Chapter 5 - Weeblycolontlc.weebly.com/uploads/1/3/3/9/13391515/solutions...Practice 5.1 1. For each pair of values, x and y:!5 1 5 10 2!5 15 3!5 So, y is directly proportional to x

Chapter 5Quick Check 1. Number of speakers : number of sets of earbuds ! 80 : 45

! 80 " 5 : 45 " 5

! 16 : 9

2. Number of headphones : number of speakers ! 60 : 80

! 60 " 20 : 80 " 20

! 3 : 4

3. 9 : 11 ! 9 # 2 : 11 # 2 ! 18 : 22

9 : 11 and 18 : 22 are equivalent ratios.

4. 33 : 1 ! 33 " 33 : 1 " 33 ! 1 : 1

33 $ 1 : 33

331

and 133

are not equivalent ratios.

5. 3 : 6 ! 3 · 3 : 6 · 3 ! 9 : 18

3 : 6 and 9 : 18 are equivalent ratios.

6. 4 : 5 is in simplest form because 4 and 5 have no common factors.

4 : 5 ! 4 # 2 : 5 # 2

! 8 : 10

4 : 5 ! 4 # 3 : 5 # 3

! 12 : 15

8 : 10 and 12 : 15 are two ratios equivalent

to 4 : 5.

7. 15100

is not in simplest form because 15 and

100 have a common factor of 5.

15100

! 15 5100 5

!

!!

320

15100

! 15 2100 2

!

!!

30200

320

and 30200

are two ratios equivalent to 15100

.

8. 7 to 14 is not in simplest form because 7 and 14 have a common factor of 7.

7 : 14 ! 7 " 7 : 14 " 7

! 1 : 2

7 : 14 ! 7 # 2 : 14 # 2

! 14 : 28

1 : 2 and 14 : 28 are two ratios equivalent to

7 : 14.

9. Average speed ! Total distance traveledTotal time

! 2 42894 5,

.kmh

! 25.69 ! 26 km/h

10. Store A: $3.20 for 16 oz of walnuts. Unit price of walnuts at Store A:

$ .3 2016

! $0.20/oz

Store B: $2.30 for 10 oz of walnuts.

Unit price of walnuts at Store B:

$ .2 3010

! $0.23/oz

$0.20/oz ! $0.23/oz; The walnuts cost less

at Store A.

11. Store C: $2.13 for 3 lb of potatoes. Unit price of potatoes at Store C:

$ .2 133

! $0.71/lb

Store D: $3.35 for 5 lb of potatoes.

Unit price of potatoes at Store D:

$ .3 355

! $0.67/lb

$0.71/lb " $0.67/lb; The potatoes cost less

at Store D.

12. P(1, 2), Q(2, 4), R(5, 6), S(5, 2), T(6, 3)13. 36 beads

(45%)

? beads (100%)

From the bar model,

45% → 36

1% → 3645

100% → 3645

# 100 ! 80

There are 80 beads altogether.

14.

? (105%)

$72 (100%)

From the bar model,

100% → 72

1% → 72100

! 0.72

105% → 0.72 % 105 ! 75.6 She paid a total of $75.60.

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Solutions Key Course 2 75

MIF_SK_C2_Ch01-10.indd 75 12/16/11 4:08 PM

Lesson 5.1Hands-On Activity (Identify Direct Proportion in an Experiment)Step 1: Answers vary.

Step 3: Answers vary.

Step 4: Answers vary.

Math Journal: H increases as L increases.

Guided Practice 1. For each pair of values, x and y:

yxmih

! !1002

50

yxmih

! !1503

50

yxmih

! !2004

50

So, the distance traveled by the school bus is

directly proportional to the number of hours it

has traveled.

The constant of proportionality is 50 and

represents the speed of the bus.

The direct proportion equation is y ! 50x. 2. For each pair of values, x and y:

151

15pitchesinning

!

302

15pitchesinning

!

503

23

16pitchesinning

!

So, the number of pitches made is not directly

proportional to the number of innings of a

baseball game.

3. 0.4y ! x

0 40 4 0 4.. .y x

!

y ! 2.5x Because the original equation 0.4y ! x can be

rewritten as an equivalent equation in the form

y ! kx, it represents a direct proportion. The constant of proportionality is 2.5.

