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.

Question

gabylovesyou · Answer

@johnweldon1993

mathstudent55 · Answer

$\dfrac{1}{time~person ~1 ~does ~job} + \dfrac{1}{time ~person ~2~does ~job} = \dfrac{1}{time ~both ~people ~do ~job ~together}$

gabylovesyou · Answer

$$\frac{ 1 }{ x } + \frac{ 1 }{ 8x } = \frac{ 1 }{ 3 }$$

gabylovesyou · Answer

?

mathstudent55 · Answer

This is how you think of these combined work problems.
Person 1 does the job in x hours.
Person 2 does the job in x - 8 hours.
Together they do the job in 3 hours.

gabylovesyou · Answer

ohhhhh so 1/x + 1/x-8 = 1/3

mathstudent55 · Answer

In 1 hour, person 1 does 1/x of the job.

In 1 hour, person 2 does $\dfrac{1}{x - 8} $ of the job.

In 1 hour, together, they do 1/3 of the job.

anonymous · Answer

"Harry can rake the leaves in the yard 8 hours faster than his little brother Jimmy can. " 
Suppose Jimmy can rake the leaves in j hours, then he rakes 1/j of the leaves each hour. 

Harry can rake the leaves in j-8 hours, so he rakes 1/(j-8) of the leaves each hour. 

Working together, they rake 1/j + 1/(j-8) of the leaves each hour. 

"If they work together, they can complete the job in 3 hours." 
⅓ of the job is completed per hour. 
Therefore, 1/j + 1/(j-8) = ⅓ 

Solve for j: 
1/j + 1/(j-8) = ⅓ 
3(j-8) + 3j = j(j-8) 
3j - 24 + 3j = j² - 8j 
j² -14j + 24 = 0 
j = 12

mathstudent55 · Answer

When you add the work of the two people in 1 hour, it euqlas the work together in 1 hour.

$\dfrac{1}{x}+ \dfrac{1}{x - 8} = \dfrac{1}{3} $

mathstudent55 · Answer

^equals

mathstudent55 · Answer

BTW, @Gabylovesyou , do you understand why Harry's time is x - 8, and not 8x?
Harry is faster. Jimmy is slower.
Jimmy takes x hours to do the job.
Harry is 8 hours faster, so Harry takes 8 hours less.
8 less than x is x- 8
In this case, x is the time Jimmy takes.
x - 8 is the time Harry takes.

gabylovesyou · Answer

oh ok... and how do i solve it ?

anonymous · Answer

i sent u the answer gaby

gabylovesyou · Answer

i still need a clearer explanation on how to solve it.. ;/ @mathstudent55 thank you tho @Cj_2COOL

anonymous · Answer

oh okay

anonymous · Answer

no problem

mathstudent55 · Answer

Jimmy does the job in x hours.
In 1 hour, Jimmy does 1/x of the job.

Harry works 8 hours faster, so Harry only takes x - 8 hours to do the job.
In 1 hour, Harry does 1(x - 8) of the job.

When working together, they take 3 hours to do the job, so working together, they do 1/3 of the job in 1 hour.

When they work together, in 1 hour they do a combined
1/x + 1/(x - 8) of the job.

We know that when working together they do 1/3 of the job in 1 hour, so we combine these two expressions to get

$\dfrac{1}{x} + \dfrac{1}{x - 8} = \dfrac{1}{3} $

Now we need to solve the equation to find how long Jimmy takes to do the job.
Then Harry takes 8 hours less.

mathstudent55 · Answer

The reason Harry's time is x - 8 and not 8x is that we are told Harry is 8 hours faster than Jimmy. 8 hours faster means he takes 8 hours less. That means if the slower brother takes x hours, the faster brother takes 8 less hours, and 8 less than x is x - 8.

If one brother were 8 times faster than the other brother, then if the faster brother takes x hours, the slower brother would tale 8x hours. 

There is a big difference between "8 hours faster" and 8 times faster".
We have 8 hours faster.

mathstudent55 · Answer

Now we can work on the equation, if you'd like.

gabylovesyou · Answer

yes :) @mathstudent55

mathstudent55 · Answer

$\dfrac{1}{x} + \dfrac{1}{x - 8} = \dfrac{1}{3}$

The LCD of the 3 fractions is 3x(x - 8).
We multiply both sides of the equation by the LCD to get rid of denominators.

$3x(x - 8)\dfrac{1}{x} + 3x(x - 8)\dfrac{1}{x - 8} = 3x(x - 8)\dfrac{1}{3}$

$3\cancel{x}(x - 8)\dfrac{1}{\cancel{x}~1} + 3x\cancel{(x - 8)}\dfrac{1}{\cancel{x - 8}~1} = \cancel{3}x(x - 8)\dfrac{1}{\cancel{3}~1}$

$3(x - 8) + 3x = x(x - 8)$

$3x - 24 + 3x = x^2 - 8x$

$x^2 - 14x + 24 = 0$

$(x - 2)(x - 12) = 0$

x = 2 or x = 12

In our case, x = 2 does not make sense for Jimmy's time because then Harry's time would be 2 - 8 = -6 hours, and he cannot do a job in negative time.

The answer x = 12 makes sense.
x is Jimmy's time.
Jimmy takes 12 hours to to the job.
Harry takes 12 - 8 = 4 hours.