Question

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

Answer 1

@Michele_Laino

Answer 2

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

Answer 3

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:
\[\frac{W}{x} + \frac{W}{{x - 8}}\]

Answer 4

I understand:)

Answer 5

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

Answer 6

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

Answer 7

So now do we do?

Answer 8

so we now find a common denominator?

Answer 9

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

Answer 10

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

Answer 11

ok, ok:)

Answer 12

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

Answer 13

uh huh! :)

Answer 14

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

Answer 15

oops..only x=12 is acceptable

Answer 16

so the first equation works?

Answer 17

Thanks:) So Jimmy can work 12 hours alone?

Answer 18

yes!

Answer 19

yes! Jimmy works for x=12 hours only

Answer 20

Thanks!

Answer 21

:)