evaluate for a = 5 and 6 = 3. Are the expressions equal? If not, which expression has the lesser value?

a) a2 + b3 or a3 + b2 b) a2b2 or (ab)2

8. Is each statement true or false? Explain.

a) 33 = 81 b) 6(–2)3 = 48 c) y2 × y4 = y6

d) ((–2)3)3 = –512 e) (–a)4 ÷ (–a)2 = a2 f) (–5)3 ÷ (–5)2 = 5

9. The standard forms of the following pairs of terms are not the same. Explain.

a) (–2)4 and –24 b) ((–3)2)3 and –36

10. Simplify.

a) 2a4b3 ÷ a2b b) –4w3x5 ÷ (–2w2x2) c) 8x6y4 ÷ (–4x3y2)

d) 22

23

3

6

rq

rq e)

hfg

hgf2

253

6

9 f)

c

dc

18

12 55

11. Simplify.

a) –9e2f4 ÷ (–6ef2) b) 4a3b2c ÷ 2bc c) 8x5y3 ÷ 2x3y

d) 22

67

8

12

ts

ts e)

2

42

3

6

km

mk f)

432

535

12

20

fed

fed

12. The perimeter of a square is 8s2t. Write and simplify an expression for the length of a side.

102 MHR Review of Prerequisite Skills Copyright 2003 McGraw-Hill Ryerson Limited

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Fractions, Percents, Decimals McGraw-Hill Ryerson Mathematics of Data Management

1. Write as percents.

a) 0.45 b) 0.5 c) 0.004 d) 0.596 e) 1 f) 2.63

2. Write as percents, to the nearest tenth of a percent.

a) 10

7 b)

5

4 c)

40

27 d)

8

7 e)

9

5 f)

8

9

3. Write as percents.

a) 73:100 b) 7:100 c) 3:5 d) 9:10 e) 3:1 f) 5:4

4. Complete the table. 5. Calculate the percent.

Fraction Decimal Percent

a) 0.65

b)

4

3

c) 0.02

d) 90%

e)

25

9

f ) 37.5%

g) 88%

h) 0.725

i) 125%

j)

2

5

a) 154 out of 200 marks b) 3 students absent out of 160 students c) 10 winning tickets out of 1000 tickets d) 100 households chosen out of 100 000

households e) What percent is 459 of 153? f) What percent of 2.5 is 8

6. If you can jump 200 cm on Jupiter, you can jump 528 cm on Earth. What percent of the height of a jump on Jupiter is the height of a jump on Earth?

7. A backpack sells for $20. Before the price increased, the backpack sold for $18.

a) What was the percent increase? b) What percent of the old price is the new price?

8. The table lists the depths of the Great Lakes.

Lake Depth(m)

Superior 405

Michigan 281

Huron 229

Erie 64

Ontario 244

a) Determine the difference in depth between Lake Erie and each of the other lakes.

b) Use the differences in part a) to write rational numbers

in the form difference in depth

depth of Lake Erie.

c) Write each rational number in part b) as a decimal.


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Graphing Data McGraw-Hill Ryerson Mathematics of Data Management

1. The bar graph shows the energy, in kilojoules per minute, used for several activities.

a) Which activity uses energy about 2.5 times as fast as house cleaning? Energy Consumption

0 10 20 30 40 50

Brisk Walking

Sw imming

Cycling

House Cleaning

Act

ivit

y

Energy (kJ/min)

b) If you walked for 20 min, how much energy would you use? c) Which two activities combined would match the energy used by swimming? d) The recommended amount of energy you should use per week is about 7000 kJ. If you cycle four days a week, for how long do you need to cycle each day to meet this target?

2. The table shows the estimated population of several of the largest species of the world. Species Estimated PopulationMammals 4 600Birds 9 500Reptiles and Amphibians 10 000Fish 24 000

Display the data on a bar graph.

3. Plot two broken-line graphs on the same set of axes to show the average monthly temperature for Victoria, B.C., and Melbourne, Australia.

Temperature (°C) Temperature (°C)

Month Victoria,

B.C. Melbourne, Australia Month

Victoria, B.C.

Melbourne, Australia

January 3 20 July 16 13 February 5 19 August 16 14 March 6 18 September 14 15 April 8 15 October 10 15 May 12 14 November 6 17 June 14 13 December 4 20

4. Complete the table. Round all percents and degrees to the nearest whole number.

Causes of Air Pollution Cause Percent Calculation of Central Angle Central Angle

Industry 15 0.15 × 360° 54° Transportation 44 Humans 0.09 × 360° Fuel 72° Other

5. The table shows the sizes of the continents. Use the data to draw a circle graph.

Continent Area (thousands of square kilometres) Continent

Area (thousands of square kilometres)

North America 24 350 Africa 30 300 South America 17 870 Australia 8 550 Europe 9 840 Antarctica 13 990 Asia 45 100


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Graphing Exponential Functions McGraw-Hill Ryerson Mathematics of Data Management

1. Given suitable conditions, a certain culture of bacteria triples every 24 h. The initial count of a sample shows 100 bacteria present.

a) Write an exponential function that models the given conditions. Define each variable used in the model.

b) Approximately how many bacteria will there be in two days? c) How many bacteria were there two days prior to the count? d) Graph the function.

2. An investment of $4500 is earning interest at 6% per annum compounded semi-annually.

a) Write an exponential function to represent the amount of the investment, A(t), in dollars, after t years.

b) Graph the function. Use the graph to determine approximately how many years it will take for the amount of the investment to double.

3. The intensity of light below the surface of a particular lake is reduced by 3% for every metre below the surface.

a) Write an exponential function that models the intensity of light at any depth below the surface. b) What percent of the original intensity of light remains at 5 m below the surface? c) Graph the function. d) Use the graph to determine how far below the surface the intensity of light is 50% of the

surface intensity. Round the answer to the nearest tenth of a metre.

4. Graph each of the following functions.

a) y = 3x b) y = 3–x c) y = 1.5(3)x

d) y = 2(0.6)x e) y = 0.3(4)x f) x

y

3

1

5. Solve for x.

a) 82x – 3 =16x – 1 b) 82

4

9

127

xx c) 13

32

4

3216

8

xx

x

d) xxx 2)16(2 1 2

6. To determine whether a person has a thyroid deficiency, a radioactive iodine with a half-life of 8.2 days is injected into the blood stream. A healthy thyroid gland absorbs all the radioactivity.

a) Determine an exponential function that models the amount of radioactive iodine that should be present in the thyroid gland of a healthy person after t days, given that the original amount injected is I0.

b) How long will it be before 87.5% of the iodine is no longer radioactive in the body of a healthy person?

c) Graph the function given that 25 mg is originally injected.

