£ ≈ ∑ Chapter 9: Test Your Proficiency Directions: Select a section to work on. Work out each...

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Chapter 9: Test Your Proficiency

Directions: •Select a section to work on.•Work out each problem on a piece of paper. •Click to check your answer.•For detailed steps click on the provided link.•Move on to the next problem or return to the menu.

Sections

9-1: Rational Expressions- products, quotients, and undefined values

9-2: Rational Expressions- sums and differences

9-6: Solving rational equations

9-6: Applications (word problems) involving rational equations

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9-1

2

1. For what value(s) of the variable is the expression undefined?5

7 10x

x x

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9-1

The expression is undefined when 2 and 5.x

See a detailed explanation.

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2


7 10x

x x


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9-1

2


7 10x

x x

A rational expression is undefined for any values of the variable that make the denominator equal to zero. Therefore, to find the value(s) that make an expression undefined, set the denominator equal to zero and solve for the variable.

2 7 10 05 2 0

5 0 or 2 05 or 2

x xx xx xx x

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Chapter 9: Test Your Proficiency9-1

2

2

2. For what value(s) of the variable is the expression undefined?

42 8

xx x


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9-1

The expression is undefined when 2 and 4.x x

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2

2


42 8

xx x



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9-1


2 2 8 04 2 0

4 0 or 2 04 or 2

x xx xx xx x

The expression is undefined for 2 and 4.x xReturn

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2

2


42 8

xx x


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Chapter 9: Test Your Proficiency9-1

2

3 2


58 15x x

x x x


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9-1

The expression is undefined when 0, 3 and 5.x x x

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2

3 2


58 15x x

x x x



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9-1


3 2

2

8 15 0

8 15 0

3 5 0

0, 3 0 or 5 00, 3 or 5

x x x

x x x

x x xx x xx x xReturn

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2

3 2


58 15x x

x x x

The expression is undefined when 0, 3 and 5.x x x


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9-1

2

4. Simplify the rational expression.5

7 10x

x x


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9-1

2


7 10x

x x

1

2x

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9-1

Step 1: Factor the numerator and denominator completely.

5

5 2x

x x

Step 2: Reduce/cancel common factors.

5

5 2x

x x

Step 3: State the reduced rational expression.

1

2x

2


7 10x

x x

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9-1

2

2

5. Simplify the rational expression.

4 124

x xx

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9-1

2

2


4 124

x xx

62

xx

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9-1

Step 1: Factor the numerator and denominator completely.

6 22 2x xx x


Step 3: Simplify the numerator and state the reduced rational expression.

62

xx

2

2


4 124

x xx

6 22 2x xx x

1 Note: Opposites reduce to -1

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9-1

2

2

6. Simplify the product.

8 423 4

x x xxx x


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9-1

8

2 2xx

2

2


8 423 4

x x xxx x

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9-1

Step 1: Factor completely.

8 4

4 1 2x x xx x x


Step 3: Simplify the denominator and state the reduced rational expression.

8

2 2xx

2

2


8 423 4

x x xxx x

8 4

4 1 2x x xx x x

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9-1

2

2 2

7. Simplify the quotient.

36 3 183 4 3

x xx x x x

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9-1

2

2 2


36 3 183 4 3

x xx x x x

2 5 63

x xx

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9-1

Step 2: Factor completely.

6 6 3 1

3 3 6x x x xx x x


Step 4: Simplify the numerator and state the reduced rational expression.

2

2 2


36 3 183 4 3

x xx x x x

Step 1: Rewrite the problem: multiply by the reciprocal

2 2

2

36 4 33 183

x x xxx x

6 6 3 1

3 3 6x x x xx x x

2 5 63

x xx

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9-2 1. Simplify each expression.3 25 3xy x


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9-2 1. Simplify the expression.3 25 3xy x

29 1015x y

xy


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9-2

Step 2: Multiply each numerator and denominator by missing LCD factors.

Step 3: Simplify each numerator separately.

Step 4: Combine like terms in the numerator (if necessary) and write over the LCD.


Step 1: Find the LCD (lowest common denominator)The LCD is the product of all different factors. LCD = 5 3 15x y xy

1. Simplify the expression.3 25 3xy x

3 25

3 53 53x yx y

xy x

29 115 15

0xyy

xy x

29 1015x y

xy

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9-2

2

2. Simplify the expression.5 2

636 xx


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9-2

2

2 736

xx

2


636 xx


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9-2



Step 4: Distribute the subtraction sign. Combine like terms in the numerator and write over the LCD.


