Final answer:
By setting up an equation based on their combined rate of work, it can be determined that it takes Mary 9 hours to make the costume alone.
Step-by-step explanation:
To solve how long it will take for Mary to make the costume alone, we can use the concept of rates in work problems. If Mary can make a costume twice as fast as Sam, we can denote Sam's rate of making a costume as 1/x and Mary's rate as 1/(x/2) or 2/x, where x represents the time it takes Sam to make the costume alone.
Combined, their rate of working together is the sum of their individual rates.
Since they can complete the costume in 6 hours together, their combined rate is 1/6 costumes per hour. So, we set up the equation:
1/x + 2/x = 1/6
To find Sam's time x, we first obtain a common denominator for the equation:
(2 + 1)/x = 1/6
3/x = 1/6
Multiplying both sides by x and then by 6 gives us:
3 = x/6
x = 18
So, Sam can make a costume alone in 18 hours, making Mary's time to make the costume alone 9 hours.