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Mathematics - Algebra

(Please see the attached file)

I have solved the first part of the problem I need help on the second part. I also have a third question asking to find the error in the problem.

Part 1 ( I have solved this part) See questions I need help on in RED below

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.

Let the time (in hours) it takes my father-in-law to weed the garden be t. The rate at which he does the weeding is:

in gardens-per-hour.

It takes my son 4 hours longer to weed the garden working by himself so it takes him (t+4) hours to weed the garden. Therefore, his weeding rate is:

in gardens-per-hour.

However, when he works with his grandfather, he works twice as fast so his rate becomes:

in gardens-per-hour.

Their combined rate when they work together is:

in gardens-per-hour.

Adding these two rational expressions, which have an LCD of t(t+4):

To find the time it takes for them to weed the garden working together (T) in hours, we divide the number of gardens (1) by the rate in gardens-per-hour:

Part 2 (I need help on this)

In the first part of the question 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?

Question #2 (I need help on this one too)

The following appears to prove that any two numbers are equal. Obviously it is wrong; can you spot the flaw?

Let a and b be any two different numbers. Define x as the difference between them:
x = b - a

Multiply both sides of the equation by (b-a):
x(b - a) = (b - a)(b - a)

bx - ax = b2 - 2ab + a2

Add -bx+ab-a2 to both sides and simplify:
bx - ax - bx + ab - a2 = b2 - 2ab + a2 - bx + ab - a2

bx - bx - ax + ab - a2 = -bx + b2 - 2ab + ab + a2 - a2

-ax + ab - a2 = -bx + b2 - ab

Both sides of this equation have a common factor:
a(-x + b - a) = b(-x + b - a)

Divide both sides by (-x+b-a):
a(-x + b - a) = b(-x + b - a)
-x + b - a -x + b - a

a = b

What was my mistake?

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A Complete, Neat and Step-by-step Solution is provided in the attached file.

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