4. x ! 1 " 2y x # 2y ! 1 " 2y # 2y x # 2y " x ! 1 " x 2y ! 1 " x

22

12 2

y x! "

y x! "12 2

Because the original equation x ! 1 " 2y cannot be rewritten as an equivalent equation in the form

y ! kx, it does not represent a direct proportion.

5. Constant of proportionality:

561

56baseballsday

!


represents the daily production rate of baseballs.

The direct proportion equation is y ! 56x. 6. Answers vary. Sample: Let x be the number of sandwiches. Let y be the amount Jason pays. Cost per sandwich: $4 per sandwich

The direct proportion equation is y ! 4x. 7. Constant of proportionality:

qp

! !2412

2

The constant of proportionality is 2.

The direct proportion equation is q ! 2p. 8. Constant of proportionality:

wh

! !183

6


The direct proportional equation is w ! 6h.

Practice 5.1 1. For each pair of values, x and y:

51

5! 102

5! 153

5!

So, y is directly proportional to x. The constant of proportionality is 5.

The direct proportion equation is y ! 5x. 2. For each pair of values, x and y:

1302

65! 1004

25! 706

23

11!

So, y is not directly proportional to x. 3. For each pair of values, x and y:

203

23

6! 406

23

6! 509

59

5!

So, y is not directly proportional to x. 4. For each pair of values, x and y:

502

25! 1004

25! 1506

25!

So, y is directly proportional to x. The constant of proportionality is 25.

The direct proportion equation is y ! 25x. 5. 3 1

2y x!

33

13

12

y x! "

y x! 16

Because the original equation 3 12

y x! can be

rewritten as an equivalent equation in the form

y ! kx, it represents a direct proportion. The constant of proportionality is 1

6.

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76 Solutions Key Course 2

MIF_SK_C2_Ch01-10.indd 76 12/16/11 4:08 PM

6. 2y ! 5 " x 2y ! 5 # 5 " x # 5 2y " x # 5

22

12

5y x! "( )

y x! 12

5( )"

Because the original equation 2y ! 5 " x cannot be rewritten as an equivalent equation in the form

y " kx, it does not represent a direct proportion. 7. p " 0.25q Because the original equation is written in the

form y " kx, it represents a direct proportion. The constant of proportionality is 0.25.

8. 4.5a " b # 12 10 $ 4.5a " 10 $ (b # 12) 45a " 10b # 120

a b! 145

10 120( )"

Because the original equation 4.5a " b # 12 cannot be rewritten as an equivalent equation in

the form y " kx, it does not represent a direct proportion.

9. For each pair of values, n and d:

201

20migal

! 402

20migal

! 603

20migal

!

So, d is directly proportional to n. The constant of proportionality is 20 and

represents the distance traveled, in miles

per gallon of gasoline.

The direct proportion equation is d " 20n.10. For each pair of values, x and y: 24

1

482

24

24

gamespoint

gamespoints

!

!

gamespoints

803

23

26!

So, y is not directly proportional to x.11. For each pair of values, x and y:

201

20balls

machine!

603

20balls

machines!

1005

20balls

machines!

So, y is directly proportional to x. The constant of proportionality is 20 and

represents the number of tennis balls produced

per machine.

The direct proportion equation is y " 20x.

12. You can tell whether two quantities are in direct proportion by checking whether the two quantities

increase or decrease by the same factor. In other

words, you can check if yx is a constant value.

13. P " 3c

Pc

cc

!3

Pc

!3

Yes. Because Pc

is a constant value, P and c are in direct proportion.

14. Constant of proportionality: dt

! !201

20


The direct proportion equation is d " 20t.

15. Constant of proportionality: wt

! !121

12


The direct proportion equation is w " 12t.

16. Constant of proportionality: yx

! !1015

23

The direct proportion equation is y x! 23

.


! !3311

3

The direct proportion equation is y " 3x.18. Answers vary. Sample: Let t be the time Karl takes to hike. Let d be the distance he hikes. Distance hiked per minute:

dt

mimin

mimin

mi/min! !3

45115

The constant of proportionality is 115

.

The direct proportion equation is d t! 115

.

19. Answers vary. Sample: Let n be the number of songs Paul downloads. Let P be the amount he pays.

Cost per song: $ $Pn songs songs

! !20

1654


.

