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Applications

Module by: Wade Ellis, Denny Burzynski. E-mail the authors

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.

The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.

Objectives of this module: be able to use the five-step method to solve various applied problems.

Overview

  • The Five-Step Method

The Five-Step Method

We are now in a position to study some applications of rational equations. Some of these problems will have practical applications while others are intended as logic developers.

We will apply the five-step method for solving word problems.

Five-Step Method

  1. Represent all unknown quantities in terms of x x or some other letter.
  2. Translate the verbal phrases to mathematical symbols and form an equation.
  3. Solve this equation.
  4. Check the solution by substituting the result into the original statement of the problem.
  5. Write the conclusion.

Remember, step 1 is very important: always

Introduce a variable.

Sample Set A

Example 1

When the same number is added to the numerator and denominator of the fraction 3 5 , 3 5 , the result is 7 9 . 7 9 . What is the number that is added?

 Step 1: Let x=x= the number being added. Step 2: 3+x 5+x = 7 9 . Step 3: 3+x 5+x = 7 9 . An excluded value is −5. The LCD is 9( 5+x ).  Multiply each term by 9( 5+x ). 9( 5+x ) ·  3+x 5+x = 9( 5+x ) ·  7 9 9( 3+x ) = 7( 5+x ) 27+9x = 35+7x 2x = 8 x = 4 Check this potential solution. Step 4: 3+4 5+4 = 7 9 . Yes, this is correct. Step 5: The number added is 4. Step 2: 3+x 5+x = 7 9 . Step 3: 3+x 5+x = 7 9 . An excluded value is −5. The LCD is 9( 5+x ).  Multiply each term by 9( 5+x ). 9( 5+x ) ·  3+x 5+x = 9( 5+x ) ·  7 9 9( 3+x ) = 7( 5+x ) 27+9x = 35+7x 2x = 8 x = 4 Check this potential solution. Step 4: 3+4 5+4 = 7 9 . Yes, this is correct. Step 5: The number added is 4.

Practice Set A

The same number is added to the numerator and denominator of the fraction 4 9 . 4 9 . The result is 2 3 . 2 3 . What is the number that is added?

Exercise 1

Step 1:   Let x x =

Step 2:


Step 3:





Step 4:



Step 5:   The number added is
.

Solution

The number added is 6.

Sample Set B

Example 2

Two thirds of a number added to the reciprocal of the number yields 25 6 . 25 6 . What is the number?

Step 1:    Let x x = the number.

 Step 2:    Recall that the reciprocal of a number x x is the number 1 x 1 x .

       2 3  · x+ 1 x = 25 6 2 3  · x+ 1 x = 25 6

Step 3: 2 3  · x+ 1 x = 25 6 The LCD is 6x. Multiply each term by 6x. 6x ·  2 3 x+6x ·  1 x = 6x ·  25 6 4 x 2 +6 = 25x Solve this nonfractional quadratic equation to obtain the potential solutions. (Use the zero-factor property.) 4 x 2 −25x+6 = 0 (4x−1)(x−6) = 0 x= 1 4 , 6 Check these potential solutions. Step 3: 2 3  · x+ 1 x = 25 6 The LCD is 6x. Multiply each term by 6x. 6x ·  2 3 x+6x ·  1 x = 6x ·  25 6 4 x 2 +6 = 25x Solve this nonfractional quadratic equation to obtain the potential solutions. (Use the zero-factor property.) 4 x 2 −25x+6 = 0 (4x−1)(x−6) = 0 x= 1 4 , 6 Check these potential solutions.

Step 4:    Substituting into the original equation, it can be that both solutions check.

Step 5:    There are two solutions: 1 4 1 4 and 6.

Practice Set B

Seven halves of a number added to the reciprocal of the number yields 23 6 . 23 6 . What is the number?

Exercise 2

Step 1:   Let x x =

Step 2:


Step 3:





Step 4:



Step 5:   The number is
.

Solution

There are two numbers: 3 7 , 2 3 . 3 7 , 2 3 .

Sample Set C

Example 3

Person A, working alone, can pour a concrete walkway in 6 hours. Person B, working alone, can pour the same walkway in 4 hours. How long will it take both people to pour the concrete walkway working together?

Step 1:  Let x x = the number of hours to pour the concrete walkway working together (since this is what we’re looking for).

