# Who said Mathematics is a perfect science.



## Bob Wemm (Mar 30, 2013)

Hi Guys and Gals,

I don't know if anyone is interested but I can cut up a piece of paper 8inches square (64 sq inches) and refit the same pieces together exactly in a different shape and they measure 65 sq. inches.

When I went to school that was impossible, and I would love to hear from anyone who is interested or has an explanation of how this is possible.

Because I cannot explain it. ??????????????????

Bob.


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## theidlemind (Mar 30, 2013)

Got pictures?
I'm curious.


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## maxwell_smart007 (Mar 30, 2013)

Doesn't make sense to me - law of conservation of matter - if you have 8x8 square inches of material to start with, you're not adding to the amount of material by changing it's arrangement in the same plane...


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## juteck (Mar 31, 2013)

This is a fun one!   I've attached a graphic I found online to show the detail.  And yes, there is an explanation .....


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## lyonsacc (Mar 31, 2013)

The drawing is misleading.

Take a look at the larger 65 sq inch picture where section C and B touch each other - from the bottom left corner to the upper right.

Part B slopes down at a rate of 3 blocks over 8 blocks or 3/8 which is .375.
Part C slopes at a rate of 2 blocks over 5 blocks or 2/5 which is .4.

So, if the larger picture was drawn more accurately there would be a small sliver of "new" space between sections C and B as well as A and D. This new space, while small, would in all likelihood add up to 1 square.

That's my guess - assuming John U's picture is what Mr. Bob was writing about.

Dave (minored in math a very long time ago)


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## PenPal (Mar 31, 2013)

Hi Bob,

You cant get kittens from a cat named George. I can explain that.

All that glitters is not Gold.

Kind regards Peter.


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## Bob Wemm (Mar 31, 2013)

Here is another one to get you thinking:-

What goes up the Chimney down,
But wont go down the Chimney up??????????

Bob.


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## ericofpendom (Mar 31, 2013)

an umbrella


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## Bob Wemm (Mar 31, 2013)

lyonsacc said:


> The drawing is misleading.
> 
> Take a look at the larger 65 sq inch picture where section C and B touch each other - from the bottom left corner to the upper right.
> 
> ...


 

Dave, the picture is not really misleading, for the extra 1 sq. inch to be there between the two triangles there would have to be at least 1/16in gap all the way along. When I did my cut and fitup there was virtually no gap.

John, there has to be an explanation but lets see what develops here before you let the cat out of the bag.

Bob.


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## Bob Wemm (Mar 31, 2013)

ericofpendom said:


> an umbrella


 
Smarty pants.

Bob.


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## skiprat (Mar 31, 2013)

This reminds me of the old fun maths problem we did at school.
How many can remember this one? 

Three people eat in a restaurant when finished, the waiter brings the bill for $30.
Each person puts in $10.
The waiter takes the money to the cashier who finds that the bill should have only been $25, so gives the waiter $5 change.

Now the waiter isn't good at maths and can't figure out how to split $5 between three so puts $2 in his pocket and gives each of them $1 each.

So, each person paid $10 and got back $1, so $9 each right?
3 x 9 = 27 plus the $2 in waiter's pocket
27 + 2 = 29

Where is the last missing dollar?????


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## its_virgil (Mar 31, 2013)

This is an accounting error, not a math one.
The meal cost 25 so 25 divided by 3 is 8 1/3 dollars each.
The dollar returned by the waiter makes the meal 9 1/3 dollars each.
So 9 1/3 times 3 plus 2 is the whole 30 dollars.

There still seems to be a dollar missing somewhere. I'm not an accountant.
Do a good turn daily!
Don







skiprat said:


> This reminds me of the old fun maths problem we did at school.
> How many can remember this one?
> 
> Three people eat in a restaurant when finished, the waiter brings the bill for $30.
> ...


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## Akula (Mar 31, 2013)

"3 x 9 = 27 plus the $2 in waiter's pocket"

the $2 is part of the $27, so it would not be added again

$27 + the $3 = $30


I like our Washington math (or my Wife LOL).