7. A bacterial culture starts with 500 bacteria. After 2 h, the estimated count is 8000. What is the doubling period for this bacterial culture?

8. A certain bacterium quintuples every month. How long will it take a single bacterium to grow to 2524 bacteria?


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Graphing Linear Functions McGraw-Hill Ryerson Mathematics of Data Management

1. The domain of each equation is {–2, –1, 0, 1, 2}. Complete a table of values. Then, graph each equation.

a) y = –2x + 5 b) 2x + y = 0

2. Graph each equation. The domain is R.

a) y = 2x + 1 b) y = –x – 3 c) x + y = –1 d) 2x – y = 3

3. Given the two tables of values, write an equation for each relation.

a) x y b) x y –2 2 –2 –6 –1 3 –1 –3 0 4 0 0 1 5 1 3 2 6 2 6

4. Given the points on the grid, write an equation to represent the relation. Then, state the domain.

5. The table shows the equivalent depths of water

and heavy wet snow.

Depth of water (cm) 5 10 15 20 25

Depth of heavy wet snow (cm) 100 200 300 400 500

a) Plot the depth of heavy wet snow versus the depth of water. b) Can you join the points to graph the equivalent depths of water and heavy wet snow? Explain. c) Use the graph to estimate the depth of heavy wet snow equivalent to 12 cm of water. d) Use the graph to estimate the depth of water equivalent to 360 cm of heavy wet snow.

6. Use the x- and y-intercepts to graph each line.

a) 2x + 3y = 6 b) 4x – y = 4 c) 3x + 5y – 30 = 0 d) x + y + 2 = 0

7. Graph each equation using the slope and the y-intercept.

a) y + l = 2x b) x + 4y = 8

8. Write an equation of a line whose x- and y-intercepts are opposite integers, but not 0, and whose x-intercept is positive.

9. Write an equation of a line whose y-intercept is 0, and whose slope has a negative value.

10. The screen of an air-traffic controller shows two planes approaching at 10 000 m. One plane is travelling in a direction described by y = 4 – 2x, and the other in a direction described by x – y = –1. Can the planes continue in the same manner? Give a reason for your answer.


Name _____________________

Graphing Quadratic Functions McGraw-Hill Ryerson Mathematics of Data Management

1. Sketch the graph of each parabola and state the direction of opening, the coordinates of the vertex, and the maximum or minimum value.

a) y = x2 + 4 b) y = 2x2 c) y = –x2 +2

d) 12

1 2 xy e) 42

3 2 xy f) y = –3(x2 – 4)

2. State how the graph of the second equation is related to the graph of the first equation.

a) y= 2x2 + 1 and y = x2 – 3 b) y= –x2 and 2

2

1xy c) y = x2 and y = –x2 + 3

3. Write an equation for the parabola created when each pair of transformations is applied to the graph of y = x2.

a) a reflection in the x-axis, followed by a vertical translation of –3 b) a vertical translation of 2, followed by a vertical stretch of scale factor 3

4. Graph manually to estimate the x-intercepts, if any, or use a graphing calculator to determine them to the nearest tenth.

a) 44

1 2 xy b) y = –3x2 – 3 c) y = 2x2 – 9 d) y = –4x2 +25

5. Write an equation for the quadratic function with the given vertex and passing through the given point.

a) vertex (0, 3), point (–2, 11) b) vertex (0, 1), point (4, –7) c) The graph of y = 4x2 + c passes through the point (–1, –1). Find c. d) The graph of y = ax2 + q passes through the points (2, 3) and (–4, –9). Find a and q.

6. Without graphing, state the coordinates of the vertex and whether this vertex represents a maximum or minimum value of the function.

a) y = –3(x + 5)2 – 6 b) 2)4(4

3 2 xy c) y = 7.5(x – 2.5)2 – 9 d) y = –(x – 9)2 +19

7. Write an equation for each parabola.

a) vertex (3, –1); a = –2 b) vertex (2, 5); congruent to 2

2

1xy

c) vertex (–4, –1); y-intercept –9 d) vertex (–5, 3); through the point (–7, 15)

8. The vertex of a parabola is (–3, 7). The y-intercept is 0. What are the x-intercepts?

9. A parabola has axis of symmetry x = 2 and passes through the points (0, 7) and (–1, –3). What is the equation of the parabola?

10. Find the value of c that will make each expression a perfect square trinomial.

a) x2 + 22x + c b) x2 – 16x + c c) x2 +7x + c d) cxx 5

42

11. Write each function in the form y = a(x – p)2 + q. Sketch the graph. Show the coordinates of the vertex and two other points on the graph, and indicate the axis of symmetry.

a) y = x2 – 4x – 1 b) y = –2x2 – 4x + l c) 432

1 2 xxy


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Mean, Median, Mode McGraw-Hill Ryerson Mathematics of Data Management 1. Find the mean of each set of numbers.

a) 14, 17, 27, 16, 26 b) 26, 17, 34, 31, 48, 36 c) 29, 14, 37, 24, 51, 31 d) 96, 62, 100, 74 e) 55, 89, 111, 33 f) 157, 127, 164, 148, 139 g) 40, 53, 47, 38, 57 h) 114, 278, 141, 295

2. The temperature, as recorded daily at 14:00 for one week in September, was 25°C, 21°C, 23°C, 19°C, 15°C, 18°C, and 19°C. Find the mean temperature for the week.

3. Find the median of each set of numbers.

a) 27, 33, 19, 45, 24 b) 86, 78, 75, 83, 81, 79 c) 103, 121, 114, 118, 132, 100

4. Find the mode of each set of numbers.

a) 2, 4, 5, 4, 3, 4, 2, 7 b) 36, 35, 37, 36, 40, 39 c) 62, 68, 64, 68, 67, 65, 64, 61

5. Mooreville School has six mathematics classes, with 23, 25, 27, 30, 21, and 24 students.

a) What is the mean class size? b) What is the median class size?