Step 2:  If person A can complete the job in 6 hours, A can complete 1 6 1 6 of the job in 1 hour.
 If person B can complete the job in 4 hours, B can complete 1 4 1 4 of the job in 1 hour.
 If A and B, working together, can complete the job in x x hours, they can complete 1 x 1 x of the job in 1 hour. Putting these three facts into equation form, we have

 1 6 + 1 4 = 1 x 1 6 + 1 4 = 1 x
Step 3: 1 6 + 1 4 = 1 x . An excluded value is 0.  The LCD is 12x. Multiply each term by 12x. 12x ·  1 6 +12x ·  1 4 = 12x ·  1 x 2x+3x = 12 Solve this nonfractional  equation to obtain the potential solutions. 5x = 12 x = 12 5      or     x=2 2 5 Check this potential solution. Step 4: 1 6 + 1 4 = 1 x 1 6 + 1 4 = 1 12 5 . Is this correct? 1 6 + 1 4 = 5 12 The LCD is 12. Is this correct? 2 12 + 3 12 = 5 12 Is this correct? 5 12 = 5 12 Is this correct? Step 5:      Working together, A and B can pour the concrete walkway in 2 2 5  hours. Step 3: 1 6 + 1 4 = 1 x . An excluded value is 0.  The LCD is 12x. Multiply each term by 12x. 12x ·  1 6 +12x ·  1 4 = 12x ·  1 x 2x+3x = 12 Solve this nonfractional  equation to obtain the potential solutions. 5x = 12 x = 12 5      or     x=2 2 5 Check this potential solution. Step 4: 1 6 + 1 4 = 1 x 1 6 + 1 4 = 1 12 5 . Is this correct? 1 6 + 1 4 = 5 12 The LCD is 12. Is this correct? 2 12 + 3 12 = 5 12 Is this correct? 5 12 = 5 12 Is this correct? Step 5:      Working together, A and B can pour the concrete walkway in 2 2 5  hours.

Practice Set C

Person A, working alone, can pour a concrete walkway in 9 hours. Person B, working alone, can pour the same walkway in 6 hours. How long will it take both people to pour the concrete walkway working together?

Exercise 3

Step 1:

Step 2:



Step 3:





Step 4:


Step 5:   Working together, A and B
.

Solution

Working together, A and B can pour the concrete walkway in 3 3 5 hr. 3 3 5 hr.

Sample Set D

Example 4

An inlet pipe can fill a water tank in 12 hours. An outlet pipe can drain the tank in 20 hours. If both pipes are open, how long will it take to fill the tank?

Step 1:  Let x x = the number of hours required to fill the tank.

Step 2:  If the inlet pipe can fill the tank in 12 hours, it can fill 1 12 1 12 of the tank in 1 hour.
If the outlet pipe can drain the tank in 20 hours, it can drain 1 20 1 20 of the tank in 1 hour.
If both pipes are open, it takes x x hours to fill the tank. So 1 x 1 x of the tank will be filled in 1 hour.
Since water is being added (inlet pipe) and subtracted (outlet pipe) we get

 1 12 − 1 20 = 1 x 1 12 − 1 20 = 1 x
Step 3: 1 12 − 1 20 = 1 x . An excluded value is 0. The LCD is 60x.  Multiply each term by 60x. 60x ·  1 12 −60x ·  1 20 = 60x ·  1 x 5x−3x = 60 Solve this nonfractional equation to obtain  the potential solutions. 2x = 60 x = 30 Check this potential solution. Step 4: 1 12 − 1 20 = 1 x 1 12 − 1 20 = 1 30 . The LCD is 60. Is this correct? 5 60 − 3 60 = 1 30 Is this correct? 1 30 = 1 30 Yes, this is correct. Step 5:      With both pipes open, it will take 30 hours to fill the water tank. Step 3: 1 12 − 1 20 = 1 x . An excluded value is 0. The LCD is 60x.  Multiply each term by 60x. 60x ·  1 12 −60x ·  1 20 = 60x ·  1 x 5x−3x = 60 Solve this nonfractional equation to obtain  the potential solutions. 2x = 60 x = 30 Check this potential solution. Step 4: 1 12 − 1 20 = 1 x 1 12 − 1 20 = 1 30 . The LCD is 60. Is this correct? 5 60 − 3 60 = 1 30 Is this correct? 1 30 = 1 30 Yes, this is correct. Step 5:      With both pipes open, it will take 30 hours to fill the water tank.

