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 July 12th, 2013, 07:28 AM #1 Newbie     Joined: Jun 2012 Posts: 25 Thanks: 0 Rational Expression word problem. Please Help This question doesn't have an answer at the back of the book and I'm unable to solve it. Chris can clean a house in 3 hours and Cheryl can clean the house in 2 hours. Their son, Tyler, can scatter toys all over the house in 4 hours. If Tyler starts scattering toys at the same time Chris and Cheryl start cleaning, when (if ever) will the house be clean?
 July 12th, 2013, 08:16 AM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 520 Math Focus: Calculus/ODEs Re: Rational Expression word problem. Please Help Since the least common multiple of 3, 2, and 4 is 12, I would look at how many houses the 3 of them will together clean in 12 hours. This is: 4 + 6 - 3 = 7 So, their house will be clean in 12/7 hr.
July 12th, 2013, 10:46 AM   #3
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Hello, AKofTroy!

Quote:
 Chris can clean a house in 3 hours and Cheryl can clean the house in 2 hours. Their son, Tyler, can scatter toys all over the house in 4 hours. If Tyler starts scattering toys at the same time Chris and Cheryl start cleaning, when (if ever) will the house be clean?

$\text{Chris does the job in 3 hours.}
\;\;\text{In one hour, he does }\frac{1}{3}\text{ of the job.}
\;\;\;\;\text{In }x\text{ hours, he does }\frac{x}{3}\text{ of the job.}$

$\text{Cheryl does the job in 2 hours.}
\;\;\text{In one hour, she does }\frac{1}{2}\text{ of the job.}
\;\;\;\;\text{In }x\text{ hours, she does }\frac{x}{2}\text{ of the job.}$

$\text{Tyler will "undo" the job in 4 hours.}
\;\;\text{In one hour, he will undo }\frac{1}{4}\text{ of the job.}
\;\;\;\;\text{In }x\text{ hours, he will undo }\frac{x}{4}\text{ of the job.}$

$\text{Working together for }x\text{ hours,}
\;\;\text{they will do }\,\frac{x}{3}\,+\,\frac{x}{2}\,-\,\frac{x}{4}\text{ of the job.}$

$\text{But in }x\text{ hours, we expect them to }complete\text{ the job (1 job).}$

$\text{Hence, we have: }\:\frac{x}{3}\,+\,\frac{x}{2}\,-\,\frac{x}{4} \:=\:1$

$\text{Multiply by 12: }\:4x\,+\,6x\,-\,3x \:=\:12 \;\;\;\Rightarrow\;\;\; 7x \:=\:12 \;\;\;\Rightarrow\;\;\;x \:=\:\frac{12}{7}$

$\text{Therefore, the house will be clean in }1\frac{5}{7}$$\text{ hours.}$

 July 13th, 2013, 10:35 AM #4 Newbie     Joined: Jun 2012 Posts: 25 Thanks: 0 Re: Rational Expression word problem. Please Help Thanks alot soroban. Word problems are the hardest in algebra. MarkFL, you could have put in more explanation.
July 13th, 2013, 11:00 AM   #5
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Quote:
 Originally Posted by AKofTroy ...MarkFL, you could have put in more explanation.
What more needed to be explained? Together, the 3 people's efforts results in 7 houses being cleaned in 12 hours...that's 12/7 hr/house.

 July 13th, 2013, 01:45 PM #6 Newbie     Joined: Jun 2012 Posts: 25 Thanks: 0 Re: Rational Expression word problem. Please Help I hadn't understood why you used LCM and where 4+6-3 came from.
July 13th, 2013, 04:19 PM   #7
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Quote:
 Originally Posted by AKofTroy I hadn't understood why you used LCM and where 4+6-3 came from.
My apologies...I used LCM(2,3,4)= 12 since it is the smallest number into which 2,3, and 4 can all go into evenly. This alleviates the need to use fractions. The way [color=#00BF00]soroban[/color] worked the problem is the way I was taught in school, but another member here demonstrated the technique I showed above and I found it much simpler to use.

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