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

Question 1022112: 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?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39620)   (Show Source): You can put this solution on YOUR website!
This is identical to question 1022114. Using each rate directly as its own variable for each person to make a system of LINEAR equations may work best so you can avoid a system of rational equations.

You must mean, Penny and Lenny together need hours to eat 1 gallon.

Each rate as GALLONS EATEN per HOUR can be assigned like this (now using the lower case letters):

Jenny j Penny p Lenny y

Putting the description into a set of equations will become
*

That complex fraction is same as 3/5, so revise the system as

Creating a matrix and doing row reduction operations would be best, using this format for each row: a*j+b*p+c*y=k for each row to take the form a b c k.

Row rearrangement is allowed, maintaining the same set of represented equations.

Page repeatedly fails to load, and browser gives error messages...
---not yet finished---

R2=R2-R1 and then R2=(-1)R2
AMP Rendering Error of '(matrix(3,4,
1,1,1,1
0,0,1,1/2
0,1,1,3/5)
)'.

R3=R3-R2
AMP Rendering Error of '(matrix(3,4,
1,1,1,1
0,0,1,1/2
0,1,0,1/10)
)'.

Swith R2 and R3
AMP Rendering Error of '(matrix(3,4,
1,1,1,1
0,1,0,1/10
0,0,1,1/2)
)'.

That tells you that
Using the R1 as its equation,

, which was what the question asked. This is in GALLONS per HOUR for Jenny.
The reciprocal is , HOURS per GALLON.

Some of the matrices are not rendering.

---
* If you look again at the initial system, OR/AND the initial matrix, you will see that solving CAN begin with elimination of both j and p, all in one step! Look at the initial system carefully and you should be able to see justification for this step!
Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
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?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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) + = , (2) ( <--- = 1 : ) + + = . (3) Now distract (2) from (3) (both sides). You will get = = . Hence, j = = hours. Answer. It will take 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.

Here I simply summarized the discussion that we, the three tutors, had on this problem in
http://www.algebra.com/algebra/homework/coordinate/word/Linear_Equations_And_Systems_Word_Problems.faq.question.1022114.html.

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