Practice Set D

An inlet pipe can fill a water tank in 8 hours and an outlet pipe can drain the tank in 10 hours. If both pipes are open, how long will it take to fill the tank?

Exercise 4

Step 1:

Step 2:



Step 3:





Step 4:


Step 5:

Solution

It will take 40 hr to fill the tank.

Sample Set E

Example 5

It takes person A 3 hours longer than person B to complete a certain job. Working together, both can complete the job in 2 hours. How long does it take each person to complete the job working alone?

Step 1:  Let x x = time required for B to complete the job working alone. Then, ( x+3 )= ( x+3 )= time required for A to complete the job working alone.
 Step 2: 1 x + 1 x+3 = 1 2 . Step 3: 1 x + 1 x+3 = 1 2 . The two excluded values are 0 and −3. The LCD is 2x( x+3 ).  2x( x+3 ) ·  1 x +2x( x+3 ) ·  1 x+3 = 2x( x+3 ) ·  1 2 2( x+3 )+2x = x( x+3 ) 2x+6+2x = x 2 +3x This is a quadratic equation that can  be solved using the zero-factor property. 4x+6 = x 2 +3x x 2 −x−6 = 0 ( x−3 )( x+2 ) = 0 x = 3,      −2 Check these potential solutions. Step 4:      If x=−2, the equation checks, but does not even make physical sense.      If x−3, the equation checks.      x=3       and      x+3=6 Step 5:      Person B can do the job in 3 hours and person A can do the job in 6 hours. Step 2: 1 x + 1 x+3 = 1 2 . Step 3: 1 x + 1 x+3 = 1 2 . The two excluded values are 0 and −3. The LCD is 2x( x+3 ).  2x( x+3 ) ·  1 x +2x( x+3 ) ·  1 x+3 = 2x( x+3 ) ·  1 2 2( x+3 )+2x = x( x+3 ) 2x+6+2x = x 2 +3x This is a quadratic equation that can  be solved using the zero-factor property. 4x+6 = x 2 +3x x 2 −x−6 = 0 ( x−3 )( x+2 ) = 0 x = 3,      −2 Check these potential solutions. Step 4:      If x=−2, the equation checks, but does not even make physical sense.      If x−3, the equation checks.      x=3       and      x+3=6 Step 5:      Person B can do the job in 3 hours and person A can do the job in 6 hours.

Practice Set E

It takes person A 4 hours less than person B to complete a certain task. Working together, both can complete the task in 8 3 8 3 hours. How long does it take each person to complete the task working alone?

Exercise 5

Step 1:

Step 2:





Step 3:





Step 4:



Step 5:

Solution

Person A, 4 hr to complete the task; person B, 8 hr complete the task.

Sample Set F

Example 6

The width of a rectangle is 1 3 1 3 its length. Find the dimensions (length and width) if the perimeter is 16 cm.

Step 1:  Let x x = length. Then, x 3 = x 3 = width.

Step 2:  Make a sketch of the rectangle.
A rectangle with the length as " x" and the breadth as a stacked fraction with the numerator " x" and the denominator " three".
The perimeter of a figure is the total length around the figure.
 x+ x 3 +x+ x 3 = 16 2x+ 2x 3 = 16 Step 3: 2x+ 2x 3 = 16. The LCD is 3. 3 · 2x+3 ·  2x 3 = 3 · 16 6x+2x = 48 8x = 48 x = 6 Check this potential solution. Step 4: 6+ 6 3 +6+ 6 3 = 16 Is this correct? 6+2+6+2 = 16 Is this correct? 16 = 16 Yes, this is correct. Since x = 6,      x 3 = 6 3 =2 Step 5:      The length = 6 cm and the width = 2 cm. x+ x 3 +x+ x 3 = 16 2x+ 2x 3 = 16 Step 3: 2x+ 2x 3 = 16. The LCD is 3. 3 · 2x+3 ·  2x 3 = 3 · 16 6x+2x = 48 8x = 48 x = 6 Check this potential solution. Step 4: 6+ 6 3 +6+ 6 3 = 16 Is this correct? 6+2+6+2 = 16 Is this correct? 16 = 16 Yes, this is correct. Since x = 6,      x 3 = 6 3 =2 Step 5:      The length = 6 cm and the width = 2 cm.