6. Find the mean and the median to the nearest tenth, for each set of values.

a) 31, 33, 23, 45, 24 b) 94, 79, 74, 107, 87 c) 186, 140, 169, 192, 111, 158, 173 d) 224, 270, 198, 251, 169, 258, 206 e) 117, 93, 104, 131, 135, 109 f) 726, 829, 803, 830, 906, 731

7. List the following.

a) Four different numbers with a median of 32 b) Seven different numbers with a median of 87

8. The table shows the marks on a math test. Find the mean, median, and mode of the marks.

Mark Frequency Mark Frequency45 3 38 742 2 37 440 1 35 3

9. The mean of the numbers 8, 12, 13, 15, 17, 19, and y is 16. Find y.

10. Write five numbers that have a) a mean of 12 and median of 13 b) a mean of 9 and mode of 10.

11. The list includes the names of eight famous Canadians and their life spans.

Norman Bethune 1890–1939 Joseph-Armand Bombardier 1907–1964 Ned Hanlan 1855–1908 Helen Sawyer Hogg 1905–1993 Margaret Laurence 1926–1987 Lester B. Pearson 1897–1972 Mary Pickford 1893–1979 Emily Stowe 1831–1903

a) Complete the list by finding the number of years each person lived. b) Determine the mean, and the median of the values found in part a). c) Do the values in part a) have a mode? d) How many people on the list lived for less than the mean value? e) How many people on the list lived for less than the median value? f) Investigate why each person on the list was famous.


Name _____________________

Number Patterns McGraw-Hill Ryerson Mathematics of Data Management

1. Eleven people are at a party. Each person shakes hands with each of the others. What is the total number of handshakes?

2. Toothpicks are used to create the following pattern. How many toothpicks would be in diagram 12? diagram 25?

3. a) How many small triangles would be in the 8th figure? the 12th figure? the 25th figure?

b) Describe the pattern in words.

4. The numbers 1, 8, 27, and 64 are the first four cubes. Find the sum of

a) the first two cubes b) the first three cubes c) the first four cubes d) Describe the pattern in the sums. e) Find the sum of the first nine cubes.

5. Find the total numbers of squares of all sizes on each grid.

a) 4 squares by 10 squares b) 5 squares by 12 squares c) chessboard (8 by 8) d) Snakes and Ladders board (10 by 10)

6. A Senet gameboard consists of a 3 by 10 grid of squares. Find the total number of squares of all sizes on a Senet game board.

7. If you place a block on a table, 5 faces are showing. If you add another block as shown, 8 faces are showing. How many faces are showing when there are 21 blocks in a row?

8. The staircase has 4 steps and is made up of 20 cubes. How many cubes are needed for a staircase with 12 steps?

9. Eight people entered a tennis tournament. Each person played each of the others once. How many games were played?

10. A triangle has no diagonals. A quadrilateral has two diagonals. A pentagon has five diagonals. How many diagonals does a 12-sided figure have?

11. What is the units digit when 17133 is written in expanded form?

12. Find the sum of

100

99

100

2

100

1

5

4

5

3

5

2

5

1

4

3

4

2

4

1

3

2

3

1

2

1 .

13. The value of n!, called n factorial, is defined as n! = n × (n – l) × (n – 2)...× 3 ×2 × l. For example, 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720. What is the last digit of 1! + 2! + 3! +...+ 99!?

14. Given that S1 = 1, S2 = 2 + 3, S3 = 4 + 5 + 6, S4 = 7 + 8 + 9 + 10, and so on, determine the value of S20.


Name _____________________

Order of Operations McGraw-Hill Ryerson Mathematics of Data Management

1. Simplify.

a) 13 + 8 – 7 b) 12 × 5 + 18 c) 15 + 9 – 3 d) 16 ÷ 4 × 7 e) 32 + 3 × 2 f) 8(5 × 2) + 62 g) 6 × 3 ÷ 9 + 9 h) 43 – (2 + 4)

i) 34

31023

j) 3 × 7 + 42 – (6 – 2)

2. Insert brackets to make each statement true.

a) 16 ÷ 4 – 2 – l = 7 b) 5 – 3 + 6 × 2 = 16 c) 4 + 6 ÷ 2 = 5 d) 1.7 × 1.3 – 0.8 = 0.85 e) 5.2 × 3.4 – 2.4 = 5.2 f) 2 – 5 × 3 + 23 = –1

3. Evaluate.

a) (+3) + (–3)(–4) b) (–2)(+5) – 4 c) 16 ÷ (–8) – (–3) d) 0 ÷ (4)(7) e) (–4)(–3) × 0 f) (+12) × (–3)(–2)

4. Evaluate.

a) 14 ÷ (–7) + 5 b) 4 × 5 + 10 c) 9 – (–4) + 7 d) 9 + 5 – 10 – 3 e) –3 + (–5) – 8 f) 12 + (–4) ÷ 4

5. Evaluate.

a) –3(–5) ÷ 5 b) (–6) × 7 + 15 c) 11 – (–5)(+3) d) –4(+2) + 4(–2) e) 16 ÷ 8 × 4 – 8 f) 50 ÷ (–10) × 7

6. Evaluate.

a) 4(9 – 5) – 9 b) –4 + 3(8 – 5) c) (–10)(5 + 2) – 2 d) 6(4 – 6) + 4(3 + 1)

7. Evaluate.

a) –3 + (–4)2 b) (2)3 – 5 c) –32 × 4 d) (–2)3 – (–1) e) –2 + (–3)2 × 2 f) (–4)(4 – 5) 2 – (–2)2

8. Evaluate.

a) 6

43 b)

12

)3(4 c)

)1(3

515

d) 76

)2(33

e) 611

)41( 2

f) 2

2

3

7)4(

9. a) Student ski lift tickets cost $25.50 per day on weekends. On weekdays, the cost is 0.8 of the weekend rate. Write an expression to calculate the cost for a group of 25 students skiing on Sunday and Monday.

b) Write an expression to show how much would be saved if 20 students went skiing on Monday and Tuesday instead of on Sunday and Monday.


Name _____________________

Ratios of Areas McGraw-Hill Ryerson Mathematics of Data Management

1. a) Which two triangles are similar? Why? P

Q

R

32

16

24

ST

U

24

16

20

V

X

W

8

16

12

b) For the pair of similar triangles, state the pairs of corresponding sides and angles. c) Are the areas of the similar triangles similar? Explain.

2. The triangles in each pair are similar. Find the unknown side lengths, and the ratio of the areas.

a) b) c)

P15 cm

R25 cm

20 cm

20 cmS

U

T t

s

V

15 m

W32 m

10 m 16 m

Y

ZX

wy

3. A school has two rectangular skating rinks. One rink measures 11 m by 9 m and the other measures 13 m by 7 m. After a snowfall, the principal asks two students to shovel the snow from the rinks. Which rink would you prefer to shovel? Give a reason for your choice.