I was going to spend $800 Million dollars buying gadgets but only spent $600 million.  I saved you $200 million


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## plantman (Mar 31, 2013)

Bob: Can you do the same thing with money? Rearange it so you have more.   Jim  S


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## ohiococonut (Mar 31, 2013)

skiprat said:


> This reminds me of the old fun maths problem we did at school.
> How many can remember this one?
> 
> Three people eat in a restaurant when finished, the waiter brings the bill for $30.
> ...


 
To confuse matters..............when my ex-sister-in-law was confronted with this question she was stuck on trying to figure out how the waiter was able to take two dollars from a five dollar bill:tongue:

If mathematics were perfect we'd have no Pi. Make mine apple please :biggrin:


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## Bob Wemm (Mar 31, 2013)

plantman said:


> Bob: Can you do the same thing with money? Rearange it so you have more. Jim S


 
Jim,

I'm working on it, but there seems to be a small hitch with the press.:biggrin:

Bob,


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## Bob Wemm (Mar 31, 2013)

Akula said:


> "3 x 9 = 27 plus the $2 in waiter's pocket"
> 
> the $2 is part of the $27, so it would not be added again
> 
> ...


 
It all depends on where you start from Hey.

The meal was $25  plus the $2 in the waiters pocket makes $27, and the three eaters had $1 each in their pocket so that makes $30.

But Where did the extra 1 Square inch come from???????

Bob.


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## triw51 (Mar 31, 2013)

Bob Wemm said:


> Here is another one to get you thinking:-
> 
> What goes up the Chimney down,
> But wont go down the Chimney up??????????
> ...


 smoke


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## its_virgil (Mar 31, 2013)

This is an accounting error, not a math one.
The meal cost 25 so 25 divided by 3 is 8 1/3 dollars each.
The dollar returned to each by the waiter makes the meal 9 1/3 dollars each.
So 9 1/3 times 3 plus 2 is the total 30 dollars.

There still seems to be a dollar missing somewhere. I'm not an accountant.
Do a good turn daily!
Don







skiprat said:


> This reminds me of the old fun maths problem we did at school.
> How many can remember this one?
> 
> Three people eat in a restaurant when finished, the waiter brings the bill for $30.
> ...


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## jcm71 (Mar 31, 2013)

Bob Wemm said:


> lyonsacc said:
> 
> 
> > The drawing is misleading.
> ...



John's explanation is correct.  With the two slopes of the cut pieces being different, it is impossible for the hypotenuse of the two rearranged larger "triangles" to be straight lines.  The rearranged drawing is an illusion.  It is not accurate.   Tape together each small triangle with its corresponding trapezoid.  Then carefully fit the opposite corners of each larger "triangle" to make the larger figure.  You'll see there is a gap going the length of the so called diagonal.   Each of the two larger "triangles" are in fact two identical,but irregular four sided figures.  The gap between the two taped pieces equals one square unit, as John suggested.


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## tbroye (Mar 31, 2013)

I will asked the Grandkids at dinner today the will figure it out.  All 5 are great at math a talent they didn't get from me.  I passed high school geometry by one point, that was enough to allow me to graduate.  I have gotten better at math as I have become older but they beat me every time.


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## its_virgil (Mar 31, 2013)

An old rancher died and left 17 horses to his three sons. The oldest was to inherit 1/2 of the horses. The middle son was to inherit 1/3 of the horses and the youngest to inherit 1/6 of them. As they were sitting on the corral fence debating how to divide the horses (without fractional horses) an old cowboy rode up and asked what they were doing. After telling the old cowboy of their problem the old cowboy put his horse in the corral. Now there were 18 horses in the corral. He told the oldest to take 9 horses but to leave his. He told the middle son to take 6 horses but leave his. And the youngest son took 2 horses and left the old cowboy's horse. The old cowboy got on his horse and rode off. What is the problem with this thinking or how did it work? 

I often gave this question as a bonus question on tests after teaching about fractions, proportions and percents.

Do a good turn daily!
Don


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## Mack C. (Mar 31, 2013)

its_virgil said:


> An old rancher died and left 17 horses to his three sons. The oldest was to inherit 1/2 of the horses. The middle son was to inherit 1/3 of the horses and the youngest to inherit 1/6 of them. As they were sitting on the corral fence debating how to divide the horses (without fractional horses) an old cowboy rode up and asked what they were doing. After telling the old cowboy of their problem the old cowboy put his horse in the corral. Now there were 18 horses in the corral. He told the oldest to take 9 horses but to leave his. He told the middle son to take 6 horses but leave his. And the youngest son took 2 horses and left the old cowboy's horse. The old cowboy got on his horse and rode off. What is the problem with this thinking or how did it work?
> 
> I often gave this question as a bonus question on tests after teaching about fractions, proportions and percents.
> 
> ...


