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 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) About Me  (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 1%262%2F3 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
* system%28j%2Bp=1%2F2%2Cp%2By=1%2F1%262%2F3%2Cj%2Bp%2By=1%29

That complex fraction is same as 3/5, so revise the system as
system%28j%2Bp=1%2F2%2Cp%2By=3%2F5%2Cj%2Bp%2By=1%29

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.


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R2=R2-R1 and then R2=(-1)R2


R3=R3-R2


Swith R2 and R3



That tells you that system%28p=1%2F10%2Cy=1%2F2%29
Using the R1 as its equation,
j%2B1%2F10%2B1%2F2=1
highlight%28j=2%2F5%29, which was what the question asked. This is in GALLONS per HOUR for Jenny.
The reciprocal is 1%2Fj=5%2F2=highlight%282%261%2F2%29, HOURS per GALLON.

Some of the matrices are not rendering.


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* 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) About Me  (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%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.

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.