4. A gardener has 28 m of fence. She wants to enclose a rectangular part other garden to grow vegetables. What is the area of the largest rectangular garden that she can enclose with each length of fence?

a) 28 m b) 40 m c) 56 m

5. The Caseys built a circular patio with diameter of 5 m, and surrounded it with a ring of lawn 3 m wide. The remaining area in the yard became a flower garden. Find the ratio of the area of the patio to the area of lawn.

6. Write a ratio in simplest form to compare the areas of the Great Lakes.

Lake Area (km2)

Superior 84 000

Michigan 58 000

Huron 63 000

Erie 26 000

Ontario 19 000

a) Lake Huron to Lake Superior b) Lake Erie to Lake Michigan c) smallest to largest


Name _____________________

Scatter Plots McGraw-Hill Ryerson Mathematics of Data Management

1. The scatter plot shows the number of competitors at each Winter Olympics from 1964 to 1998. Describe any relationship you see.

Winter Olympics Competitors

1000

1200

1400

1600

1800

2000

2200

2400

64 68 72 76 80 84 88 92 96Year

Num

ber

of C

ompe

titor

s

2. The table gives average heights and average masses of different types of dogs.

Type of Dog Height (cm) Mass (kg) Belgian Sheepdog 63 30Cocker Spaniel 42 14Collie 63 27English Springer Spaniel 51 23Irish Setter 69 32Irish Terrier 48 12Japanese Chin 30 4Labrador Retriever 60 30Newfoundland 71 65Saint Bernard 70 57

Draw a scatter plot of mass versus height. Describe any relationship you see.

3. A class recorded their marks in science and in French, and drew a scatter plot of the data. From these data, does it appear that a student’s science mark is a good indication of the French mark? Explain.

Class Marks

3035404550556065707580859095

100

40 50 60 70 80 90 100Science

Fre

nch4. The table gives the total population of

whooping cranes in North America over several years.

Year Population Year Population 1962 45 1978 751964 35 1980 771966 40 1982 731968 50 1984 821970 60 1986 110 1972 52 1988 1301976 68 1990 140

a) Draw a scatter plot of population versus year. b) Use your scatter plot to estimate the total whooping crane population in 1994 and 1998. c) Use your research skills to find the actual numbers in 1994 and 1998. Compare these number

with your estimates.


Name _____________________

Sigma Notation McGraw-Hill Ryerson Mathematics of Data Management

1. Write each series in expanded form. Then, determine the sum.

a)

5

0

22

1

k

k b) c)

6

1

)23(k

k

9

1

)5(k

k

d) e)

6

2

1)3(4k

k

5

3 2

15

m

m

f)

3

3

2 )7(k

k

2. Write each series in expanded form. Then, find the sum.

a) b) c)

8

1

)6(k

k

6

1

)14(k

k

5

1

)716(k

k

d) e) f)

6

1

)2005.2(k

k

15

10

)3100(k

k

9

3

)2.18.7(k

k

3. Write each series in expanded form. Determine the sum of each.

a) b) c) d)

3

1

1)2(4k

k

4

1

)1.0(10i

i

5

1

19k

k

8

3 2

1200

m

m

e) f) g) h)

10

1

11k

k

7

1

1)5(3k

k

8

1

12 )3(k

k

10

3

1)15.0(3.0k

k

4. Write each series using sigma notation with lower limit 1. Determine the sum of each.

a) 64

3

16

3

4

3312 b) 0.5 + 1 + 2 +...+ 256

c) 16

33612 d) 0.11 + 0.33 + 0.99 +... + 721.71

e) 256

25

4

2525100 f) 1.99 + 19.9 + 199 +...+ 1 990 000

5. Determine the explicit formula for the terms in each series. Then, write the series using sigma notation with the specified lower limit.

a) 2 + 6 + 10 + 14 + 18 + 22, lower limit 1

b) 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40, lower limit 1

c) –2 – 5 – 8 – 11 – 14 – 17 – 20, lower limit 3

d) 1.5 + 3.5 + 5.5 + 7.5 + 9.5 + 11.5, lower limit 1

e) 4.4 + 2.4 + 0.4 – 1.6 – 3.6 – 5.6 – 7.6, lower limit 2

f) 120 + 100 + 80 + 60 + 40 + 20, lower limit –5


Name _____________________

Similar Triangles McGraw-Hill Ryerson Mathematics of Data Management

1. The triangles in each pair are similar. Find the unknown side lengths.

a) b) c)

2. Explain why ABC is similar to AEF. 3. Find a.

E

B

FCA

4. Nida is 1.8 m tall and casts a shadow 1.5 m long. At the same time, a microwave relay tower casts a shadow 32 m long. Draw and label two triangles depicting the information. Determine the height of the tower.

5. A park designer placed survey posts, as shown in the diagram below. Measurements were made as indicated. How far apart are the posts?

21 m

post B

post A

18 m

pond

2 m

6. Explain why similar triangles cannot be used to measure the height of the tree in the diagram.

7. Observe your surroundings and identify a situation where similar triangles are used.

a) Design a problem based on that situation. What do you need to solve your problem? b) Draw an appropriate diagram. c) Solve the problem. d) Give your problem to a classmate to solve.

8. a) Explain why these two triangles are similar.

b) State the pairs of corresponding angles and corresponding sides.


Name _____________________

Simplifying Expressions McGraw-Hill Ryerson Mathematics of Data Management

1. Simplify.

a) a

a

ab

ba

3

6

2

4 222

b) 32

2

20

15

5

3

p

pq

pq

qp c)

dc

dc

cd

dc3

2232

8

12

4

6

d) x

y

x2

3

3

e) 4

3

4

3

y

x f)

8

23

2

5

mm g)

5

2

3

)3(3 aa

h)

s

qq

4

5

2

32

2. Simplify each expression.

a) 239

15

a

a b)

2

2

14

35

y

zy c)

2

234

4

16724

x

xxx d)

44

33

72

54

yx

yx

3. Express in simplest form.

a) )3(15

3

x

x b)

2

)2(

x

xx c)

)2)(3(

)3)(3(

xx

xx d)

158

252

2

xx

x

4. Simplify.

a) 155

124

x

x b)

13

3 2

y

yy c)

34

2

12

92

2

2

2

xx

xx

xx

x d)

253

2322

2

pp

pp

5. Simplify.

a) 52

25

y

y b)

)3)(3(

)3)(3(

xx

xx c)

x

x

6

6 d)

b

b

1015

2114

6. Simplify.

a) 20

52

xx

x b)

127

32

aa

a c)

107

1532

xx

x d)

12

1522

2

xx

xx

e) 252

4113

12

2532

2

2

2

aa

aa

aa

aa f)

34

2

23

322

xxxx

7. For what values of x are the following expressions not defined?

a) x

5 b)

xx

x

3

5 c)

7

7

x

x d)

78

32

xx

x

8. Write expressions for the following.

a) The time it takes to travel (4 – 4x) km at 100 km/h. b) The cost of one widget if it costs $250(x2 – 6x + 8) to produce 500(5x – 2) widgets.

