Question 143802
It takes 2 hours and 24 minutes for Gail and Bob to paint a room together.
If painting alone, it would take Bob 2 hours more to paint the room than
it would take Gail. How long does it take Gail to paint the room?
:
Convert 2 hr 24 min to hrs; 2 + 24/60 = 2.4 hrs
:
Let x = time required for G to paint the room alone
:
It says it takes B two hrs longer to do the job, therefore:
(x+2) = B's time to paint it alone
:
Let the completed job = 1
:
Write an equation representing the fraction painted by each person:
;
{{{2.4/((x+2))}}} + {{{2.4/x}}} = 1
Multiply equation by x(x+2) to eliminate the denominators:
x(x+2)*{{{2.4/((x+2))}}} + x(x+2)*{{{2.4/x}}} = x(x+2)(1)
cancel the denominators and you have:
:
2.4x + 2.4(x+2) = x(x+2)
:
2.4x + 2.4x + 4.8 = x^2 + 2x
:
4.8x + 4.8 = x^2 + 2x
:
Arrange as a quadratic equation:
x^2 + 2x - 4.8x - 4.8 = 0
:
x^2 - 2.8x - 4.8 = 0
:
Use the quadratic formula to find the solution:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}}
In this equation: a=1; b=-2.8; c=-4.8
{{{x = (-(-2.8) +- sqrt(-2.8^2 - 4 * 1 *-4.8 ))/(2*1) }}}
{{{x = (2.8 +- sqrt(7.84 + 19.2))/(2) }}}
You can do the math, the positive solution should be x = 4