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

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      Log On


   

Question 1022114: 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 3 solutions by josgarithmetic, ikleyn, Edwin McCravy:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
* ADDED EASIER SOLUTION BELOW - SEE ASTERISK (*).


Rates of more than one eater at the same time are additive.

j, time for Jenny
p, time for Penny
y, time for Lenny
To eat the one gallon on each's own




Notice that the numerators in all terms are 1, and only a variable is used in every denominator on the left hand members.
An alternative is to not to focus on the times as variables, but to focus on the RATES directly as variables, although each rate will still be a fraction.

You might be more comfortable reassigning new variables and forming
system%28J%2BP=1%2F2%2CP%2BY=1%2F%281%262%2F3%29%2CJ%2BP%2BY=1%29
Now you have a system of LINEAR equations instead of RATIONAL equations. Your goal is to solve for J or 1%2FJ.


--
I used matrix operations, not shown here, and found system%28P=1%2F10%2CY=1%2F2%29, so you can find J using J%2B1%2F10%2B1%2F2=1.
Tell me if you really need to see the matrix row operation steps.
Jenny takes highlight%282%261%2F2%29 hours to eat one gallon of ice cream by herself.



*--------EVEN EASIER FOR THIS EXERCISE---------

Notice that two of the equations have an expression, 1%2Fj%2B1%2Fp.
First a small adjustment to be made to the system,
system%281%2Fj%2B1%2Fp=1%2F2%2C1%2Fp%2B1%2Fy=3%2F5%2C1%2Fj%2B1%2Fp%2B1%2Fy=1%29

There is a very easy elimination you can perform. Working with first and third equations to do this...
1%2Fj%2B1%2Fp%2B1%2Fy-%281%2Fj%2B1%2Fp%29=1-1%2F2
1%2Fj%2B1%2Fp%2B1%2Fy-1%2Fj-1%2Fp=1%2F2
highlight%281%2Fy=1%2F2%29------giving one of your answer parts, meaning highlight%28y=2%29, two hours for Lenny to eat one gallon on his own.

Now look at the second equation of the system. You already found value for y.
1%2Fp%2B1%2Fy=3%2F5
1%2Fp=3%2F5-1%2Fy
1%2Fp=3%2F5-1%2F2
1%2Fp=6%2F10-5%2F10
1%2Fp=1%2F10
highlight%28p=10%29, meaning Penny needs 10 hours on her own to eat one gallon of ice cream.

Now return to the first equation of the system, only with j for Jenny being unknown.
1%2Fj%2B1%2Fp=1%2F2
1%2Fj=1%2F2-1%2Fp
1%2Fj=1%2F2-1%2F10
1%2Fj=%285-1%29%2F10
1%2Fj=4%2F10
highlight%28j=5%2F2%29-----Meaning Jenny needs 2.5 hours to eat one gallon ice cream.

Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
The previous tutor just pointed you an approach and a way to write the equations.
I will show you how to solve them elementary.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~` 

Rates of more than one eater at the same time are additive.

j, time for Jenny
p, time for Penny
y, time for Lenny
To eat the one gallon on each's own



Notice that the numerators in all terms are 1, and only a variable is used in every denominator on the left hand members.
An alternative is to not to focus on the times as variables, but to focus on the RATES directly as variables, 
although each rate will still be a fraction.

You might be more comfortable reassigning new variables J = 1%2Fj, P = 1%2Fp and Y = 1%2Fy and forming

J + P     = 1%2F2,     (1)    
P +     Y = 3%2F5%29,     (2)          ( <----- 3/5 = 1%2F%281%262%2F3%29 )
J + P + Y = 1%29.     (3)

Now you have a system of LINEAR equations instead of RATIONAL equations. 

Add equations (1) and (2). You will get

J + 2P + Y = 1%2F2+%2B+3%2F5 = 5%2F10+%2B+6%2F10 = 11%2F10.   (4)

Now distract equation (3) from (4). You will get

P = 11%2F10-1 = 1%2F10.   (You just have the rate for Lenny!)

Next, from (1) you get J = 1%2F2 - P = 1%2F2-1%2F10 = 4%2F10 = 2%2F5.

This is Jenny's rate.

Hence, it will take 1%2FJ = 1%2F%28%282%2F5%29%29 = 5%2F2 = 21%2F2 hours for Jenny to eat a whole gallon of ice cream by herself.

The problem is solved.


Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Here's the easiest way to solve the problem.  No algebra needed.
Just basic math.  Not only that but also too much information is 
given.  We don't even need the first sentence at all.

The times given in the other sentences are 1 2/3 hours, and 
1 hour.

1 2/3 hours is 1 hour 40 minutes or 100 minutes.

So the times given are 100 minutes and 60 minutes.

The easiest common multiple of those is 600 minutes or 10 hours.
So let's take everything up to 10 hours.

Penny and Lenny can eat a whole gallon of ice cream together
in 1 (2/3) hours.
That's 100 minutes, so in 600 minutes (or 10 hours) Penney 
and Lenny can eat 6 gallons.

All three together can eat a whole gallon of ice cream
together in 1 hour. 
So in 10 hours they can all three eat 10 gallons.

Since in 10 hours Penney and Lenny would eat 6 of those 10 
gallons, Jenny would eat the other 4.  Jenny can eat 4 
gallons in 10 hours, so Jenny could eat 1 gallon in 1/4th 
of 10 hours or 2 1/2 hours.  

See?  We didn't need the first sentence, or any algebra, 
at all. :)

Edwin