9. A solid cylinder has a height x, equal to its radius.

a) Find the ratio of the volume to the surface area. Simplify, if possible. b) If the expression found in part a) equals 3 for a particular cylinder, what is the height of the

cylinder? c) If the volume of that cylinder is 7.2 cm3, find its surface area.

10. Find the ratio of the volume to the surface area of the rectangular prism shown. Simplify, if possible.

Copyright 2003 McGraw-Hill Ryerson Limited Review of Prerequisite Skills MHR 115 4x + 8

x 2

2x + 4

Name _____________________

Solving Equations McGraw-Hill Ryerson Mathematics of Data Management

1. Solve.

a) 5t = –30 b) –9r = 36 c) –56n = –8

d) 15

10k

e) 7

12s

f) 2.5r = –7.5

2. Solve.

a) 4t – 6 = –10 b) –3 = –2 – x c) 5 + 2n = –15 d) 3p – 7 = 14 e) –14 = 4y + 3y f) 8a – 3a = 15

3. Solve.

a) 3114

y

b) 4

1

4

3

2

n c) 6a – 3a + 5 = 14

d) 3b + 2b – b = 15 – 7 + 4 e) 3r + r + 2r – 6 = 11 + 2 – 7

4. Solve.

a) 3x + 1.2 = 3.9 b) 4k – 2.5 = 1.5 c) 4 –3.2d = 13.6 d) 0.6g – 1.6 = 0.8 e) 6.3 = 0.1 – n f) 12 – 2.4a = 2.4

5. In 1992, the first year the Toronto Blue Jays won the World Series, the Jays played 162 regular- season games and 12 championship games. They won 34 more games than they lost. Solve the equation x + (x + 34) = 162 + 12 to find the number of games the Jays won and lost.

6. Solve.

a) 2a = 15.9 – 8.7 – 30 b) 3q –1.5q = 12 – 4.5q c) 5k + 1 = 3k + k – 3.8 d) 12 + 7j = 14.2 + 9j + 1.8

7. Washing a car for 20 min uses x kilojoules of energy. Doing yard work for 20 min uses about 2x kilojoules of energy. The number of kilojoules used doing yard work is equal to the number used washing a car plus 65. Find the number of kilojoules of energy used to perform each activity.

8. Seven times a number is the same as 12 more than 3 times the number. Find the number.

9. Solve.

a) 3(x + 1) = 24 b) 2(x – 2) = 8 c) –4(y + 3) = –16 d) 10 = 5(t + 2) e) 3(x + 2) = –9 –2x f) 3(n – 2) – 19 = 5 + 2(n + 5)

10. Solve.

a) 4(x + 2) – 3(x + l) = 2(x + 2) b) 3(n – 5) – (2n + 2) = 2(n – l) c) 2(a – 4) – 3(a – 2) = 4(a + l) + 4 d) 5(c + 4) = 4(2c – 3) – 7

11. The sides of a triangle are given, in centimetres, by the expressions 3(n – 2), 4(n + 3), and 2(n + 4). The perimeter of the triangle is 140 cm. Find the length of each side.

12. Solve.

a) k – 1.2 = 1.8 b) 2 = –0.5d c) –1.6 = 2 – 0.4y

d) 1.2 + 1.4y = 1.5y + 0.63 e) aaa

6

5

3

2

4

3 f)

2

1

4

34

cc


Name _____________________

Substituting Into Equations McGraw-Hill Ryerson Mathematics of Data Management

1. Solve each system of equation by substitution. Check each solution.

a) x – y = 1 b) 2x – y = 13 c) 3a + 4b = 15 d) 2x + 3y = 5 3x + y = 11 x + 2y = –6 a + b = 5 x – 4y = –14

e) 2c – d + 2 = 0 f) 4x – y = 3 g) a + 4b = 3 h) 3e – f – 2 = 0 3c + 2d + 10 = 0 6x – 2y = 5 5b = –2a + 3 5e + 2f = 3

i) 5 = 2y – x j) x + 7y = 1 k) 32

1 xy l) 3x – 2y = –12

7 = 3y – 2x 3x – 14y = –7 y = 5 – x x – 4y = 8

2. Solve each linear system by graphing and by substitution. Which method do you prefer? Why?

a) x + y = 6 b) 2x + y = –4 c) 2x + y = 5 d) 6y + 3x = –4 x – y = 42 4 + 3y = –6 2y = 2x + 11 x – 2y = –2

3. Try to solve each system of equations by substitution. Then, analyse each system to decide how many solutions there are. Describe how the number of solutions is related to the results when you try to solve by substitution.

a) x – 4y = 8 b) y = 5 – 2x c) x + y – 4 = 0 2x – 8y = 8 3x = 2y + 11 2x = 8 – 2y d) x + y + 6 = 0 e) 6x = 3y + 2 f) y = 3x – 2

2x – y – 3 = 0 y – 2x + 4 = 0 9x – 3y – 6 = 0

4. The highest point in British Columbia, f metres above sea level, is on Fairweather Mountain. The highest point in Manitoba, b metres above sea level, is on Baldy Mountain. The heights are related by the following system of equations:

f – b = 3831

f = 6b – 329

a) Interpret each equation in words. b) Solve the system of equations to find the height of each mountain.

5. Simplify each linear system, and then solve it by substitution. Check each solution.

a) 2(x – 4) + y = 6 b) 2(x – 1) – 3(y – 3) = 0 c) 2(3x – l) – (y + 4) = –7 3x – 2(y – 3) = 13 3(x + 2) – (y – 7) = 20 4(1 – 2x) – 3(3 – y) = –12

6. The sum of two numbers is 752. Their difference is 174. What are the two numbers?

7. The longest cable suspension bridge in the world is the Akashi Kaikyo Bridge in Japan. It has one long span and two short spans. Each short span is half the length of the long span. The total length of the bridge is 3560 m. What is the length of each span?