Practice Set F

The width of a rectangle is 1 12 1 12 its length. Find the dimensions (length and width) if the perimeter is 78 feet.

Exercise 6

Step 1:



Step 2:




Step 3:




Step 4:



Step 5:

Solution

length = 36 ft, width = 3 ft.

Exercises

For the following problems, solve using the five-step method.

Exercise 7

When the same number is added to both the numerator and denominator of the fraction 3 7 , 3 7 , the result is 2 3 . 2 3 . What is the number?

Solution

The number added is 5.

Exercise 8

When the same number is added to both the numerator and denominator of the fraction 5 8 , 5 8 , the result is 3 4 . 3 4 . What is the number?

Exercise 9

When the same number is added to both the numerator and denominator of the fraction 3 8 , 3 8 , the result is 1 6 . 1 6 . What is the number?

Solution

The number added is −2. −2.

Exercise 10

When the same number is added to both the numerator and denominator of the fraction 7 9 , 7 9 , the result is 2 3 . 2 3 . What is the number?

Exercise 11

When the same number is subtracted from both the numerator and denominator of 1 10 , 1 10 , the result is 2 3 . 2 3 . What is the number?

Solution

The number subtracted is −17. −17.

Exercise 12

When the same number is subtracted from both the numerator and denominator of 3 4 , 3 4 , the result is 5 6 . 5 6 . What is the number?

Exercise 13

One third of a number added to the reciprocal of number yields 13 6 . 13 6 . What is the number?

Solution

x= 1 2 ,6 x= 1 2 ,6

Exercise 14

Four fifths of a number added to the reciprocal of number yields 81 10 . 81 10 . What is the number?

Exercise 15

One half of a number added to twice the reciprocal of the number yields 2. What is the number?

Solution

2

Exercise 16

One fourth of a number added to four times the reciprocal of the number yields −10 3 . −10 3 . What is the number?

Exercise 17

One inlet pipe can fill a tank in 8 hours. Another inlet pipe can fill the tank in 5 hours. How long does it take both pipes working together to fill the tank?

Solution

3 1 13  hours 3 1 13  hours

Exercise 18

One pipe can drain a pool in 12 hours. Another pipe can drain the pool in 15 hours. How long does it take both pipes working together to drain the pool?

Exercise 19

A faucet can fill a bathroom sink in 1 minute. The drain can empty the sink in 2 minutes. If both the faucet and drain are open, how long will it take to fill the sink?

Solution

two minutes

Exercise 20

A faucet can fill a bathtub in 6 1 2 6 1 2 minutes. The drain can empty the tub in 8 1 3 8 1 3 minutes. If both the faucet and drain are open, how long will it take to fill the bathtub?

Exercise 21

An inlet pipe can fill a tank in 5 hours. An outlet pipe can empty the tank in 4 hours. If both pipes are open, can the tank be filled? Explain.

Solution

No. x=−20 x=−20 hours.

Exercise 22

An inlet pipe can fill a tank in a a units of time. An outlet pipe can empty the tank in b b units of time. If both pipes are open, how many units of time are required to fill the tank? Are there any restrictions on a a and b b ?

Exercise 23

A delivery boy, working alone, can deliver all his goods in 6 hours. Another delivery boy, working alone, can deliver the same goods in 5 hours. How long will it take the boys to deliver all the goods working together?

Solution

2 8 11  hours 2 8 11  hours

Exercise 24

A Space Shuttle astronaut can perform a certain experiment in 2 hours. Another Space Shuttle astronaut who is not as familiar with the experiment can perform it in 2 1 2 2 1 2 hours. Working together, how long will it take both astronauts to perform the experiment?

Exercise 25

One person can complete a task 8 hours sooner than another person. Working together, both people can perform the task in 3 hours. How many hours does it take each person to complete the task working alone?

Solution

First person: 12 hours; second person: 4 hours

Exercise 26

Find two consecutive integers such that two thirds of the smaller number added to the other yields 11.

Exercise 27

Find two consecutive integers such that three fourths of the smaller number added to the other yields 29.

