SOLUTION: Jenny and Penny can eat a whole gallon of ice cream together in 2 hours. Penny and Lenny can eat a whole gallon of ice cream together in 1 2/3 hours. All three together can eat a w

Algebra ->  Coordinate Systems and Linear Equations  -> Linear Equations and Systems Word Problems -> SOLUTION: Jenny and Penny can eat a whole gallon of ice cream together in 2 hours. Penny and Lenny can eat a whole gallon of ice cream together in 1 2/3 hours. All three together can eat a w      Log On


   

Question 1068277: Jenny and Penny can eat a whole gallon of ice cream together in 2 hours. Penny and Lenny can eat a whole gallon of ice cream together in 1 2/3 hours. All three together can eat a whole gallon of ice cream together in 1 hour. How many hours would it take Jenny to eat a whole gallon of ice cream all by herself?
My Question: I know there is already a post on this.Unfortunately, I did not understand where their equations were derived from. It would be nice if you could specify there and be detailed. Thanks in advance!
Found 2 solutions by Fombitz, ikleyn:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
Rate x Time = Output
Let say that eating a gallon of ice cream requires an output (or input) of 100 units (it's an arbitrary number, it could have been 1 or 27 or any other non-zero number).
J-Jenny's rate
P-Penny's rate
L-Lenny's rate
So,
%28J%2BP%29%282%29=100
1.J%2BP=50
and
%28P%2BL%29%281%2B2%2F3%29=100
%28P%2BL%29%285%2F3%29=100
2.P%2BL=60
and
%28J%2BP%2BL%29%281%29=100
3.J%2BP%2BL=100
Subtracting 1 from 3,
J%2BP%2BL-J-P=100-50
L=50
So,
P%2B50=60
P=10
and
J%2B10%2B50=100
J%2B60=100
J=40

Answer by ikleyn(52864) About Me  (Show Source):
You can put this solution on YOUR website!
.
This is the link to the solution you could not find in the archive:
https://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1022112.html


Below for your convenience I copied and pasted that solution. 


Let j = time for Jenny to eat a whole gallon of ice cream all by herself
    p = time for Penny - - - - " - - - 
    y = time for Lenny.

Then from the condition you have these three equations (standard equations for rates)

1%2Fj + 1%2Fp      = 1%2F2,    (1)

     1%2Fp + 1%2Fy = 3%2F5,    (2)     ( <--- 3%2F5 = 1 : 12%2F3 )

1%2Fj + 1%2Fp + 1%2Fy = 1.    (3)


Now distract (2) from (3) (both sides). You will get 

1%2Fj = 1+-+3%2F5 = 2%2F5.

Hence, j = 5%2F2 = 21%2F2 hours.

Answer. It will take 21%2F2 hours for Jenny to eat a whole gallon of ice cream all by herself.

Notice:  The equation  (1)  was not used in the solution at all.
              The corresponding part of the condition is not necessary and is excessive.


There is a wide variety of solved joint-work problems with detailed explanations in this site. See the lessons
    - Rate of work problems
    - Using Fractions to solve word problems on joint work
    - Solving more complicated word problems on joint work
    - Using quadratic equations to solve word problems on joint work
    - Solving rate of work problem by reducing to a system of linear equations
    - Selected joint-work word problems from the archive
    - Joint-work problems for 3 participants
    - OVERVIEW of lessons on rate-of-work problems

Read them and get be trained in solving joint-work problems.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lessons are the part of this textbook under the topic "Rate of work and joint work problems" of the section "Word problems".