My father-in-law used to have a garden in our backyard. We worried about him weeding the whole garden by himself so we usually sent one of our children out to help. Now, if my son worked on the garden by himself it would take him 4 hours longer to weed the garden than his grandfather took. However, if he worked with my father-in-law, he worked twice as fast. Develop a formula for how long it took for both of them to weed the whole garden working together. Simplify/reduce to lowest terms.

In DQ #2 you developed a formula for how long it took my father-in-law and my son to weed the garden together. If it took an hour and a quarter for them to weed the garden together, how long did it take my son working by himself?



Set up:
The problem contains the following facts:
Grandfather weeds at the rate of 1 job / x hours
Son weeds when with his grandfather at the rate of 2 jobs/ x+4 hours

The problem asks to develop a formula reduced to the lowest terms.
Let 1/x represent the rate of the grandfather.
Let 2/(x+4) represent the rate of the son when weeding with his grandfather.
Let ‘y’ represent the time they take to complete the weeding job.

If we find the sum of the products of rate and time of the son and grandfather and set it equal to 1 weeding job, we develop the equation.

y/x+2y/(x+4)=1

Solution -- simplify/reduce to lowest terms:

y/x+2y/(x+4)=1

x(x+4)(y/x)+(2y/(x+4))x(x+4)=x(x+4)1 Find the LCD x(x+4) and use the Distributive prop.

y(x+4)+2xy=x(x+4) Distributive property

(xy+4y)+2xy=x^2+4x Group by Associative property of Add.

3xy+4y=x^2+4x combine like terms


(3xy+4y)-(3xy+4y)=〖(x〗^2+4x)-(3xy+4y) Simplify equation by combining like
terms and set to 0.

x^2-4x-3xy+4y=0 factor out x in two middle terms

x^2-(3y+4)x+4y=0 Simplified/reduced to lowest terms


Final answer:
The following equation represents the original equation simplified/reduced to the lowest terms.

Let ‘1.25 hours’ represent the time they take to complete the weeding job.

We use the equation developed in the DQ #2 and substitute 1.25 hours for ‘y’ and solve for ‘x’.
x^2-(3y+4)x+4y=0 Substitute 1.25 hours for ‘y’
x^2-(3(1.25)+4)x+4(1.25)=0 Use distributive property to simplify
equation
x^2-(3.75x+4x)+6=0 simplify by combining like terms
x^2-7.75x+6=0 Simplify using quadratic equation
a=1 b=-7.75 c=6
x=(-(-7.75)±√(〖(-7.75)〗^2-4(1)(6)))/(2(1)) Substitute values in to quadratic equation
x=(7.75±√(60.0625-24))/2 Simplify right side of the equation.

Wow, wow wow, don't go too fast! You made a mistake!


Son weeds when with his grandfather at the rate of 2 jobs/ x+4 hours

The son weeds alone at the rate of 1 job/ x+4 hours

Together, they weed at the rate of 1 job/ [(x+4)/2]

EDIT: oh no... now I'm all confuzzled :(

Unknown is correct. The 2 should be in the denominator.

The father in laws rate is x.

The sons rate by himself is x+4

But, he works twice as fast when they work together:

\frac{t}{x}+\frac{t}{2(x+4)}=1

If t=5/4, then:

\frac{5}{4x}+\frac{5}{8(x+4)}=1

Solving for x, we get \approx 1.413 \;\ or -3.538

Of course, the negative solution is extraneous and we accept the positive one.

Therefore, the sons rate by himself is 5.413 hours or about 5 hours and 25 minutes.