Solution

16,17 16,17

Exercise 28

The width of a rectangle is 2 5 2 5 its length. Find the dimensions if the perimeter is 42 meters.

Exercise 29

The width of a rectangle is 3 7 3 7 the length. Find the dimensions if the perimeter is 60 feet.

Solution

width=9 ft; length=21 ft width=9 ft; length=21 ft

Exercise 30

Two sides of a triangle have the same length. The third side is twice as long as either of the other two sides. The perimeter of the triangle is 56 inches. What is the length of each side?

Exercise 31

In a triangle, the second side is 3 inches longer than first side. The third side is 3 4 3 4 the length of the second side. If the perimeter is 30 inches, how long is each side?

Solution

side 1=9 inches; side 2=12 inches; side 3=9 inches side 1=9 inches; side 2=12 inches; side 3=9 inches

Exercise 32

The pressure due to surface tension in a spherical drop of liquid is given by P= 2T r , P= 2T r , where T T is the surface tension of the liquid and r r is the radius of the drop. If the liquid is a bubble, it has two surfaces and the surface tension is given by

 P= 2T r + 2T r = 4T r P= 2T r + 2T r = 4T r
(a) Determine the pressure due to surface tension within a soap bubble of radius 2 inches and surface tension 28.
(b) Determine the radius of a bubble if the pressure due to surface tension is 52 and the surface tension is 39.

Exercise 33

The equation 1 p + 1 q = 1 f 1 p + 1 q = 1 f relates the distance p p of an object from a lens and the image distance q q from the lens to the focal length f f of the lens.
A convex lens with a horizontal line passing through its center. There is a small vertical line placed over the horizontal line on the left side of the lens, and is labeled "object." The distance between the lens and the object is indicated by a curly brace between them, and is labeled "p." There is a small vertical line placed below the horizontal line on the right side of the lens, and is labeled as "image." The distance between the lens and the image is indicated by a curly brace between them and is labeled as "q."
(a) Determine the focal length of a lens in which an object 10 feet away produces an image 6 feet away.
(b) Determine how far an object is from a lens if the focal length of the lens is 6 inches and the image distance is 10 inches.
(c) Determine how far an image will be from a lens that has a focal length of 4 4 5 4 4 5 cm and the object is 12 cm away from the lens.

Solution

(a) f= 15 4  ft f= 15 4  ft  (b) p=15 inches p=15 inches  (c) q=8 cm q=8 cm

Exercise 34

Person A can complete a task in 4 hours, person B can complete the task in 6 hours, and person C can complete the task in 3 hours. If all three people are working together, how long will it take to complete the task?

Exercise 35

Three inlet pipes can fill a storage tank in 4, 6, and 8 hours, respectively. How long will it take all three pipes to fill the tank?

Solution

1 11 13  hours 1 11 13  hours

Exercise 36

An inlet pipe can fill a tank in 10 hours. The tank has two drain pipes, each of which can empty the tank in 30 hours. If all three pipes are open, can the tank be filled? If so, how long will it take?

Exercise 37

An inlet pipe can fill a tank in 4 hours. The tank has three drain pipes. Two of the drain pipes can empty the tank in 12 hours, and the third can empty the tank in 20 hours. If all four pipes are open, can the tank be filled? If so, how long will it take?

Solution

30 hours 30 hours

Exercises For Review

Exercise 38

((Reference)) Factor 12 a 2 +13a−4. 12 a 2 +13a−4.

Exercise 39

((Reference)) Find the slope of the line passing through the points ( 4,−3 ) ( 4,−3 ) and ( 1,−6 ). ( 1,−6 ).

Solution

m=1 m=1

Exercise 40

((Reference)) Find the quotient: 2 x 2 −11x−6 x 2 −2x−24 ÷ 2 x 2 −3x−2 x 2 +2x−8 . 2 x 2 −11x−6 x 2 −2x−24 ÷ 2 x 2 −3x−2 x 2 +2x−8 .

Exercise 41

((Reference)) Find the difference: x+2 x 2 +5x+6 − x+1 x 2 +4x+3 . x+2 x 2 +5x+6 − x+1 x 2 +4x+3 .

Solution

0

Exercise 42

((Reference)) Solve the equation 9 2m−5 =−2. 9 2m−5 =−2.

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