Please help me understand how to solve these problems. Thank you Mary and Tom gather and sterilize various items in the hospital. It takes Mary 2 hours for this activity and it takes Tom 3 hours. When working together, How long will it take them? John is a CPA and can research and evaluate a client's financial activity in 5 hours. If he works with Ellen, they can evaluate the activity in 2 hours. If Ellen worked separately, How long would the project take? A saline solution pump can fill a vessel in 5 hours. Another pump can empty the tank in 6 hours. If both pumps are operating simultaneously, how many hours are required to fill the vessel?

Summary:

 

Question 1.) When Mary and Tom works together they can finish the job in 1.2 hours

Question 2.) When Ellen is to do the task alone she can finish it after 3.3333 hours ( 10/3 hours )

Question 3.)  If both pumps are working , it will take 30 hours to fill the vessel.

Step-by-step explanation

Work Done:

 

Question 1.) Mary and Tom gather and sterilize various items in the hospital. It takes Mary 2 hours for this activity and it takes Tom 3 hours. When working together, How long will it take them?

 

• Step 1.) we know that the work done is equal to rate of work times the time

 

(rate)(time)=work

 

• Step 2.) now we can find the rate of Mary and Tom

 

Mary:(rate)(2)=1

Mary:2(rate)(2)?=21?

Mary:rate=21?

 

Tom:(rate)(3)=1

Tom:3(rate)(3)?=31?

Tom:rate=31?

 

• Step 3.) now if we combine their work done we can create the equation:

 

Mary+Tom=1

(rate)(time)+(rate)(time)=1

(21?)(time)+(31?)(time)=1

21?(time)+31?(time)=1

 

• Step 4.) then combine like terms

 

21?(time)+31?(time)=1

65?(time)=1

 

• Step 5.) then divide both sides by 5/6 to get the value of time

 

65?(time)=1

65?65?(time)?=65?1?

time=65?1?

time=(1)∗56?

time=56?

time=1.2−hours

 

Answer: When Mary and Tom works together they can finish the job in 1.2 hours

 

 

 

 

 

Question 2.) John is a CPA and can research and evaluate a client's financial activity in 5 hours. If he works with Ellen, they can evaluate the activity in 2 hours. If Ellen worked separately, How long would the project take?

 

 

• Step 1.) lets take the rate of work for John

 

John:(rate)(5)=1

John:5(rate)(5)?=51?

John:rate=51?

 

• Step 2.) and since it is given that the time they work together to finish the task is 2 hours , we can get the rate of Ellen

 

John+Ellen=1

(51?)(2)+(rate)(2)=1

52?+(rate)(2)=1

52?+(rate)(2)−52?=1−52?

(rate)(2)=53?

2(rate)(2)?=253??

rate=53?∗21?

rate=103?

 

• Step 3.) then we can find the time Ellen finish the task alone by solving her work done alone

 

Ellen:(rate)(time)=1

Ellen:(103?)(time)=1

Ellen:(103?)(time)(10)=1(10)

Ellen:(3)(time)=10

Ellen:3?(3)(time)?=310?

Ellen:time=310?:or:3.3333−hours

 

Answer: When Ellen is to do the task alone she can finish it after 3.3333 hours ( 10/3 hours )

 

 

 

 

 

Question 3.) A saline solution pump can fill a vessel in 5 hours. Another pump can empty the tank in 6 hours. If both pumps are operating simultaneously, how many hours are required to fill the vessel?

 

• Step 1.) lets get the work done

 

Fill:(rate)(time)=1

Fill:(rate)(5)=1

Fill:5(rate)(5)?=51?

Fill:rate=51?

 

Empty:(rate)(time)=1

Empty:(rate)(6)=1

Empty:6(rate)(6)?=61?

Empty:rate=61?

 

• Step 2.) now if we combine them we can get the time it takes to finish 1 tank , BUT take not since one of the pumps is draining the vessel this means it will have a minus sign

 

Fill−Empty=1

(rate)(time)−(rate)(time)=1

(51?)(time)−(61?)(time)=1

 

combine like terms

(51?)(time)−(61?)(time)=1

51?−61?=301?

 

(51?)(time)−(61?)(time)=1

(301?)(time)=1

 

multiply both sides by (30) to find the time

 

(301?)(time)=1

(301?)(time)(30)=1(30)

time=30−hours

 

Answer: If both pumps are working , it will take 30 hours to fill the vessel.

 

 

 

Summary:

 

Question 1.) When Mary and Tom works together they can finish the job in 1.2 hours

Question 2.) When Ellen is to do the task alone she can finish it after 3.3333 hours ( 10/3 hours )

Question 3.)  If both pumps are working , it will take 30 hours to fill the vessel.