# Pan Balance Equations

• Mar 30th 2010, 06:52 PM
ParisTNK
Pan Balance Equations
Help!

There are two pan balances, the first pan balance has 2 full juice boxes on the left side and on the right side are 18 grapes along with a half-full juice box. The pan balance is equal.

The second pan balance has an empty juice box on the left side and 6 grapes on the right side. The pan balance is equal.

Now looking at the two pan balances I need to determine the following: A full juice box weighs as much as how many grapes. How much does the contents of one juice box, without the box, weigh in grapes?

I know that what you do to one side of the equation you need to do to the other side. So by removing the 1/2 juice box from the right side of the first balance and also removing a 1/2 juice box from the left side would make 1 1/2 juice boxes equal to 18 grapes.

I need help finding out the possible equations for this problem. I do not just want an answer rather helpful guidance for a better understanding.

Paristnk
• Mar 31st 2010, 04:43 AM
Problem was slightly trickier than I originally thought...

Right let's look at this in terms on equations...

Break it down into 3 parts, the juice box (empty), the juice itself, and the grapes.
Now I'm assuming when you say half full juice box you mean a full sized box that is half full, not a juice box that is only half the size.

You have...

2 juice boxes + 2 juice = 18 grapes + 1 juice box + 1/2 juice on the first pan

On the second pan you have 1 juice box = 6 grapes...

So we can substitute that into the first pan to simplify things.

So I'll let you do that and if you need anymore hints I'll reply again.
• Mar 31st 2010, 03:04 PM
ParisTNK
Great way of looking at this. I have to have this completed by tonight so I hope you are on the site. You have really simplified things for me. Thanks a million. I hope I am on the right track.

If 2 juice boxes + 2 juice = 18 grapes +1 juice box + 1/2 juice

and 1 juice box = 6 grapes. ( I love the way you simplified this, yes 1/2 does mean the amount of juice in the box)

So...

The first pan would look like this

2 juice boxes = 12 grapes

The first pan would be 12 grapes + 2 juice = 18 grapes + 6 grapes + 1/2 juice
or

12 grapes +2 juice = 24 grapes + 1/2 juice

or

1 1/2 juice = 12 grapes

Does that look right. Am I on the right track to coming up with an equation an answer?

Kory
• Mar 31st 2010, 03:19 PM
Yep that's what I got as well.
• Mar 31st 2010, 04:44 PM
ParisTNK
Thank you for the reassurance. However, I feel I am sounding very wordy in my assignment and I have an answer but I do not have an equation that will work. That is where I need some more assistance. Sorry for the inconvenience. I am going to copy my answer rationale to see if it makes sense to you?

Kory

It is as follows

I immediately looked at the first pan balance and removed the half juice box from the left side and removed a half of a juice box from the right side because I remembered learning what you do to one side you need to do to the other side. Then, the balance would show 1 ½ juice boxes are equivalent to 18 grapes, but then I realized this did not take into consideration the full box I removed from the left side but not from the right side. So I reviewed my thought process and decided to break it down into 3 parts, the juice box (empty), the juice itself, and the grapes. I converted the first pan into 2 juice boxes + 2 juice = 18 grapes + 1 juice box + 1/2 juice and on the second pan there would be 1 juice box = 6 grapes.

If 2 juice boxes + 2 juice = 18 grapes +1 juice box + 1/2 juice
and 1 juice box = 6 grapes then 2 juice boxes = 12 grapes.
I can substitute the amount of grapes into the spot of empty juice boxes.

Then the first pan would be 12 grapes + 2 juice = 18 grapes + 6 grapes + 1/2 juice
or 12 grapes +2 juice = 24 grapes + 1/2 juice

12 grapes + 2 juice = 24 grapes +1/2 juice
-12 grapes-1/2 juice= -12grapes –1/2 juice
1 1/2 juice = 12 grapes
12 grapes divided by 1 ½ juice = 8 grape
1 juice = 8 grapes
½ juice = 4 grapes
1 ½ juice = 12 grapes

Then I checked my work by using the first pan balance.
The left side of the pan balance was 2 full juice boxes which would ultimately equal 8 grapes + 8 grapes for the juice and 6 grapes + 6 grapes for the juice which equals a total of 28 grapes. The right side was 18 grapes and a half full juice box which would equal the original 18 grapes + 6 grapes for the box and 4 grapes for the ½ filled amount of juice. When added 18 + 6 + 4 equals 28. So I was able to determine:

1 full juice box = 8 grapes for the juice + 6 grapes for the box itself
1 full juice box = 14 grapes

The contents of one juice box = 8 grapes
The empty juice box = 6 grapes as the second pan balance showed.
• Mar 31st 2010, 04:52 PM
Quote:

Originally Posted by ParisTNK
-12 grapes-1/2 juice= -12grapes –1/2 juice

Not sure what that line does.