Step 1: Find the LCD (lowest common denominator)

The LCD is the product of the common factors andother factors.

LCD = 6 6x x

2

2 736

xx

2


636 xx

2

6 65

66

6x x xxx

5 2 12

6 6 6 6x

x x x x

common other factor

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9-2

2


2 42x x

xx x


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9-2

2

2

2 52 2 4x xx x

2


2 42x x

xx x

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9-2



Step 4: Combine like terms in the numerator and write over the LCD.

Step 1: Find the LCD (lowest common denominator)

The LCD is the product of the common factors andother factors.

LCD = 2 1 2x x

2

2

2 52 2 4x xx x

common factorother factors

2


2 42x x

xx x

1 32 1 2 2

122 1

x xx

xx x x

22

2 1 2 22 4 3

2 1x x

x xxxx

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9-2

3

2

3 2

4. Simplify the expression.

53 6

6 243

xxxx x


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9-2

210 203

x xx

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3

2

3 2


53 6

6 243

xxxx x



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9-2

Step 2: Factor each numerator and denominator completely and reduce/cross cancel common factors. Remember: you can only cross cancel in a multiplication problem.

Step 3: Multiply the remaining factors in the numerator and denominator.

Note: the final answer cannot have any parentheses and both numerator and denominator should be in standard form.

Step 1: Rewrite as a division problem and use the rule for division—multiply by the reciprocal

25 2 2 10 2 10 201 3 3 3x x x x x xx x x

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3

3 2

2


53 6

36 24

xx

x xx

3 3 2 3 2

2 3 2

5 3 5 6 243 6 3 66 24 3x x x x xx xx x x

3

2

6 2 253 2 3

x xxx x x

x 2 1

111

210 203

x xx


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9-2

5. Simplify the expression.5

3 22 35 4

xy xy

y


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9-2

2

2

20 15024 45x yxy x

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3 22 35 4

xy xy

y



53 22 35 4

xy xy

y

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9-2

Step 2: Multiply each term in the expression by the LCD; this is equivalent to multiplying by a ratio of 1; reduce as much as possible.

Step 3: Multiply the remaining factors in the numerator and denominator.

Note: the final answer cannot have any parentheses and both numerator and denominator should be in standard form.

Step 1: Find the LCD for all of the denominators.

20 5 3012 2 3 15

x x yx y y x

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3(5)(4)( )( ) 60LCD x y xy

3020

12 15

2

2

20 15024 45x yxy x


3 22 35 4

xy xy

y

60xy 60xy

60xy 60xy


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9-6 11 7 91. Solve. 4 2x x


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9-6 11 7 91. Solve. 4 2x x


2

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9-6

Step 2: Multiply each term by the LCD. Reduce and rewrite the equation.

Step 3: Check for extraneous solutions. Then state the solution set.

Step 1: Find the LCD. LCD = 4x

2


11 7 91. Solve. 4 2x x

11 7 94 2

11 14 3611

4 4 41 1

22

1

2

x x xx x

xxx

11 7 9. ?4 2

11 7 9?4 4 218 9

2

2 2

4

Check

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9-62

5 5 122. Solve. 3 39 x xx


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9-6


8

2

5 5 122. Solve. 3 39 x xx

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9-6

Step 2: Multiply each term by the LCD. Reduce and rewrite the equation.


Step 1: Find the LCD. LCD = 3 3x x

8


2

5 5 122. Solve. 3 39 x xx

5 5 123 3 3 3

5 5 3 12 3

5 5 15 12 365 20 12 3

3 3 3 3 3 31 1 1

656 7

8

x xx x x x x x

x x

x xx xx x

xx

2

5 5 12?3 39

5 5 12?55 5 115 55 60?

55 55 5560 60

5 5

8

5

88

5

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9-62

4 123. Solve. 4 2 2 8

xx x x x


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9-6

10

2

4 123. Solve. 4 2 2 8

xx x x x

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9-6

Step 2: Multiply each term by the LCD. Reduce and rewrite the equation. Then solve the equation.



10

2

2

2

4 124 2 4 2

4 12 8 20 0

4

4 2 4 2 4 21 1 1

2 4

8 4 12 10 2 0

8 8 12 10 0 or 2 010 or 2

xx x x x

x x x

x x x x x

x x x

x x x x x x

x

x x

x x

2

4 123. Solve. 4 2 2 8

xx x x x

2

1010 10 10 10

4 12?4 2 2 8

4 10 12?6 12 7248 60 12?