8. ABC Plumbing charges $70 for a service call, plus $50/h for the time worked. Quality Plumbers charges $52 for a service call, plus $54/h. In what situation is each company cheaper?

9. The total number of species of owls and pigeons that raise their young in Canada is 17. Subtracting one from five times the number of species of pigeons gives the number of species of owls. Find the number of species of each bird that raise young in Canada.

10. The receipts from 550 people attending a play were $9184. The tickets cost $20 for adults and $12 for students. Find the number of adult tickets and then number of student tickets sold.


Name _____________________

Tree Diagrams McGraw-Hill Ryerson Mathematics of Data Management

Draw a tree diagram to list all the possibilities for the tasks in questions 1 and 2.

1. If you must choose two elective courses, one from three possible science courses and one from four possible humanities courses, how many different course selections are there?

2. A combination-meal menu has four choices for the appetizer, two choices for the entree, and three choices for dessert. How many different meals can be chosen?

3. How many different licence plates can be made that consist of three digits followed by three letters?

4. Molecular biologists have discovered that a gene contains a long molecule that is made up of smaller molecules called nucleotides. Nucleotides are composed of a sequence of the bases: adenine, cytosine, guanine, and thymine. These bases are often referred to by the letters A, C, G, and T. How many different genetic codes can be made if the sequence is a) three bases long? b) four bases long? c) n bases long?

5. Louis Braille (1809–1852) devised a system of six raised or not-raised dots to signify a letter of the alphabet. For example, the letter N is denoted by the arrangement shown, where a solid dot represents a raised dot.

●● ○● ●○

a) How many different letters or characters may be coded in Braille? b) If you wanted to code 94 distinct characters, such as those found on a computer keyboard, how

many dots would the system need?

6. Canadian postal codes have six characters, with the first, third, and fifth characters being letters, and the other characters being digits. a) How many postal codes are possible? b) The first two characters locate the regional post office. How many locations can the last four

characters pinpoint?

7. Samuel Morse (1791–1872) developed a character coding system that was used to send messages along telegraph lines and also for light signalling between ships. Morse code is based on a series of dots and dashes (short or long bursts). The most frequently used letter, E, is coded as a single dot. The least frequently used letter, Q, is coded as dash, dash, dot, dash. a) How many letters can be coded by a single burst? by two bursts? by three bursts? by four bursts? b) How many characters, in total, can be coded by up to four bursts?

8. In Canada and the United States, telephone area codes consist of three digits. Suppose the first digit is from 2 through 9 inclusive, the second digit is either 0 or 1, and the third digit is any digit except 0. a) How many different area codes are possible with these criteria? b) Suppose the restrictions on the first digit remain, but the restrictions on the second and third

digit are removed so that any digit is allowed. How many different codes are possible now?

9. If there are 17 576 000 different licence plates consisting of three digits followed by three letters, how many more licence plates can be issued using a seven-character format, with four letters followed by three digits?


Practice/Assessment Masters Answers Review of Prerequisite Skills

Evaluating Expressions

1. a) 13x b) r

d c) 6 + w + 19 d) 7 – k

2. a) 51.2 b) 8.1 c) 67 d) 35 3. a) 33.1 b) 72.9 c) –7.6


4. a) –12 b) 2 c) –9 d) –104 5. a) –102 b) –335 c) 45 d) 816 6. a) 99 m, 75 m b) 0 m. After 20s, the rocket has fallen back to its starting point. 7. a) 72.9 kg b) 111.4 kg 8. a) 9.75 b) 195 c) 404.5 d) –15.25 9. a) s: 10 cm, 25 cm, 9 m, 2.55 m b) A: 17.3 cm2, 102.5 cm2, 9.4 m2, 1.1 m2 Exponent Laws 1. a) 25 b) 38 c) 46 d) 104 e) 97 f) 85 g) x7 h) y6 i) z5 2. a) 2 b) 3 c) 2 d) 1 e) 3 f) 4 g) 10 h) 1 3. a) 52 b) 43 c) 3 d) 93 e) 7 f) 22 g) m2 h) p2 4. a) 2 b) 2 c) 6 d) 4 e) 2 f) 7 g) 4 h) 8 5. a) 36 b) 28 c) 712 d) 68 e) 56 f) 415 g) x9 h) r10 6. a) 3 b) 2 c) 4 d) 4 e) 3 f) 3 g) 4 h) 3 7. a) 52, 134; a2 + b3 is less than a3 + b2 b) 225, 225; the expressions are equal 8. a) False; 34 = 81 b) False; 6(–2)3 = –48 c) True; exponent law d) True; ((–2)3)3 = (–2)9 = –512 e) True; (–a)4 ÷ (–a)2 = (–a)2 = a2 f) False; (–5)3 ÷ (–5)2 = –5 9. a) (–2)4 is –2 to the exponent 4, while –24 is the negative of +2 to the exponent 4 b) ((–3)2)3 = (–3)6 ≠ –36, by similar reasoning to that given in part a). 10. a) 2a2b2 b) 2wx3 c) –2x3y2 d) 2q

e) 2 33

2f g h

f) 4 52

3c d

11. a) 23

2ef b) 2a3b c) 4x2y2 d) 5 43

2s t

e) 2km2

f) 35

3d f

12. side = 8s2t ÷ 4 = 2s2t Fractions, Percents, Decimals 1. a) 45% b) 50% c) 0.4% d) 59.6% e) 100% f) 263% 2. a) 70% b) 80% c) 67.5% d) 87.5% e) 55.6% f) 112.5% 3. a) 73% b) 7% c) 60% d) 90% e) 300% f) 125%

4. a) 13

,20

65% b) 0.75, 75% c) 1

,50

2%

d) 9

,10

0.9 e) 0.36, 36% f) 3

,8

0.375 g) 22

,25

0.88

h) 29

, 72.5% 40

i) 5

,4

1.25 j) 2.5, 250%

5. a) 77% b) 1.875% c) 1% d) 0.1% e) 300% f) 320% 6. 264% 7. a) 11.1% b) 111% 8. a) Superior: 341 m; Michigan: 217 m; Huron:

165 m; Ontario: 180 m b) 341 217 165 180

, , ,64 64 64 64

c) 5.33, 3.39, 2.58, 2.81 Graphing Data 1. a) swimming b) 440 kJ c) house cleaning and brisk walking d) 62.5 min. 2.

Estimated Population of Selected Large Species

0

5000

10000

15000

20000

25000

30000

MammalsBird

s

Reptiles and Amphibians

Fish

Species

Es

tim

ate

d P

op

ula

tio

n

3.