The direct proportion equation is P n!54

.

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MIF_SK_C2_Ch01-10.indd 77 12/16/11 4:08 PM

20. Sample: The cost per day of an advertisement for the first 3 days is the same in both tables:

$20 per day.

They are different because, for the Daily Post, Ct

is a constant value for all pairs of values. But

for the Evening Star, Ct

is not a constant value.

So, the total cost of an advertisement is directly

proportional to the number of days for the Daily Post but not for the Evening Star.

Lesson 5.2Guided Practice 1. The graph is a straight line through the origin,

and does not lie along the x- or y-axis. So, it represents a direct proportion.

Because the graph passes through (1, 20), the

constant of proportionality is 20.

The direct proportion equation is y ! 20x. 2. Although the graph is a straight line that does not

lie along the x- or y-axis, it does not pass through the origin. So, the graph does not represent a

direct proportion.

3. Although the graph passes through the origin and does not lie along the x- or y-axis, it is not a straight line. So, it does not represent a direct

proportion.

4. a) Constant of proportionality: 501

50mih

!

The constant of proportionality is 50. So,

Ms. Gray’s driving speed is 50 miles per hour.

b) The direct proportion equation is y ! 50t. c) It means that Ms. Gray travels 350 miles

in 7 hours.

d) From graph, the distance traveled is 150 miles. e) From graph, it takes her 8 hours to travel

400 miles.

Practice 5.2 1. Although the graph is a straight line and does not

lie along the x- or y-axis, it does not pass through the origin. So, it does not represent a direct

proportion.

2. The graph is a straight line through the origin, and it does not lie along the x- or y-axis. So, it represents a direct proportion. Because the

graph passes through (1, 500), the constant of

proportionality is 500.

The direct proportion equation is y ! 500x.

3. Although the graph is a straight line that does not lie along the x- or y-axis, it does not pass through the origin. So, it does not represent a direct

proportion.

4. The graph is a straight line through the origin, and it does not lie along the x- or y-axis. So, it represents a direct proportion. Because the

graph passes through (1, 100), the constant of

proportionality is 100.

The direct proportion equation is y ! 100x.


50!


represents the cost of staying at a motel

per night.

b) 1 week ! 7 days From the graph, it costs $350 to stay at a

motel for a week.

6. First you observe whether it is a straight line. Then you check whether it passes through the

origin. Finally, you make sure that it does not lie

on the x- or y-axis. 7. a) Yes, the amount of pesos is directly

proportional to the amount of U.S. dollars

because the graph is a straight line through

the origin and does not lie along the x- or y-axis.

b) From the graph, you get 36 pesos for 3 U.S. dollars.

c) From the graph, you get 2 U.S. dollars for 24 pesos.

d) The exchange rate is 1 U.S. dollar for 12 pesos.

e) Constant of proportionality: 121

12!

The direct proportion equation is y ! 12x. 8.

5

0

1015202530

1 2 3 4 5 6x

y

Number of Days

Num

ber

of

Cer

amic

Pot

s

Ceramic PotsProduction

a) Constant of proportionality: 51

5!


represents daily production rate of ceramic pots.

b) It means Beth makes 20 ceramic pots in 4 days.

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MIF_SK_C2_Ch01-10.indd 78 12/16/11 4:08 PM

c) From the graph, Beth can make 15 pots in 3 days.

d) From the graph, Beth will take 6 days to make 30 pots.

Lesson 5.3Guided Practice 1. Method 1:

Let x be the number of hours it takes to make 250 cars.

6045

250carshours

carshours

!x

6045

250!

x

x ! 60 " 250 ! 45 60x " 250 ! 45

6060

11 25060

x!

,

x " 187.5 It takes 187.5 hours to make 250 cars.

Method 2: Let x be the number of hours. Let y be the number of cars. Constant of proportionality:

yx ! !6045

43

Direct proportion equation: y ! 43x

When y " 250 and y x! !43

43

250, ! x

43

! x " 250

4343

! x " 25043

x " 187.5 It takes 187.5 hours to produce 250 cars.

2. P is the number of pears at the orchard. C is the number of boxes used.

a) Number of pears per box: 20010

" 20

The direct proportion equation is P " 20C. b) When C " 8 and P " 20C, P " 20 ! 8 P " 160 160 pears are packed into 8 crates.