The 1st son got 9/17 horses, the 2nd son got 6/17, and the youngest son got 2/17 horses. Not exactly as the old rancher that died had laid it out in his will, but close enough I guess!


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## Mr Vic (Mar 31, 2013)

its_virgil said:


> An old rancher died and left 17 horses to his three sons. The oldest was to inherit 1/2 of the horses. The middle son was to inherit 1/3 of the horses and the youngest to inherit 1/6 of them. As they were sitting on the corral fence debating how to divide the horses (without fractional horses) an old cowboy rode up and asked what they were doing. After telling the old cowboy of their problem the old cowboy put his horse in the corral. Now there were 18 horses in the corral. He told the oldest to take 9 horses but to leave his. He told the middle son to take 6 horses but leave his. And the youngest son took 2 horses and left the old cowboy's horse. The old cowboy got on his horse and rode off. What is the problem with this thinking or how did it work?
> 
> I often gave this question as a bonus question on tests after teaching about fractions, proportions and percents.
> 
> ...


 
1/2 = 3/6
1/3 = 2/6
1/6 = 1/6
      +___
        6/6


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## its_virgil (Mar 31, 2013)

Sorry, I made a mistake. The 1/6 should have been 1/9. Then 1/2 + 1/3 + 1/9 = 9/18 + 6/18 + 2/18 = 17/18 . The fractional parts must equal one (1).  I guess if the sons were happy then all is good. My typo ruined the question.
Don



Mr Vic said:


> its_virgil said:
> 
> 
> > An old rancher died and left 17 horses to his three sons. The oldest was to inherit 1/2 of the horses. The middle son was to inherit 1/3 of the horses and the youngest to inherit 1/6 of them. As they were sitting on the corral fence debating how to divide the horses (without fractional horses) an old cowboy rode up and asked what they were doing. After telling the old cowboy of their problem the old cowboy put his horse in the corral. Now there were 18 horses in the corral. He told the oldest to take 9 horses but to leave his. He told the middle son to take 6 horses but leave his. And the youngest son took 2 horses and left the old cowboy's horse. The old cowboy got on his horse and rode off. What is the problem with this thinking or how did it work?
> ...


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## PenMan1 (Mar 31, 2013)

pwhay said:


> Hi Bob,
> 
> You cant get kittens from a cat named George. I can explain that.
> 
> ...



Peter:
I respectfully disagree! You can, in fact, get kittens from a cat named George. I propose that possibly one half of all cats named George by three year old children, can, in fact, produce kittens DAMHIKT


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## turbowagon (Mar 31, 2013)

Infinite chocolate!


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## Bob Wemm (Mar 31, 2013)

Where can I buy one of those CHOCOLATE BLOCKS.

Bob.


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## Bob Wemm (Mar 31, 2013)

OK, it seems that you cannot turn 64 into 65.
Dave, please accept my apologies, the diagram is indeed  very misleading.
I have attached two images, firstly 8 inches square with the division into 4 pieces.
The second image is a graph of 2 in. squares and the appropriate enlarged pieces drawn in.
There is indeed a "Spare" piece between the two triangles. When I tried this years ago I obviously didn't cut my paper straight enough.

My excuse is I only went to year 10 at school, and I'm sticking to that.

I hate it when I'm wrong. That's the second time.

At least we have learned about never ending Chocolate blocks, Horse divisions and a few other things. Thanks for participating. Oh, I nearly forgot George the Cat.

Bob.:biggrin::biggrin:


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## CharlesJohnson (Mar 31, 2013)

On the restraint. Pure misdirection. I think every one had a lot of fun contributing.  We "do' Have a lot of fun!


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## switch62 (Apr 1, 2013)

lyonsacc is right, very small amount of space is introduced due to the difference in slopes.
You can see it in this accurate picture. 