72 72 7212 1272 72

2

22 2 2

4 12?4 2 2 8

4 2 12

2

6 0 0

These expressions are undefined, therefore the solution is extraneous and must be thrown out.

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9-6

2

6 18 44. Solve. 3 58 15x xx xx x


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9-6

2,3


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2

6 18 44. Solve. 3 58 15x xx xx x


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9-6

Step 2: Multiply each term by the LCD. Reduce and rewrite the equation. Then solve the equation.

Step 3: Check for extraneous solutions (denominator = 0). Then state the solution set.


10

2

2

2

2

6 18 43 58 15

(6 18) 4 5 6 0

5 6 18 4 1

3 5 3 5 3 51

2 3 2 0

18 4 12 3 0 or

1 1

5

2 03 or 2

3

x x x xx xx xx x

x x x x

x x x x x x

x x

x x

x

x

x xx

x

x

23 33 (3

6 18 43 58 15)

x x

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2

6 18 44. Solve. 3 58 15x xx xx x

den.= 6den.= 48 den.= 8

23 33 (3

6 18 43 58 15)

x x


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9-6 Word Problems


1. Tammy can plant a garden in 3 hours. Her sister can plant the garden in 5 hours. How long will it take to plant the garden if they work together? Round to the nearest tenth if necessary.

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9-6 Word Problems

It will take about 1.9 hours to plant the garden if they work together.

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1. Tammy can plant a garden in 3 hours. Her sister can plant the garden in 5 hours. How long will it take to plant the garden if they work together? Round to the nearest tenth if necessary.



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9-6 Word Problems

Step 2: Set up the equation: Tammy’s work + Sister’s work = Total work

Step 3: Solve the rational equation. Step 4: State your conclusion.

Step 1: Set up the table.

1. Tammy can plant a garden in 3 hours. Her sister can plant the garden in 5 hours. How long will it take to plant the garden if they work together?

Rate (1/time alone) Time working together Work (r·t)

Tammy x

Sister x

1315

3x

5x

13 5x x

13 5

5 3 158 15

15 1.98

15 15 151 1 1

x x

x xx

x

It will take about 1.9 hours to plant the garden if they work together.

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9-6 Word Problems


2. Sukvir can do the yard work in 4 hours. If his brother helps him the yard work will be done in 1.5 hours. How long would it take to do the yard work if his brother worked alone? Round to the nearest tenth if necessary.

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9-6 Word Problems


It would take his brother 2.4 hours to do the yard work if he works alone.

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2. Sukvir can do the yard work in 4 hours. If his brother helps him the yard work will be done in 1.5 hours. How long would it take to do the yard work if his brother worked alone? Round to the nearest tenth if necessary.


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9-6 Word Problems

Step 2: Set up the equation: Sukvir’s work + Brother’s work = Total work




Rate (1/time alone) Time working together Work (r·t)

Sukvir 1.5

Brother 1.5

141x

1.54

1.5x

1.5 1.5 14 x

4 4 41.5 1.5 141.5 6 4

2.5 6

1

2

1

.4

1x x x

xx xx

x

It will take his brother 2.4 hours to do the yard work if he works alone.

2. Sukvir can do the yard work in 4 hours. If his brother helps him the yard work will be done in 1.5 hours. How long would it take to do the yard work if his brother worked alone?

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9-6 Word Problems


3. The current in a river is 7 miles per hour. In her motorboat Stella can travel 84 miles downstream in the same amount of time she can travel 56 miles upstream. What is the speed of her motorboat in still water? Round to the nearest tenth if necessary.

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9-6 Word Problems


The speed of Stella’s motorboat in still water is 35 mph.

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3. The current in a river is 7 miles per hour. In her motorboat Stella can travel 84 miles downstream in the same amount of time she can travel 56 miles upstream. What is the speed of her motorboat in still water? Round to the nearest tenth if necessary.


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9-6 Word Problems

Step 2: Set up the equation: Upstream Time = Downstream Time




Distance Rate

Upstream 56

Downstream 84

7x

847x

56 847 7x x

56 847 756 7 84 7

56 392 84 58828 98

7 7 7 71 1

035

x xx x x

x x

x

x xxx

3. The current in a river is 7 miles per hour. In her motorboat Stella can travel 84 miles downstream in the same amount of time she can travel 56 miles upstream. What is the speed of her motorboat in still water?