If you want an equation do...

b = juice box
j = juice
g = grapes

Then

2b + 2j = 18g + b + 0.5j (1)

and b = 6g. (2)

Substituting (2) into (1) gives,

12g + 2j = 18g + 6g + 0.5j

Hence moving all the j terms to one side, all the g terms to another and combining them gives

1.5j = 12g

Hence j = 8g, which means the juice from 1 box weighs as much as 8 grapes.
• Mar 31st 2010, 04:57 PM
ParisTNK
You are absolutely so wonderful I cannot express my thanks in enough words. Thank you.. Thank you...Thank you. I sure hope you are able to impact other lives with your wonderful mathematical brain for your career along with this great website! Thank you!

Paristnk
• Mar 31st 2010, 05:14 PM
No worries, use this button on the bottom right of a helpful post to say thanks to someone.

http://www.mathhelpforum.com/math-he...ost_thanks.gif
• Mar 31st 2010, 07:43 PM
ParisTNK
Can I pick your brain again??? If you are busy just let me know, I have ample time to research this further.
I can figure it out by using a graph but when that is taken away I am confused?

Do the graphs of y = x and y = x + 2 intersect with each other?

How could you answer this question without actually graphing each equation?

How would you describe the relationship between the symbolic representation for these lines and their graphs?
• Apr 1st 2010, 03:55 AM
Edit:
• Apr 1st 2010, 04:18 AM
To find where two graphs intersect each other, set them equal to each other and solve to find a value of $x$. When you find this value plug it back into any of your $y$ formulas to get the $y$ value.

I won't do your one but as an example...

Do the graphs $y = x$ and $y = 2x - 1$ intersect? Where do they intersect?

Well as I said, set them equal to each other and solve...

=> $x = 2x - 1$

Now we subtract x from each side to get...

$0 = x -1$

Then add one to each side...

$1 = x -1 + 1 = x$

Hence we are left with

$1=x$ or just write $x=1$ for convenience...
Since we found a solution the graphs intersect but let's continue...

This is the $x$ value that the graphs intersect.

Now just plug this back into any formula to get the $y$ value. I'll put it into both to show you they both give the same answer...

For the $y=x$ graph we get $y = 1$.

For the $y=2x - 1$ graph we get $y = 2 - 1 = 1$.

That was just to show you they both gave the same $y$ answer, you only have to put the $x$ value back into one of them to get your $y$ value.

So anyway, we have our coordinate now which is $x=1$, $y=1$.

Hence the graphs intersect at $(1,1)$. See attached pic.

Red line represents $y=x$, blue line $y=2x-1$. See that they intersect at $(1,1)$

http://img163.imageshack.us/img163/4984/12501540.jpg

Now for your graph... You will probably solve it and think, what? This can't be right, or you may not be able to solve it... Just remember that not every pair of graphs intersect...
• Apr 2nd 2010, 11:40 AM
ParisTNK
Thanks again.
I did go ahead and graph the two equations. I noticed they do not intersect and they are parallel. The one equation y=x goes through the middle of the graph. I am stuck here.

y=x y=x+2

I can substitute x+2 into the other equation for Y

x+2 =x+2 but then I get 0=0

or when I substitute y in for x I get x=x+2 and then I subtract x and then there is no x and a postive 2. I am confused or maybe on the right track...I am not sure.
• Apr 2nd 2010, 11:44 AM
Quote:

Originally Posted by ParisTNK
Thanks again.
I did go ahead and graph the two equations. I noticed they do not intersect and they are parallel. The one equation y=x goes through the middle of the graph. I am stuck here.

y=x y=x+2

I can substitute x+2 into the other equation for Y

x+2 =x+2 but then I get 0=0

or when I substitute y in for x I get x=x+2 and then I subtract x and then there is no x and a postive 2. I am confused or maybe on the right track...I am not sure.

What you want to do is solve

$x = x+2$.

Clearly this is not true for any x, hence the graphs do not cross.
• Apr 3rd 2010, 05:58 PM
ParisTNK
Ultimately,

If I show..

y=x and y=x+2

or

x=x+2
or

x-x=2

The equation just cancelled the x out? Therefore, only one line intercepts the y axis 2 and the other line does not cross the y axis because there is no y intercept.

So the two lines are parallel.

I guess I am not sure if my explanation really answers the questions. Nor if I am truly understanding the whole concept.

When I substitute a point into the equations to take it a step further. Such as (2,2) or (3,3) it doesn't work. Does it not work because there is not a point where the two lines meet therefore the lines are parallel?