Average Monthly Temperatures for Victoria, B.C., and Melbourne

Australia

0

5

10

15

20

25

Janu

ary

Febru

ary

Mar

chApril

May

June

July

August

Septe

mber

Octobe

r

Novem

ber

Decem

ber

Month

Tem

per

atu

re (

deg

rees

C

elsi

us)

Victoria Melbourne

Causes of Ai4. r Pollution

Cause Percent of l

Calculation CentraAngle

Central Angle

Industry 15 0.15 × 360° 54° Transportation 44 0.44 × 360° 158° Humans 9 0.09 × 360° 32° Fuel 20 0.2 × 360° 72° Other 12 0.12 × 360° 43°

count, and y is the bacteria count

00 c) 11 d)

Graphing Exponential Functions 1. a) y = (100)3x where x is the number of days after the initialafter x days. b) 9

2. a) A(t) = 4500(1.03)2t b) Doubling time = 10 years

3. a) I(d) = 0.97d, where d is the depth below the surface, and I is the intensity of light at d metres below the surface, measured as a fraction of original intensity at the surface. b) 86% c)

d) 22.8 m Area of the Continents (1000s of

square kms)4. a)

North America

b)

17870

9840

45100

30300

8550

13990 24350South America

Europe

Asia

Africa

Australia

Antarctica

5.


c) c)

d)

7. 0.5 hours 8. 48 months Graphing Linear Equations

1. a) x y = –2x + 5 –2 9

–1 7 0 5 1 3

2 1 e)

f) b) x y = –2x

–2 4 –1 2

0 0 1 –2

2 –4

5. a) x = 2.5 b) x = 4 c) x = 1.45 d) x = 4 or x = 1

6. a) 8.2

1 0

1

2

t

I I

where I0 is the initial amount of

iodine, and I1 is the amount of iodine present in the thyroid gland after t number of days.

b) 24.6 days


2. a)

b)

c)

d)

3. a) y = x + 4 b) y = 3x 4. y = 0.5x; Domain: {–2 x 4, x R} 5. a)

b) Yes—the points plotted indicate a linear graph c) 240 cm d) 18 cm 6. a)

b)

c)

d)

7. a)


b)

e) down; (0, –4); maximum: –4

8. answers will vary (example: y = x – 1)

9. answers will vary (example: y = –x) 10. No. The planes will meet at the point where x = 1, y = 2 (this is where the lines intersect). Graphing Quadratic Functions 1. a) up; (0, 4); minimum: 4

f) down; (0, 12); maximum: 12

b) up; (0, 0); minimum: 0

2. a) Vertical compression of scale factor 0.5, followed by a vertical translation of –4 b) Vertical compression of scale factor 0.5 c) Reflection in the x-axis, followed by a vertical translation of 3 3. a) y = –x2 – 3 b) y = 3x2 + 2

4. a) 4, –4 b) none c) 3 2 3 2

,2 2

d) 2.5, –2.5

5. a) y = 2x2 + 3 b) y = –0.5x2 + 1

c) down; (0, 2); maximum: 2

c) y = 4x2 – 5; c = –5 d) y = –x2 + 7; a = –1, q = 7 6. a) (–5, –6), max. b) (–4, 2), min. c) (2.5, –9), min. d) (9, 19), max.

7. a) y = –2(x – 3)2 – 1 b) 21( 2) 5

2y x

c) y = –0.5(x + 4)2 – 1 d) y = 3(x + 5)2 + 3 8. 0, –6 9. y = –2(x – 2)2 + 15

d) up; (0, –1); minimum: –1 10. a) 121 b) 64 c)

49

4 d)

4

25

11. On graphs, sample points will vary, except for the vertex.


a) y = (x – 2)2 – 5;

b) y = –2(x + 1)2 + 3;

c) y = 0.5(x – 3)2 – 0.5;

Mean, Median, Mode 1. a) 20 b) 32 c) 31 d) 83 e) 72 f) 147 g) 47 h) 207 2. 20°C 3. a) 33 b) 80 c) 116 d) 293.5 4. a) 4 b) 36 c) 64, 68 5. a) 25 b) 24.5 6. a) 31.2, 31 b) 88.2, 87 c) 161.3, 169 d) 225.1, 224 e) 114.8, 113 f) 804.2, 816 7. a) Answers will vary. b) Answers will vary. 8. mean: 38.9; median: 38; mode: 38 9. y = 28 10. a) Answers will vary. b) Answers will vary. 11. a) Bethune: 49, Bombardier: 57, Hanlan: 53, Hogg: 88, Laurence: 61, Pearson: 75, Pickford: 86, Stowe: 72 b) mean: 67.6; median: 66.5 c) no mode (each value occurs just once each) d) 4 e) 4 Number Patterns

1. 55 2. 25; 51 3. a) 64; 144; 625 b) each successive figure has the next perfect square number of small triangles (the nth figure has n2 small triangles) 4. a) 9 b) 36 c) 100 e) 2025 5. a) 40 b) 60 c) 64 d) 100 6. 56 7. 65 8. 156 9. 28 10. 54 11. 7 12. 2475 13. 3 14. 4010 Order of Operations 1. a) 14 b) 78 c) 18 d) 28 e) 15 f) 116 g) 11 h) 58 i) 9 j) 33 2. a) 16 ÷ (4 – 2) – 1 = 7 b) (5 – 3 + 6) × 2 = 16 c) (4 + 6) ÷ 2 = 5 d) 1.7 × (1.3 – 0.8) = 0.85 e) 5.2 × (3.4 – 2.4) = 5.2 3. a) 15 b) –14 c) 1 d) 0 e) 0 f) 72 4. a) 3 b) 30 c) 20 d) 1 e) –16 f) 11 5. a) 3 b) –27 c) 26 d) –16 e) 0 f) –35 6. a) 7 b) 5 c) –72 d) 4 7. a) 13 b) 3 c) –36 d) –7 e) 16 f) –8 8. a) –2 b) –1 c) –1 d) –7 e) 5 f) 1 9. a) 3 × 5 ÷ (2 + 3) = 3 b) (4 + 6) ÷ 2 = 5 c) (2 – 5) × 3 + 23 = –1 d) 1.7 × (1.3 – 0.8) = 0.85 e) (42 + 5 – 1) ÷ 5 = 4 f) 5.2 × (3.4 – 2.4) = 5.2 10. a) 25 × (25.5 + 25.5 × 0.8) b) 20 × (25.5 + 25.5 × 0.8 – 2 × 25.5 × 0.8) Ratios of Areas 1. a) ΔPQR is similar to ΔVXW; the ratios of their sides are the same b) sides: PQ, VX; QR, XW; PR, VW; angles: P = V; R = W; Q = X c) yes. ΔPQR has area 4 times that of ΔVXW (it has double the base and height)

2. a) 3

80t cm,

3

100s cm; 9:16 b) b = 5 m,

d = 16 m; 1:16 c) y = 20 m, w = 24 m; 25:64 3. 13 m by 7 m rink; this rink has a smaller area. 4. a) 49 m2 b) 100 m2 c) 196 m2 5. 25:39 6. a) 3:4 b) 13:29 c) 19:84 Scatter Plots 1. For years up to 1980, the graph indicates a strong negative linear correlation between the number of competitors through each successive Olympic year.


