# Word Problem. Am I wrong, or is my Algebra book?

In my algebra textbook the following problem is given:

Ken and Bettina Wikendt live in Minneapolis, Minn. Following a severe snowstorm, Ken and Bettina must Clear the driveway and sidewalk. Ken can clear the snow by himself in 4 hours. Bettina can clear the snow by herself in 6 hours. After Bettina has been working for 3 hours, Ken is able to join her. How much longer will it take them working together to move the rest of the snow.

The book gives the answer as:

t/6 + (t+3)/4 = 1,
thus t = 3/5 of an hour to complete the rest of the job.

When I worked the problem I set it up as follows:

(t+3)/6 + t/4 = 1.
Which works out as t = 1 and 1/5 hours to complete the rest of the job.

At first I figured I just worked the problem wrong, but after reviewing it, I'm not sure if I'm wrong or the book is wrong.

I used "t" as the time they have been working together. So "t+3" equals the total amount of time Bettina has been working.

So using the formula "amount of work = rate * time" with Bettina's rate of 1/6 the total job per hour, and a working time of "t+3" that means the amount of work she did was "(t+3)* 1/6" or simply "(t+3)/6"
For Ken's the amount of work equals "t * 1/4" or simply "t/4"

The total job, "1" should be Ken's work plus Bettina's work, which is "(t+3)/6 + (t/4) = 1"
Working this I get 1 and 1/5 hours for "t"

I even tried working the problem from a diffent angle.
Since after 3 hours Bettina will have completed half the job ("3 * 1/6 = 3/6")
I can figure out the time left by simply firguring out how much time it takes Ken and Bettina to finish the other half of the job. Or simply: "t/6 + t/4 = 1/2". Again, I get 1 and 1/5 hours, NOT 3/5 of an hour.

Is the book wrong or am I? Thanks,
Ben

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dextercioby
Homework Helper
YOU ARE CORRECT!.

Daniel.

P.S.Both lines of logic are correct.

HallsofIvy
Homework Helper
The book (or you in copying the problem) has mixed up Ken and Bettina.
The equation t/6+ (t+3)/4= 1 would be correct if it were Ken working for the extra 3 hours. If we use the equation (t+3)/6+ t/4= 1. Multiplying through by 12, 2(t+3)+ 3t= 5t+ 6= 12 so 5t= 6 and t= 6/5.

Here's how I would analyse this problem:

It would take Ken 4 hours to clear the sidewalk by himself so he works at the rate of 1/4 "sidewalk per hour".

It would take Bettina 6 hours to clear the sidewalk by herself so she works at the rate of 1/6 "sidewalk per hour". She works by herself for 3 hours so she will have cleared half the sidewalk.

Now Ken and Bettina work together to clear the rest of the sidewalk. Their rates add so they work at 1/4+ 1/6= 3/12+ 2/12= 5/12 "sidewalk per hour". To clear 1/2 a sidewalk at the rate of 5/12 sidewalk per hour requires (1/2)/(5/12)= (1/2)(12/5)=
6/5= 1 and 1/5 hour: 1 hour and 12 minutes.

Hmmm, "two great minds"...

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Thanks!
The problem as shown above is word for word from the book, as is the answer they gave.

I knew something was screwy. I just didn't know if it was me or the book. :tongue2:

Ben

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