How long would it take to paint a fence? (rational polynomial equation)

Question

Q: Danny and Josh are going to help paint Josh’s neighbor’s fence. Josh painted the fence by himself two years ago. Four years ago, Danny painted the same fence and did it 9 hours faster than Josh. Together, Danny and Josh finished painting the fence in just 7 hours.

a. What would be the equation to represent this?

b. What would you get after solving?

c. Do you have to find the LCD?

d. Why is a least common denominator important for solving rational polynomial equations?

Answer 1

J=Josh's time

J-9=Danny's time

Josh paints 1/J of the fence in one hour

Danny paints 1/(J-9) of the fence in one hour

they take 7 hours together to paint the fence

7(1/J)+7(1/[J-9])=1 (job finished)

LCD=J(J-9)

multiply both sides by the LCD to get...

7(J-9)+7(J)=J(J-9)

7J-63+7J=J^2-9J

J^2-9J-14J+63=0

J^2-23J+63=0

using the quadratic formula....

[23±√(529-252)]/2

[23±√277]/2

(23±16.64)/2

(23+16.64)/2 (using -16.64 gives a negative time for Danny)

39.64/2

19.82 hours is Josh's time alone

10.82 hours is Danny's time alone

the difference is 9 hours

also 7/19.82 + 7/10.82=0.3531+0.6469=1 (job finished)

Answer 2

j represents the time Josh needs

j - 9 represents the time Danny needs

(7 / j) + (7 / (j - 9)) = 1

7(j - 9) + 7j = j(j - 9) multiply by LCD

7j - 63 + 7j = j² - 9j

14j - 63 = j² - 9j

0 = j² - 23j + 63

By Quadratic Formula

j = (23 + √277) / 2

j = 19.821

Answer 3

If Josh paints the whole fences in J hours then he will paint 1/J of the fences in one hour.

If Danny paints the whole fence in D hours then he will paint 1/D of the fences in one hour. So we have two equations with two unknowns:

J - D =9 since Josh takes more time to do the job.

1/D + 1/J = 1/7 since together they will finish in 7 hours so in one hour they will finish only 1/7 of the whole work.

Plug J = D + 9 into the second equation:

1/D + 1/(D+9) = 1/7 after getting the common denominator and cross multiply we will get:

(D+9+D)/(d(D+9)) = 1/7 or:

D^2 + 9D = 14D + 63 or D^2 - 5D - 63 = 0

Using the Quadratic formula and ignoring the negative answer we will get:

D = 10.82... and J = 9 + 10.82... = 19.82...

To check the answers you see that the difference is 9 hours and 1/10.82... + 1/19.82... = 0.1428755.. which is 1/7.