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Practice set b

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

Step 1:   Let x =

Step 2:


Step 3:





Step 4:



Step 5:   The number is .

There are two numbers: 3 7 , 2 3 .

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Sample set c

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

1 6 + 1 4 = 1 x
Step 3: 1 6 + 1 4 = 1 x . An excluded value is 0 .  The LCD is 12 x . Multiply each term by 12 x . 12 x · 1 6 + 12 x · 1 4 = 12 x · 1 x 2 x + 3 x = 12 Solve this nonfractional  equation to obtain the potential solutions . 5 x = 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 .

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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?

Step 1:

Step 2:



Step 3:





Step 4:


Step 5:   Working together, A and B .

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

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Sample set d

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 = 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 of the tank in 1 hour.
If the outlet pipe can drain the tank in 20 hours, it can drain 1 20 of the tank in 1 hour.
If both pipes are open, it takes x hours to fill the tank. So 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
Step 3: 1 12 1 20 = 1 x . An excluded value is 0 . The LCD is 60 x .  Multiply each term by 60 x . 60 x · 1 12 60 x · 1 20 = 60 x · 1 x 5 x 3 x = 60 Solve this nonfractional equation to obtain  the potential solutions . 2 x = 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 .

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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?

Step 1:

Step 2:



Step 3:





Step 4:


Step 5:

It will take 40 hr to fill the tank.

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Sample set e

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 = time required for B to complete the job working alone. Then, ( 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  2 x ( x + 3 ) .  2 x ( x + 3 ) · 1 x + 2 x ( x + 3 ) · 1 x + 3 = 2 x ( x + 3 ) · 1 2 2 ( x + 3 ) + 2 x = x ( x + 3 ) 2 x + 6 + 2 x = x 2 + 3 x This is a quadratic equation that can  be solved using the zero-factor property . 4 x + 6 = x 2 + 3 x 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 .

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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