MATH HELP?

..

number of jobs = 1

Time:
since ‘x’ is given as a variable in the choices, we should reserve it for Kate and use ‘y’ for Jean as follows:
Kate: => x
Jean: => y

but y = x - 5

Combined rate:
1/x+ 1/y
= (x + y)/(xy)

Number of jobs done:
number of days X combined rate = number of jobs
6 [(x + y)/(xy)] = 1
6x + 6y = xy
6x + 6(x - 5) = x(x - 5)
6x + 6x - 30 = x² - 5x
12x - 30 = x² - 5x
x² - 17x + 30 = 0
———————- Ans: A

Let x days be the time taken for Jean to paint the house by herself.
Then, time taken for Kate to paint the house by herself = (x - 5) days

Fraction of the work when Kate and Jean paint together for 1 day:
[1 / x] + [1 / (x - 5)] = 1/6
[(x - 5) / x(x - 5)] + [x / x(x - 5)] = 1/6
[(x - 5) + x] / x(x - 5) = 1/6
(2x - 5) / (x² - 5x) = 1/6
1 (x² - 5x) = 6 (2x - 5)
x² - 5x = 12x - 30
x² - 17x + 30 = 0

...... The answer is: A. x² - 17x + 30 = 0


[x days are taken for Jean to paint the house by herself.]
[(x - 5) days are taken for Kate to paint the house by herself.]

　
x = number of days it would take Kate to paint the house by herself
x+5 = number of days it would take Jean to paint the house by herself

Together, they paint the house in 6 days:
1/x + 1/(x+5) = 1/6

Multiply both sides by 6x(x+5)
6(x+5) + 6x = x(x+5)
12x + 30 = x^2 + 5x
x^2 − 7x − 30 = 0

None of the choices listed is correct.

——————————————————————————————

However if we let:
x = number of days it would take Jean to paint the house by herself
x−5 = number of days it would take Kate to paint the house by herself

then we get:
1/x + 1/(x−5) = 1/6
6(x−5) + 6x = x(x−5)
12x − 30 = x^2 − 5x
x^2 − 17x + 30 = 0

This matches choice A.

However, I am at a loss why they would ask for an equation that represents the time it takes KATE to paint the house by herself, but then use variable x to represent the time it takes JEAN to paint the house by herself. MAKES NO SENSE!

——————————————————————————————

Using original equation (with x representing Kate's time), we get:
x^2 − 7x − 30 = 0
(x − 10) (x + 3) = 0
x = 10 or x = −3

Negative values make no sense, so
x = 10
x+5 = 15

Kate: 10 days by herself
Jean: 15 days by herself

------------------------------

Using second equation (with x representing Jean's time), we get:
x^2 − 17x + 30 = 0
(x − 2) (x − 15) = 0
x = 2 or x = 15

But x = 2 makes no sense, since Jean takes 5 days longer than Kate to paint the house by herself (so x must be > 5)
x = 15
x−5 = 10

Jean: 15 days by herself
Kate: 10 days by herself

------------------------------

Either way we end up with the same answers.
But we can clearly see that in the second case, when solving for x, we were solving for JEAN's time, and not for Kate's time as was asked.

Let x be the number of hours it would take for Kate to paint the house herself.
Her rate is 1/x of the house per hour.

Let x+5 be the number of hours it would take for Jean to paint the house herself.
Her rate is 1/(x+5) of the house per hour.

Together, their combined time would be 6 hours for a rate of 1/6 of the house per hour.

So the equation would be:
1/x + 1/(x+5) = 1/6

But we can multiply everything by the common denominator of 6x(x+5), to get rid of fractions:
6(x+5) + 6x = x(x+5)

Expand that:
6x + 30 + 6x = x² + 5x
12x + 30 = x² + 5x
x² + 5x - 12x - 30 = 0
x² - 7x - 30 = 0

That can be factored:
(x - 10)(x + 3) = 0

That leaves two possible answers:
x = 10
or
x = -3

But we can't have a negative time.

Answer:
x = 10

Summary:
It takes Kate 10 hours and it takes Jean 15 hours.

None of those equations match the correct answer.

Answers:
x² - 7x - 30 = 0
or
(x - 10)(x + 3) = 0