In the change/waiter problem the $9 they each paid was for the meal (25/3) and the money the waiter kept (2/3).
Each person paid $25/3 + $2/3 = $27/3 = $9 


The horses problem, the cowboy in adding his horse made the number of horses 18 which is evenly divisible by 2, 3, and 6. It also caused all the results to round up and then round back down when they took away his horse from the results. Though the final result was a bit unfair for the youngest son, he should of gotten 2.83 (17/6) horses but only got 2. Perhaps the split should of been 8, 6, 3 

Love to have a chocolate block like that. Unfortunately it's a trick, the bottom of the diagonally cut sections grow as they move to make full rectangles of chocolate.

Just love solving puzzles :biggrin:

TonyO


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## Bob Wemm (Apr 1, 2013)

Sometimes things just do not go right.
Sorry about forgetting to post the photos.
I would probably forget my head if it wasn't joined on.
Bob.
:redface::redface::redface:





Bob Wemm said:


> OK, it seems that you cannot turn 64 into 65.
> Dave, please accept my apologies, the diagram is indeed very misleading.
> I have attached two images, firstly 8 inches square with the division into 4 pieces.
> The second image is a graph of 2 in. squares and the appropriate enlarged pieces drawn in.
> ...


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## lyonsacc (Apr 1, 2013)

No worries Bob.

My 11 and 9 year old are with me at work today.  I am going to pose the problem to them to see if they can figure it out.

Dave


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## Jjartwood (Apr 1, 2013)

Where are my meds?


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## Smitty37 (Apr 1, 2013)

its_virgil said:


> An old rancher died and left 17 horses to his three sons. The oldest was to inherit 1/2 of the horses. The middle son was to inherit 1/3 of the horses and the youngest to inherit 1/6 of them. As they were sitting on the corral fence debating how to divide the horses (without fractional horses) an old cowboy rode up and asked what they were doing. After telling the old cowboy of their problem the old cowboy put his horse in the corral. Now there were 18 horses in the corral. He told the oldest to take 9 horses but to leave his. He told the middle son to take 6 horses but leave his. And the youngest son took 2 horses and left the old cowboy's horse. The old cowboy got on his horse and rode off. What is the problem with this thinking or how did it work?
> 
> I often gave this question as a bonus question on tests after teaching about fractions, proportions and percents.
> 
> ...


Easily.  The total number of horses left was 17 but the division in the will was 17/18 so you could add one horse and give each of the sons more than they had actually been left and still have one horse left at the end.


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## Smitty37 (Apr 1, 2013)

its_virgil said:


> Sorry, I made a mistake. The 1/6 should have been 1/9. Then 1/2 + 1/3 + 1/9 = 9/18 + 6/18 + 2/18 = 17/18 . The fractional parts must equal one (1).  I guess if the sons were happy then all is good. My typo ruined the question.
> Don
> 
> 
> ...


Why wouldn't they be happy they all got more than they should have 1.e. son #1 was entitled to 1/2 of 17 horses or 8 1/2 horses and got 9 - son #2 was entitled to 1/3 of 17 horses or 5 2/3 horses and got 6 - son #3 was entitled to 1/9 of 17 horses or 1 8/9 horses and got 9.  What's to be unhappy about.


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## Bob Wemm (Apr 1, 2013)

Some people are never happy, no matter what they get.

We were talking to our daughter the other day about winning the lotto. Betty said she would give Nikki $1,000,000 if we won. Nikki said, "Is that all"???

Bob.


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## lyonsacc (Apr 1, 2013)

Bob Wemm said:


> Some people are never happy, no matter what they get.
> 
> We were talking to our daughter the other day about winning the lotto. Betty said she would give Nikki $1,000,000 if we won. Nikki said, "Is that all"???
> 
> Bob.


 
Gee Bob,

If you won the lotto I'd be willing to be your second cousin for a mere $50,000.


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## wouldentu2? (Apr 2, 2013)

Do that with a gallon of gas and I am interested.


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## maxwell_smart007 (Apr 2, 2013)

> Why wouldn't they be happy they all got more than they should have 1.e. son #1 was entitled to 1/2 of 17 horses or 8 1/2 horses and got 9 - son #2 was entitled to 1/3 of 17 horses or 5 2/3 horses and got 6 - son #3 was entitled to 1/9 of 17 horses or 1 8/9 horses and got 9.  What's to be unhappy about.