When P " 500 and P " 20C, 500 " 20C

50020

2020

!C

25 " C There are 500 pears packed in 25 crates.

Number of Crates (C) 8 200 25 Number of Pears (P) 160 10 500

3. Let x be the percent increase. Method 1:

100 percent

32percent

8$ $!

x

10032 8

!x

x ! 32 " 100 ! 8 32x " 800 32x # 32 " 800 # 32 x " 25

Method 2: Ratio of percents " Ratio of handbag prices

x percent : 100 percent " $8 : $32

x100

832

!

100 ! x100

832

! ! 100

x " 25 The percent increase in the price of the handbags

was 25%.

Practice 5.3

1. a) Constant of proportionality: mn

! !147

2

The equation that relates m and n is m " 2n. b) When n " 16 and m " 2n, m " 2 ! 16 m " 32 c) When m " 30 and m " 2n, 30 " 2n

302

22

!n

15 " n

2. a) Constant of proportionality: pq

! !630

15

The equation that relates p and q is p q!15

.

b) When p " 10 and p ! 15q, 10 ! 1

5q

5 5 1015

! " !q

q " 50

c) When q " 7 and p ! 15q, p ! "1

57

p ! 125

75

or

3. Constant of proportionality: ba

! !124

3

The equation is b " 3a. When b " 15 and b a! 3 , 15 " 3 ! a 15

333

! a

5 " a When a " 19 and b ! 3a, b " 3 ! 19 " 57

a 4 5 19b 12 15 57

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MIF_SK_C2_Ch01-10.indd 79 12/16/11 4:08 PM

4. Constant of proportionality: ab

! !410

25

The equation is a b! 25

.

When b ! 25 and a b! 25

, a ! 25

25!

a ! 10

When a ! 16 and a b! 25

, 16 25

! b

5 16 5 25

! " ! b 80 2! b

802

22

!b

40 ! b

a 4 10 16b 10 25 40

5. Let w be the weight of a person. Let a be the amount of blood in the body. a) Constant of proportionality:

aw

! !4

128132


.

b) The equation is a w!32

.

c) When a ! 5 and a w!32

, 532

!w

5 32 32

16032

! " !

"

w

w The weight of the person is 160 lb.

6. a) Constant of proportionality: Hn

!1210

!65


.

b) The equation is H n! 65

.

c) When n ! 24 and H n! 65

, H ! "65

24

! 28.8

The height of a stack of 24 books is

28.8 inches.

7. a) Weight per soccer ball: 5412

4 5oz oz! .

The weight per soccer ball is 4.5 oz.

b) The equation is m ! 4.5n. c) When n ! 30 and m ! 4.5n, m ! "4 5 30. m ! 135

8. a) Cost per CD case: Cn

!$ .2 346

! $ .0 39

The cost per CD case is $0.39.

b) The equation is C ! 0.39n. c) When n ! 7 and C ! 0.39n, C ! "0 39 7. C ! 2 73.

9. Let C be the cost of two dozen oranges.

2

5 24dollars dollars

!C

25 24

!C

2 24 5! " !C

5C ! 48

55

485

C!

C ! 9.6 The cost of two dozen oranges is $9.60.

10. Let C be the cost of renting a car for 1 week.

180

3 7

dollars dollars!

C

1803 7

!C

180 7 3! " !C 3C ! 1,260

33

1 2603

C!

,

C ! 420 The cost of renting a car for 1 week is $420.

11. Let n be the amount of gas he used.

248 78

gallons gallons!

n

248 78

!n

48 2 78! " !n 48 156n !

4848

15648

n!

n ! 3.25 He will use 3.25 gallons of gas if he drives

78 miles.

12. Let x be the glasses of juice needed.

31 15glassespeople

glassespeople

!x

31 15

!x

x ! "3 15 x ! 45 The caterer should have 45 glasses of juice ready.

13. Let x be the amount of ground beef needed.

105 8

ouncespeople

ouncespeople

!x

105 8

!x

5 10 8! " !x 5x ! 80

55

805

x!

x ! 16 16 ounces of ground beef should be used.

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MIF_SK_C2_Ch01-10.indd 80 12/16/11 4:08 PM

14. Let x be the amount of tax he paid. Ratio of taxes ! Ratio of salary

x : 30 ! 3,680 : 100

x

xx

303 680100

100 3 680 30100 110

!

