Question 1 Answer:
Aunt 1 and Grandma 1 would fill gift bags.
Mom and Aunt 2 would make centerpieces.
You and Grandma 2 would blow up balloons.
Since you are pairing up to complete the tasks, these pairs each have the shortest times in their respective categories and therefore are the most logical pairing to complete tasks.
Question 2 Answer:
We use algebra and our previous pairings to determine the length of each task.
Gifts Bags --> 6/7 hours
x = time together
[tex]\frac{1}{x}[/tex] = rate of completion
Aunt 1 = [tex]\frac{1}{1.5}[/tex] Grandma 1 = [tex]\frac{1}{2}[/tex]
[tex]\frac{2}{3} +\frac{1}{2} = \frac{1}{x}[/tex]
[tex]\frac{7}{6} = \frac{1}{x}[/tex]
[tex]x = \frac{6}{7}[/tex]
Centerpieces --> 7/4 hours
x = time together [tex]\frac{1}{x}[/tex] = rate of completion
Mom = [tex]\frac{1}{3.5}[/tex] Aunt 2 = [tex]\frac{1}{3.5}[/tex]
[tex]\frac{2}{7} + \frac{2}{7} = \frac{1}{x}[/tex]
[tex]\frac{4}{7} = \frac{1}{x}[/tex]
[tex]x = \frac{7}{4}[/tex]
Balloons --> 15/16 hours
x = time together [tex]\frac{1}{x}[/tex] = rate of completion
You = [tex]\frac{1}{1.5}[/tex] Grandma 2 = [tex]\frac{1}{2.5}[/tex]
[tex]\frac{2}{3} + \frac{2}{5} = \frac{1}{x}[/tex]
[tex]\frac{16}{15} = \frac{1}{x}[/tex]
[tex]x = \frac{15}{16}[/tex]
Shortest amount of time to complete all tasks is:
[tex]\frac{6}{7} + \frac{7}{4} + \frac{15}{16} = \frac{397}{112}[/tex] ≈ 3.54 hours
Converting hours to hours and minutes --> 3 hours 32 minutes
Therefore they must arrive by 5:28pm to complete the tasks in time to leave at 9:00pm.