The issue is the last son didn't get what he was deserving.  

1/6 of the horses x 17 horses = 2.83 horses.  He was shorted by almost an entire horse.  

The other two took a portion more than they were entitled.


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## Smitty37 (Apr 2, 2013)

maxwell_smart007 said:


> > Why wouldn't they be happy they all got more than they should have 1.e. son #1 was entitled to 1/2 of 17 horses or 8 1/2 horses and got 9 - son #2 was entitled to 1/3 of 17 horses or 5 2/3 horses and got 6 - son #3 was entitled to 1/9 of 17 horses or 1 8/9 horses and got 9.  What's to be unhappy about.
> 
> 
> 
> ...


The initial problem was stated wrong Andrew the eldest son was to get 1/2 of the horses...2nd son 1/3 of the horses and youngest son 1/9 (not 1/6 as originally stated) of the horses.  Add the fractions and you find they were left only 17/18th of the horses. Working the arithmetic from there you will see that each of the sons got more horses than he was entitled to because when the 18th horse was added all of the "fractional horses" created when dividing 17 rounded up.  But since the sons were only left 17/18th of the horses there was still one left over which the cowboy rode off into the sunset.


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## Russianwolf (Apr 2, 2013)

skiprat said:


> This reminds me of the old fun maths problem we did at school.
> How many can remember this one?
> 
> Three people eat in a restaurant when finished, the waiter brings the bill for $30.
> ...



Actually is 3x9 = 27 + the $3 in THEIR pockets is $30. The Waiter's $2 is in the $27.


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## Smitty37 (Apr 2, 2013)

There is no missing dollar...The diners spent $27.00. $25 for lunch and $2 for the unauthorized TIP.  Remind me not to go to lunch with anyone who was confused by this problem.:biggrin::biggrin::tongue:


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## Russianwolf (Apr 2, 2013)

maxwell_smart007 said:


> > Why wouldn't they be happy they all got more than they should have 1.e. son #1 was entitled to 1/2 of 17 horses or 8 1/2 horses and got 9 - son #2 was entitled to 1/3 of 17 horses or 5 2/3 horses and got 6 - son #3 was entitled to 1/9 of 17 horses or 1 8/9 horses and got 9.  What's to be unhappy about.
> 
> 
> 
> ...



1/2 = 9/18 *17 = 8.5
1/3 = 6/18 *17 = 5.66
1/9 = 2/18 *17 = 1.88
1/18 = 1/18 *17 = .999

18/18 = 1 *17 = 17

except that the State is now going to want the horse that was not willed to anyone, and as you can see 1/18th of 17 horses is closer to a whole horse than the fractions of the others. :wink:

l I know is

1 2 3 4 5 ....... 10 9 8 7 6....... 5+6 yep, I still I eleven fingers.


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## Smitty37 (Apr 2, 2013)

Russianwolf said:


> maxwell_smart007 said:
> 
> 
> > > Why wouldn't they be happy they all got more than they should have 1.e. son #1 was entitled to 1/2 of 17 horses or 8 1/2 horses and got 9 - son #2 was entitled to 1/3 of 17 horses or 5 2/3 horses and got 6 - son #3 was entitled to 1/9 of 17 horses or 1 8/9 horses and got 9.  What's to be unhappy about.
> ...


Why convert to decimal? Just work it with fractions.


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## Rangertrek (Apr 2, 2013)

*Magic Box*

OK, check this one out.

Azulejos - YouTube

Similar to some of the above solutions.  I like the chocolate better!


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## maxwell_smart007 (Apr 2, 2013)

Smitty37 said:


> maxwell_smart007 said:
> 
> 
> > > Why wouldn't they be happy they all got more than they should have 1.e. son #1 was entitled to 1/2 of 17 horses or 8 1/2 horses and got 9 - son #2 was entitled to 1/3 of 17 horses or 5 2/3 horses and got 6 - son #3 was entitled to 1/9 of 17 horses or 1 8/9 horses and got 9.  What's to be unhappy about.
> ...



Are you sure it was stated wrong?  The idea is likely that the last person gets the shaft, while the other two get extra....