" ! "

!

,

,,,

,

,

400

1104

100100

110 400100

x

x

!

!

He paid $1,104 in taxes.

15. Let x be the amount of sales tax. Ratio of dollar amounts ! Ratio of percents

x : . :540 8 25 100!

x

x

xx

5408 25100

100100

100 8 25 540100 4 455

!

" ! "

!

.

.,

!!

!

4 45510044 55

,

.x She must pay $44.55 sales tax.

16. Let x be number of cans of blue paint needed. New ratio of yellow paint to blue paint ! Original

ratio of yellow paint to blue paint

2 4 3: :x !

2 43

2 3 46 4

xx

x

!

" ! "

!

64

44

1 5

!

!

x

x.

He needs 1.5 cans of blue paint.

17. Let x be the amount of time needed. New ratio of area to time needed ! Original ratio

of area to time needed

227.2 : x ! 113.6 : 4

227 2 113 64

227 2 4 113 6908 8 113 6

113

. .

. .. .

xx

x

!

" ! "

!

.. ...

.

.

6 908 8113 6113 6

908 8113 6

xx

!

!

x ! 8

He needs 8 hours to paint the wall.

18. a) Let x be the number of hours needed.

25 10

2

hoursmodel boats

hoursmodel boats

!x

55 10

55

205

5 2 105 20

4

!

" ! "

!

!

!

x

x

xx

x It will take her 4 hours to paint 10 model

boats.

b) Let y be the number of model boats.

25

10

2

hoursmodel boats

hoursmodel boats

!y

5510

22

502

2 5 102 50

25

!

" ! "

!

!

!

y

y

yy

y She can paint 25 model boats in 10 hours.

19. Let x be the sales for that month. Ratio of commissions ! Ratio of percents

265 32 5 5 100. : . :x !

265 32 5 5100

265 32 100 5 526 53

. .

. .,

xx

!

" ! "

22 5 55 5 26 532

!

!

.. ,

xx

5 55 5

26 5325 5

..

,.

x!

x ! 4,824 His sales for that month were $4,824.


0 025! .

The direct proportion equation is

I P! 0 025. . b) When P ! 1,000 and I P! 0 025. ,

I P!! "

!

0 0250 025 1 00025

.

. ,

When I ! 56 and I P! 0 025. , 56 ! 0.025P

56

0 0250 0250 025...

!P

P ! 2,240

P 600 1,000 2,240I 15 25 56

21. The value of y will triple as well because x and y are in direct proportion.

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MIF_SK_C2_Ch01-10.indd 81 12/16/11 4:08 PM

22. To find out which store has the best deal, compare the unit price.

Since 1 lb ! 16 oz,

Store A: Unit price ! $2.40/lb

! 2 4016. ! $0.15/oz

Store B: Unit price ! $1.28/8 oz

! 1 288. ! $0.16/oz

Store C: Unit price ! $1.08/6 oz

! 1 086. ! $0.18/oz

Since the unit price at Store A is the lowest,

Store A has the best deal.

Lesson 5.4Hands-On Activity (Recognize Inverse Proportion)Step 1 and Step 2:

v 1 2 3 4 6 12

h 12 6 4 3 2 1

v " h 12 12 12 12 12 12

Math Journal: The value of v " h gives the area of the rectangle and it is a constant. As the value of

v increases, the value of h decreases.

Guided Practice 1. For each pair of values, x and y: 1 " 180 ! 180 2 " 90 ! 180 3 " 60 ! 180

The value of x increases as the value of y decreases, and the product of x and y is a constant value. So, y is inversely proportional to x.


2. For each pair of values, x and y: 40 " 9 ! 360

50 " 7 13

! 366 23

60 " 6 ! 360

The value of x increases as the value of y decreases but the product of x and y is not a constant value. So, y is not inversely proportional to x.

3. 35

6yx

!

35

53

6 53

yx

! " !

y

x!

10

y x xx

! " !10

xy ! 10 The original equation can be rewritten as two

equivalent equations in the form y kx

! and

xy k! . So, the equation represents an inverse proportion. The constant of proportionality is 10.

