Final answer:
Mary would have taken approximately 5.85 hours to complete the job alone. We found this by setting up an equation with Mary's time as x and Shelly's as x+3, then solved using the information given and applying the quadratic formula to determine Mary's solo work time.
Step-by-step explanation:
To find out how many hours it would have taken Mary to do the job alone, we need to set up an equation based on the information given. Let's assume Mary takes x hours to complete the job alone. Since Shelly takes three hours longer, she would take x + 3 hours to complete the job alone. We know they worked together for 3 hours, and then Mary finished the job in another hour.
When they work together, they combine their work rates. So, for every hour they work together, they complete 1/x + 1/(x + 3) of the job. In 3 hours, they would complete 3(1/x + 1/(x + 3)) of the job. Since Mary finished the remaining part of the job in 1 hour, which is 1/x of the job, we have:
3(1/x + 1/(x + 3)) + 1/x = 1
To solve for x, we find a common denominator and simplify the equation:
3(x + 3 + x)/(x(x + 3)) + 1/x = 1
We get:
3(2x + 3) = x(x + 3)
6x + 9 = x^2 + 3x
Moving all terms to one side gives us the quadratic equation:
x^2 - 3x - 9 = 0
Using the quadratic formula:
x = [-(-3) ± √((-3)^2 - 4(1)(-9))]/(2(1))
x = [3 ± √(9 + 36)]/2
x = [3 ± √ 45]/2
x = [3 ± 3√5]/2
We take the positive root because time cannot be negative, so:
x = (3 + 3√5)/2
Thus, it would have taken Mary (3 + 3√5)/2 hours to do the job alone, which is approximately 5.85 hours when calculated.