anonymous one year ago Harry can rake the leaves in the yard 8 hours faster than his little brother Jimmy can. If they work together, they can complete the job in 3 hours. Using complete sentences, explain each step in figuring out how to determine the time it would take Jimmy to complete this job on his own

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1. anonymous

@Michele_Laino

2. anonymous

x(x-8)/(x+x-8) = 3 solve for x

3. Michele_Laino

the working rate of Jimmy is W/x, where W is the work to be done. The working rate of Harry is W(x-8) whereas the working rate when both Jimmy and Harry work together is: $\frac{W}{x} + \frac{W}{{x - 8}}$

4. anonymous

I understand:)

5. Michele_Laino

so, using your data we can write: $\Large \frac{W}{x} + \frac{W}{{x - 8}} = \frac{W}{3}$ or after a simplification: $\Large \frac{1}{x} + \frac{1}{{x - 8}} = \frac{1}{3}$ please solve that last equation for x

6. Michele_Laino

oops..the working rate of Harry is W/x-8

7. anonymous

So now do we do?

8. anonymous

so we now find a common denominator?

9. Michele_Laino

the common denominator, is: 3x(x-8)

10. Michele_Laino

and we got this equivalent equation: $\Large 3\left( {x - 8} \right) + 3x = x\left( {x - 8} \right)$ with the condition: x-8>0 since x-8 is a time which has to be positive

11. anonymous

ok, ok:)

12. Michele_Laino

I simplify that equation, nd I get this: $\Large \begin{gathered} 3x - 24 + 3x = {x^2} - 8x \hfill \\ {x^2} - 14x + 24 = 0 \hfill \\ \end{gathered}$

13. anonymous

uh huh! :)

14. Michele_Laino

the solution of that quadratic equation are: $\Large \begin{gathered} x = \frac{{14 + \sqrt {{{\left( { - 14} \right)}^2} - 4 \times 1 \times 24} }}{{2 \times 1}} = 12 \hfill \\ \hfill \\ x = \frac{{14 - \sqrt {{{\left( { - 14} \right)}^2} - 4 \times 1 \times 24} }}{{2 \times 1}} = 2 \hfill \\ \end{gathered}$ please check my values. Only x012 is acceptable

15. Michele_Laino

oops..only x=12 is acceptable

16. anonymous

so the first equation works?

17. anonymous

Thanks:) So Jimmy can work 12 hours alone?

18. Michele_Laino

yes!

19. Michele_Laino

yes! Jimmy works for x=12 hours only

20. anonymous

Thanks!

21. Michele_Laino

:)