4. y # 3x ! 5 y x x x! " # "3 3 5 3 y ! 5 $ 3x The original equation cannot be rewritten as two


! and

xy k! . So, the equation does not represent an inverse proportion.

5. a) Answers vary. Sample: Use the point (3, 20) from the graph to find the constant

of proportionality:

x y! " !3 20 ! 60 The constant of proportionality is 60.

The inverse proportion equation is xy ! 60. b) It means that 6 volunteers can clean the

beach in 10 hours.

6. a) Constant of proportionality: x y! " !5 3! 15 The constant of proportionality is 15.

b) Inverse proportion equation:

xy ! 15 or yx

!15

The inverse proportion equation is

xy ! 15 or yx

!15 .

c) When x ! 10 and yx

!15 , y ! !15

101 5.

7. Let x be the number of trucks. Let y be the number of hours. Constant of proportionality: x y! " !15 28 ! 420

Inverse proportion equation: xy ! 420 When y ! 20 and xy ! 420, 20 " x ! 420 20x ! 420

2020

42020

x !

x ! 21 21 trucks are needed to paint the highway in

20 hours.

Practice 5.4 1. For each pair of values, x and y : 25 2 50! " 10 5 50! " 5 10 50! " The value of x decreases as the value of y increases,

and the product of x and y is a constant value. So, x and y are in inverse proportion. The constant of proportionality is 50.

2. For each pair of values, x and y: 7 30 210! " 5 60 300! 3 70 210! " The value of x decreases as the value of y increases

but the product of x and y is not a constant value. So, x and y are not in inverse proportion.

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MIF_SK_C2_Ch01-10.indd 82 12/16/11 4:08 PM

3. For each pair of values, x and y: 4 16 64! " 6 24 144! " 8 32 256! " The value of x increases as the value of y

increases and the product of x and y is not a constant value. So, x and y are not in inverse proportion.

4. For each pair of values, x and y: 6 2 12! " 3 4 12! " 1 12 12! " The value of x decreases as the value of y increases,

and the product of x and y is a constant value. So, x and y are in inverse proportion. The constant of proportionality is 12.

5. 10 5xy

!

10

10

10 5

5

5

1010

51012

x

x y y

xy

xy

y

y

xy

xyx

!

" ! "

!

!

!

!!12x

y x!

12

The original equation can be rewritten as two


! or

xy ! k. So, the equation represents an inverse proportion. The constant of proportionality is 1

2.

6. y x20

!

20 20

2020

! " !

"

y x

y x The original equation cannot be rewritten as

two equivalent equations in the form y kx

!

or xy ! k. So, the equation does not represent an inverse proportion.

7. y x! "17

12

y x x x

y x

! " # "

# "

17

17

12

17

12

17

The original equation cannot be rewritten as two


! or xy ! k. So, the equation does not represent an inverse

proportion.

8. 0 1 5. xy

!

0 1

0 1 5

5

0 10 1

50 1

.

... .

x y y

xyy

xy

! " !

"

"

xy

y

xyx x

x

"

"

"

50

50

50

The original equation can be rewritten as two


! or xy ! k. So, the equation represents an inverse proportion.


9. Use (4, 0.5) to find the constant of proportionality:

x y! " !"

4 0 52

.


10. Use (2, 5) to find the constant of proportionality:

x y! " !

"

2 510


11. Use (0.5, 8) to find the constant of proportionality:

x y! " !

"

0 5 84.


12. Use ( 12

, 30) to find the constant of proportionality:

x y! " !

"

12

30

15 The constant of proportionality is 15.

13. Two quantities are in inverse proportion when one quantity decreases and the other increases in

such a way that the product of the two quantities

remains constant. So, you can tell whether two

quantities are in inverse proportion by checking

whether the product of the two quantities remains

constant.

14. a) Use the point (6, 2) from the graph to find the constant of proportionality:

n " b ! 6 " 2 ! 12



nb ! 12 or bn

!12 .

b) Total number of batches of bagel dough produced in one hour.

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MIF_SK_C2_Ch01-10.indd 83 12/16/11 4:08 PM

c) It means that 6 workers each mix 2 batches of dough in one hour to meet the needs of

customers.

