Please give answer and show your work, mostly need help with question 2

Question

Use the information below to help you know how long it will take to get everything done.

Person	Time to fill 100 gift bags	Time to make 20 centerpieces	Time to blow up 1000 balloons for balloon archway
You	2.5 hours	5 hours	1.5 hours
Mom	3 hours	3.5 hours	5 hours
Aunt 1	1.5 hours	4 hours	3 hours
Aunt 2	3.5 hours	3.5 hours	5 hours
Grandma 1	2 hours	6 hours	7 hours
Grandma 2	2.5 hours	4.5 hours	2.5 hours

1) You plan to have 2 people work on each task. How would you pair them up to complete each task in the shortest amount of time? Explain using rational expressions.

2) You plan to have 2 people work on each task. How would you pair them up to complete each task in the shortest amount of time? Explain using rational expressions.

Answer 1

The work rate is the amount done divided by the time required. The people with the fastest rates for each task will result in the shortest overall time. The fastest rate means the smallest time for each task.

For the gift bags:

Aunt 1 and Grandma 1 have the shortest times:

Equation:

(# bags done by Aunt 1) + (# bags done by Grandma 1) = 100 gift bags done

# bags done by Aunt 1 = (Aunt 1 rate)(time doing gift bags) and Aunt 1 rate = 100/1.5 = 200/3 (bags/hr)

# bags done by Grandma 1 = (Grandma 1 rate)(time doing gift bags) and Grandma 1 rate = 100/2 = 50 bags/hr

So the Equation above becomes:

(200/3)t + 50t = 100 where "t" is the time doing gift bags. Solving:

(250/3)t + (150/3)t = 100

(350/3)t = 100

350t = 300

t = 300/350 = 6/7 hrs (approx 0.857 hrs)

For centerpieces:

The people with the shortest times (and hence the highest rates) are Mom and Aunt 2 and both have a rate of (20 centerpieces)/3.5 hrs = 40/7 centerpieces/hr

Equation:

(40/7)t + (40/7)t = 20 where "t" is the time doing centerpieces. Solving:

(80/7)t = 20

80t = 140

t = 140/80 = 1.75 hrs

For balloons:

The people with the shortest times (and hence the highest rates) are You and Grandma 2. Your rate is 1000/1.5 = 2000/3 balloons/hr. Grandma 2 has a rate of 1000/2.5 = 400 balloons/hr

Equation:

(2000/3)t + 400t = 1000 where "t" is the time doing balloons. Solving:

(2000/3)t + (1200/3)t = 1000

(3200/3)t = 1000

3200t = 3000

t = 3000/3200 = 15/16 hrs (approx 0.938 hrs)

Total shortest time then is approx 3.54 hrs

Answer 2

If You and Grandma2 work together to blow up the 1000 balloons, the calculation is:

1/1.5 + 1/2.5 = 1/x, where x is the combined time to do the job. CD is 3.75

2.5/3.75 + 1.5/3.75 = 4.0/3.75 = 1/x Cross-multiply.

4x = 3.75

x = 0.9375 hrs for the two-person team to blow up 1000 balloons (whew!)

Answer 3

You & Grandma2 on balloons, because your two times sum to 4 hours, minimal time, faster than any other two persons alone each. 1/1.5 + 1/2.5 = 1/T, 2/3 +2/5 =1/T, 16/15 =1/T, T=15/16 = T=Total time together which < than for any other two people.

Mom & Aunt 2 on centerpieces because those two individual times sum to 7 hours, minimal time compared to any other two. Time together = 3.5/2 = 1.75 < than for any other two together

Aunt1 and Grandma1 on gift bags, because those two individual times are 3.5 hours, minimal time compared to any other two. Together their time T is 6/7th of an hour. 1/1.5 + 1/2= 1/T, 2/3+1/2 =1/T, 1/T = 7/6, T =6/7

minimal total time = 14.5 hours < time for any other combination of 2 individuals.