DistanceTime = Rate

7x

567x

The speed of Stella’s motorboat in still water is 35 mph.

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9-6 Word Problems4. Cameron drove to visit his grandmother at an average speed of 56 mph. He stayed for 2 hours and when he returned home using the same route his average speed was only 42 mph. If he was away from home for 5.75 hours, how far does Cameron live from his grandmother? Round to the nearest tenth if necessary.

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9-6 Word Problems

Cameron lives 90 miles from his grandmother.

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4. Cameron drove to visit his grandmother at an average speed of 56 mph. He stayed for 2 hours and when he returned home using the same route his average speed was only 42 mph. If he was away from home for 5.75 hours, how far does Cameron live from his grandmother? Round to the nearest tenth if necessary.



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9-6 Word Problems

Step 2: Set up the equation: To Grandmother’s Time + Going Home Time = Total Driving Time



Distance Rate

To Grandmother’s x

Going Home x

56

42x

5.75 2 3.7556 42 56 42x x x x

DistanceTime = Rate

42

56x

4. Cameron drove to visit his grandmother at an average speed of 56 mph. He stayed for 2 hours and when he returned home using the same route his average speed was only 42 mph. If he was away from home for 5.75 hours, how far does Cameron live from his grandmother?

168 16

5.75 256 42

3.7556 42

3.7556 42 1

3 4 6307 63

8 1

0

1 1

9

681

0

x x

x x

x x

x xxx

Cameron lives 90 miles from his grandmother.

End of Self Check Quiz

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9-6 Word Problems


5. A bird can fly 36 miles with the wind in the same time it can fly 18 miles against the wind. If the speed of the bird in calm air is 10 mph, find the speed of the wind. Round to the nearest tenth if necessary.

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9-6 Word Problems

The speed of the wind is about 3.3 mph.

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9-6 Word Problems

Step 2: Set up the equation: With the wind Time = Against the wind Time

Step 3: Solve the rational equation. (note: you can cross multiply here because it is a proportion.)


Distance Rate

With the wind 36

Against the wind 18

10 x

18

10 x

36 1810 10x x

36 10 18 10

360 36 180 18180 54

3.3

x xx xx

x


DistanceTime = Rate

10 x

3610 x

The speed of the wind is about 3.3 mph.

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9-6 Word Problems


6. A zebra runs 6.2 miles in the same time a cheetah runs 9.7 miles. If the cheetah runs 18 mph faster than the zebra, how fast is each animal running? Round to the nearest tenth if necessary.

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9-6 Word Problems

The zebra runs about 31.9 mph and the cheetah runs about 49.9 mph.

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9-6 Word Problems

Step 2: Set up the equation: Zebra’s Time = Cheetah’s Time

Step 3: Solve the rational equation. (note: you can cross multiply here because it is a proportion.)


Distance Rate

Zebra 6.2

Cheetah 9.7

x

9.7

18x

6.2 9.718x x

6.2 18 9.7

6.2 111.6 9.7111.6 3.5

31.918 31.9 18 49.9

x xx x

xxx


DistanceTime = Rate

18x

6.2x

The zebra runs about 31.9 mph and the cheetah runs about 49.9 mph.

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9-6 Word Problems7. Claire rows 20 miles upstream and returns. The speed of the river is 3 mph. If the upstream trip took Claire twice as long as the downstream trip, what is Claire’s rate in still water? Round to the nearest tenth if necessary.

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9-6 Word Problems

Claire’s rate in still water is 9 mph.

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7. Claire rows 20 miles upstream and returns. The speed of the river is 3 mph. If the upstream trip took Claire twice as long as the downstream trip, what is Claire’s rate in still water? Round to the nearest tenth if necessary.


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9-6 Word Problems

Step 2: Set up the equation: Upstream Time = 2Downstream Time

Step 3: Solve the rational equation.


Distance Rate

Upstream 20

Downstream 20

3x

20

3x

20 2 20

3 1 3x x

20 2 203 1 3

20 403 3

20( 3) 40( 3)20 60 40 12020 180

9

x x

x xx xx xx

x

7. Claire rows 20 miles upstream and returns. The speed of the river is 3 mph. If the upstream trip took Claire twice as long as the downstream trip, what is Claire’s rate in still water? Round to the nearest tenth if necessary.

DistanceTime = Rate

3x

203x

Claire’s rate in still water is 9 mph.

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Date post:	06-Jan-2018
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£ ≈ ∑ Chapter 9: Test Your Proficiency Directions: Select a section to work on. Work out each...

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