15. a) Use the point (3, 8) from the graph to find the constant of proportionality:

w ! l " 3 ! 8 " 24 The constant of proportionality is 24.


wl " 24 or lw

!24 .

b) Area of a rectangle. c) It means that the rectangle with a fixed area

of 24 square units has a length of 8 units

when the width is 3 units.

16. a) Constant of proportionality: x y! " !5 2 ! 10



xy ! 10 or yx

!10


xy ! 10 or yx

!10 .

c) When y " 4 and xy ! 10 ,

xx

x

x

! "

"

"

"

4 104 10

2 5

44

104.

17. a) Constant of proportionality:

x y! " !12

13

! 16


.


xy ! 16

or yx

!16


xy ! 16

or yx

!16

.

c) When x ! 15

and yx

!16

, y !"

1

6 15

y ! 5

6

18. As y is inversely proportional to x, when the value of x decreases, the value of y increases proportionally. When the value of x is halved, the value of y is doubled.

19. Let n be the number of gardeners. Let t be the time in hours. Constant of proportionality: n ! t " 3 ! 4 " 12

Inverse proportion equation: nt " 12 When n " 1 and nt " 12, 1 12! "t t ! 12 It will take 1 gardener 12 hours to mow 9 lawns.

20. Let s be the download speed. Let t be the time in seconds. Constant of proportionality: s ! t " 256 ! 720 " 184,320

Inverse proportion equation: st " 184,320 When s " 512 and st " 184,320, 512 ! t " 184,320

512 184 320512512

184 320512

tt

!

!

,,

st ! !360 6 min It will take 6 minutes to download the same file.

21. Both tables show the amount paid by each person in the house per day; The products of x and y are the same until each house has 2 people. Because

xy is constant for Guest House A, the amount paid by each person in the house per day is

inversely proportional to the number of people in

the house. Because xy is not a constant for Guest House B, the amount paid by each person in the

house per day is not inversely proportional to the

number of people in the house.

22. The equation is in the form xy " k. So, xy " 50 represents an inverse proportion.

23. As the equation is in the form mv

k! , mv

! 3

represents a direct proportion.

24. As the equation is in the form y kx

! , yx

!200

represents an inverse proportion.

25. The amount of sales tax depends on the number of shirt purchased.

Sales tax amount " Shirt price ! Sales tax:

The sales tax amount is directly proportional to

the number of shirts purchased.

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MIF_SK_C2_Ch01-10.indd 84 12/16/11 4:08 PM

Brain@Work 1. Let x be the cost of the book bought in Germany (in Euro) plus VAT.

107100 25 99

!x.

100 25 99 107100 2 780 93100100

2 780 9

! " !

"

"

xxx

., ., . 33100

27 809327 81

xx

" ..!!

Let y be the cost of the book when converted to U.S. dollars.

1

0 72627 81

..

!y

y ! "0 726 27 81. .

y ! 20 19.! Let z be the cost of the book bought in US plus tax.

106100 23 99

!y.

100 23 99 106! " !y .

100 2 542 94y ! , .

100100

2 542 94100

25 429425 43

y

yy

!

!

, .

.

.!! Tom should buy the book in Germany, as it is

cheaper.

2. Amount of gasoline he plans to use for the remaining journey ! 12 " 5

! 7 gal

Distance left to travel ! 350 " 150

! 200 mi

Let x be the number of gallons of gasoline he needs to travel a distance of 200 miles.

150 : 200 ! 5 : x

150200

5!

x 150 # x ! 5 # 200

150 1 000

6

150150

1 00015023

x

x

x

!

!

!

,,

He needs 6 23

gallons to travel the remaining

journey. So, Johnny is able to drive to Town

Q before stopping for gasoline.

Chapter Review/Test 1. For each pair of values, x and y :

yx

! !4 53

32

. yx

! !7 55

32

. yx

! !10 57

32

.

So, y is directly proportional to x. 2. For each pair of values, x and y : xy ! 2 # 50 ! 100 xy ! 4 # 25 ! 100 xy ! 8 # 12.5 ! 100 The value of x increases as the value of y

decreases, and the product of x and y is a constant value. So, y is inversely proportional to x.

3. For each pair of values, x and y :

yx

! !126

2

yx

! 98

yx

! !3 524

748

.

So, y is not directly proportional to x. For each pair of values, x and y: xy ! 6 # 12 ! 72 xy ! 8 # 9 ! 72 xy ! 24 # 3.5 ! 84 The value of x increases as the value of y

decreases but the product of x and y is not a constant value. So, x and y are not in inverse proportion.