Following 1980, a moderate positive linear correlation is exhibited. 2. The scatter plot indicates an exponential relationship between mass and height.

Types of Dogs

0

10

20

30

40

50

60

70

0 20 40 60 80

Height (cm)

Mas

s (k

g)


3. Yes. Each student’s mark in Science is very close to the mark achieved in French. 4. a)

North American Whooping Crane Population

0

20

40

60

80

100

120

140

160

1960 1970 1980 1990 2000

Y ear

b) 150, 160 Sigma Notation

1. a) 19.5 b) 51 c) 0 d) 39 364

27 e)

35

32 f) 21

2. a) –12 b) 78 3. a) 28 b) 1.111 c) 7381 d) 49.218 75 e) 10 f) 58 593.72 g) 5 380 840.125 h) 0.007 963 955

4. a) b)

c)

d) e)

f)

52

1

3(4) 15.98k

k

3

1

( 3)2 23.812k

102

1

2 511.5k

k

7

5k

91

1

0.11(3) 1082.51k

k

7

1

1

1.99(10) 2 211 1k

k

6

1

(4 2)k

k

62

1

25(4) 5.33k

k

10.89

8

1

5k

k b) 5. a) c)

d) e) f)

9

3

(7 3 )k

k

2 )k0

5

(20 20k

k

6

1

(2 0.5)k

k

8

2

(8.4k )

Similar Triangles

80

3t

100

3s 1. a) cm, cm b) b = 5 m, d = 16 m

c) y = 20 m, w = 24 m 2. ΔABC is similar to ΔAEF as the interior angles of each are the same for both. 3. 3 4. height of tower, h = 38.4 m

1.8 m

1.5 m

h

32 m 5. 18.9m 6. The interior angles are not the same for both triangles. 7. Answers will vary. 8. a) They have the same three interior angles. b) P = T, Q = U, R = S; PQ and TU, QR and US, PR and TS Simplifying Expressions

l. a) 4a2b2 b) 3

2

4

5

p

q

c) –c2d d)

6 3

3

x xy

y

e)

3 3

4

x y

y

f)

8

18 m g)

5

136 a h)

s

sqsq

4

64 5

2. a) 5

13a b)

5

2

y

z c) x2 – 18x + 4 d)

3

4xy

3. a) 5 15

x

x b)

2x

x

c)

3

2

x

x

d) 5

3

x

x

4. a) 4

5 b) y c)

2

( 3)( 2)

1

x x

x

d) 2 1

3 1

p

p

5. a) –1 b) 3

3

x

x

c) –1 d) 7

5

6. a) 1

4x b)

1

4a c)

3

2x d)

5

4

x

x

e) 2( 2)

( 1)( 4)

a

a a

f) 5 13

( 2)( 3)( 1)

x

x x x

7. a) 0 b) 0, 1, –1 c) –7 d) –1, –7

8. a) 1

25

x b)

2 6 8

10 4

x x

x

9. a) x:4 b) 12 c) 21.84 cm2

10. 22 8

7 2

x

x

Solving Equations

1. a) t = –6 b) r = –4 c) 1

7n d) k = –150

e) s = 84 f) r = –3 2. a) t = –l b) x = 1 c) a = 3 d) p = 7 e) y = –2 f) a = 3 3. a) y = 32 b) n = 1 c) a = 3 d) b = 3 e) r = 2 4. a) x = 0.9 b) k = l c) d = –3 d) g = 4 e) n = –6.2 f) a = 4 5. won 104, lost 70 6. a) a = 1.44 b) q = 2 c) k = –4.8 d) y = –2 7. 65 kilojoules for washing a car, 130 kilojoules for yard work 8. 3 9. a) x = 7 b) x = 6 c) y = 1 d) t = 0 e) x = –3 10. a) x = 1 b) n = –15 c) a = –2 d) c = 13 11. 36 cm, 68 cm, 36 cm 12. a) 3 b) d = –4 c) y = 9 d) y = 5.7

e) 10

11a

f) c = 17

Substituting Into Equations 1. a) x = 3, y = 2 b) x = 4, y = –5 c) a = 5, b = 0 d) x = –2, y = 3 e) c = –2, d = –2 f) x = 0.5, y = –1

g) a = –l, b = l h) 7

,11 11

e f

1

i) x = 1, y = 3

j) x = –1, 2

7y k)

4 1,

3 3x y

1

l) x = –6.4, y = –3.6 2. a) x = 24, y = –18 b) x = –3, y = 2

c) x = 1.5, y = 2 d) 5 1

,3 6

x y

3. a) No solution b) x = 3, y = –1 c) Infinite number of solutions, x = 4 – y d) x = –1, y = –5 e) No solution. f) Infinite number of solutions, y = 3x – 2 4. a) f – b = 3831: The difference in height above sea level between the highest point in B.C. and the highest point in Manitoba is 3831 metres.

f = 6b – 329: The highest point in B.C. is six times higher above sea level than the highest point in Manitoba less 329 metres. b) f = 4663 m, b = 832 m c) x = –1, y = –5 6. 289, 463 7. Quality plumbers is cheaper when time worked is less than 4.5 hours. ABC is cheaper when time worked exceeds 4.5 hours. 8. 21°, 69° 9. 323 adult tickets, 227 student tickets 10. a) (1, 4, –2) b) (2, –1, 3) Tree Diagrams 1. 12 different course combinations. (The number of end branches in the diagram.)

2. 24 different meal combinations. (The number of end branches in the diagram.)

3. 17 576 000 4. a) 64 b) 256 c) 4n 5. a) 64 b) 7 6. a) 439 400 000 b) 67 600 7. a) 2; 4; 8; 16 b) 30 8. a) 144 b) 800 9. 16 777 216


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