We let the time that Jane takes to finish the job be x, so for Mary it is x+3.
This makes the rates of Jane and Mary be 1/x and 1/(x+3) respectively.
1/[1/x + 1/(x+3)] = 2
1 = 2/x + 2/(x+3)
1 = (4x+6)/(x²+3x)
x²+3x = 4x+6
x²-x-6 = 0
Method 1: Guess and Check
We look for two number with a product of 6 and an absolute difference of 1
6 - 1 = 5 X
3 - 2 = 1 √
So: (x-3)(x+2)=0
Therefore x is either 3 or -2 but we cannot have a negative time so we only consider 3. Therefore Mary's time is 6 hours. It takes her 6 hours to do it alone.
Method 2: Quadratic Formula
x = 1 ± √(1+24) = 1 ± 5 = -2 or 3
2 2
Again, we cannot have a negative value. So Jane's time is 3 hours and Mary's is 6 hours. So the answer is 6 hours.