4. yx

! !2 55

12

. yx

! !510

12

yx

! !7 515

12

.

So, y is directly proportional to x. 5. Although the graph is a straight line that does not

lie along the x- or y-axis, it does not pass through the origin. So, it does not represent a direct

proportion.

Also, it is not a curve that never crosses the

horizontal and vertical axes. So, it does not

represent an inverse proportion.

6. The graph is a straight line through the origin, and does not lie along the x- or y-axis. So, it represents a direct proportion.

7. The graph is a curve that never crosses the horizontal and vertical axes, so it represents an

inverse proportion.

8. The graph shows two straight lines. So, it does not represent a direct or an inverse proportion.

9. Because the original equation y x! "12

5 cannot be rewritten as an equivalent equation in

the form y ! kx, it does not represent a direct proportion. The original equation cannot be

rewritten as two equivalent equations in the

form y kx

! or xy ! k. So, it does not represent an inverse proportion.

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MIF_SK_C2_Ch01-10.indd 85 12/16/11 4:08 PM

10. y x4

7!

4 7 4

284

! " !

"

y x

y x

Because the original equation y x4

7! can be

rewritten as an equivalent equation in the

form y ! kx, it represents a direct proportion.11. "3 # x ! y

y ! x " 3 Because the original equation "3 # x ! y

cannot be rewritten as an equivalent equation in

the form y ! kx, it does not represent a direct proportion. The original equation cannot be

rewritten as two equivalent equations in the form

y kx

! or xy ! k. So, it does not represent an inverse proportion.

12. yx23

!

y xxy! " !

"

3 26

Because the original equation yx23! can be

rewritten as an equivalent equation in the

form y ! kx, it represents a direct proportion.


!164

! 4


The equation is y ! 4x. When x ! 2 and y ! 4x, y ! 4 $ 2

! 8

When y ! 25 and y ! 4x, 25 ! 4x

254

44

6 25

!

!

x

x.

x 2 4 6.25y 8 16 25


!2 55. ! 0.5

The constant of proportionality is 0.5.

The direct proportion equation is y ! 0.5x. When x ! 3 and y ! 0.5x, y ! 0.5 $ 3

! 1.5

When y ! 3 and y ! 0.5x, 3 ! 0.5x

30 5

0 50 5

6.

..

!

!

x

x

x 3 5 6y 1.5 2.5 3

15. Constant of proportionality: xy ! 2 $ 30 ! 60 The constant of proportionality is 60.

The direct proportion equation is xy ! 60. When x ! 4 and xy ! 60, 4 $ y ! 60

4y ! 60

44

60415

y

y

!

!

When y ! 10 and xy ! 60, x $ 10 ! 60 10x ! 60

1010

60106

x

x

!

!

x 2 4 6y 30 15 10

16. Constant of proportionality: xy ! 5 $ 1.6 ! 8 The constant of proportionality is 8.

The inverse proportion equation is xy ! 8. When x ! 2.5 and xy ! 8, 2.5 $ y ! 8

2 5 8

3 2

2 52 5

82 5

.

.

.. .

y

y

y

!

!

!

When y ! 2 and xy ! 8, x $ 2 ! 8

2 8

4

22

82

x

x

x

!

!

!

x 2.5 4 5y 3.2 2 1.6


!567

! 8

The direct proportion equation is y ! 8x. When x ! 4 and y ! 8x, y ! 8 $ 4 ! 3218. Constant of proportionality: xy ! 4 $ 12 ! 48 The inverse proportion equation is xy ! 48. When y ! 8 and xy ! 48, x $ 8 ! 48

8 48

6

88

488

x

x

x

!

!

!


4!


represents the unit cost per gallon of gas.

b) The direct proportion equation is y ! 4x. c) When y ! 24 and y ! 4x, 24 ! 4x

244

44

6

!

!

x

x From the graph, he bought 6 gallons of

gas for $24.

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Chapter 5 - Weeblycolontlc.weebly.com/uploads/1/3/3/9/13391515/solutions...Practice 5.1 1. For each pair of values, x and y:!5 1 5 10 2!5 15 3!5 So, y is directly proportional to x