My rationale for this is that 1/2 plus 1/3 plus 1/6 is exactly one...


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## Smitty37 (Apr 2, 2013)

maxwell_smart007 said:


> Smitty37 said:
> 
> 
> > maxwell_smart007 said:
> ...


Trust me Andrew it is 1/9th - I first encountered this problem in elementary school roughly 65 years ago, and have seen it dozens of times since - the object was to show that nobody gets screwed and the "wise old man", "wandering cowboy", "small town lawyer" or whoever you want to come along gets to keep his own horse. We were given this as a fractional arithmetic problem - the student was to figure out that the dead person only left his sons 17/18ths of his horses......


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## Smitty37 (Apr 2, 2013)

A man driving a flock of sheep met another person who asked "How many sheep are in that flock?" the shepherd said "Well if I had as many more and half as many more and two and a half, I would have a hundred".  How many sheep did he have.  If you want to make it interesting do it forgetting that you know anything about algebra .... strictly arithmetic.  I got this as a sixth or 7th grade arithmetic problem.


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## Bob Wemm (Apr 3, 2013)

Only 39.

And I like the  10 9 8 7 6 and five on this hand, makes 11 fingers. Got 11 toes as well.

Bob.


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## Waluy (Apr 3, 2013)

Smitty37 said:


> A man driving a flock of sheep met another person who asked "How many sheep are in that flock?" the shepherd said "Well if I had as many more and half as many more and two and a half, I would have a hundred".  How many sheep did he have.  If you want to make it interesting do it forgetting that you know anything about algebra .... strictly arithmetic.  I got this as a sixth or 7th grade arithmetic problem.



I am coming up with 20. But then again I can't do it with out using algebra, I have always solved using variables even before they taught me what they were.


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## Smitty37 (Apr 3, 2013)

Waluy said:


> Smitty37 said:
> 
> 
> > A man driving a flock of sheep met another person who asked "How many sheep are in that flock?" the shepherd said "Well if I had as many more and half as many more and two and a half, I would have a hundred".  How many sheep did he have.  If you want to make it interesting do it forgetting that you know anything about algebra .... strictly arithmetic.  I got this as a sixth or 7th grade arithmetic problem.
> ...


In arithmetic.
Using only arithmetic it is what he has now + what he has now + 1/2 what he has now + 2 1/2 = 100
so adding  2 1/2 what he has now +2 1/2 = 100 
so 2 1/2 what he has now = 100 - 2 1/2 = 97 1/2

so what he has now = 97 1/2 / 2 1/2 = 195/2 / 5/2 = 195/2*2/5 = 390/10 = 39

using rules or elementary arithmetic. 

Algebra is simple x=number of sheep
x+x+.5x + 2.5 = 100
2.5 x = 100-2.5
2.5x = 97.5
x= 97.5/2.5 = 39


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## Bob Wemm (Apr 3, 2013)

HOORAY, i got it right.

Bob.


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## Russianwolf (Apr 4, 2013)

Smitty37 said:


> Using only arithmetic it is what he has now (x) + what he has now(x) + 1/2 what he has now(x) + 2 1/2 = 100
> so adding  2 1/2 what he has now (x) +2 1/2 = 100
> so 2 1/2 what he has now(x) = 100 - 2 1/2 = 97 1/2
> 
> ...



actually you are doing algebra both times, you are just using fractions to solve one and decimals to solve the other. Anytime you have a number that you don't know and are solving for, its algebra. you can us a letter or words to define it, but its still the same.

As far as solving it fractionally you could have also done it this way

x = 97 1/2 / 2 1/2 = 195/2 / 5/2 = 195 / 5 = 39


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## Waluy (Apr 4, 2013)

Smitty37 said:


> Waluy said:
> 
> 
> > Smitty37 said:
> ...



Ah I see where I made my mistake for some reason I was reading the last part as an extra two and a half times what he had, not that he had an extra two and a half sheep. So I was coming up with:
x+x+.5x+2.5x=100 
5x=100
x=100/5 = 20


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## Smitty37 (Apr 4, 2013)

Russianwolf said:


> Smitty37 said:
> 
> 
> > Using only arithmetic it is what he has now (x) + what he has now(x) + 1/2 what he has now(x) + 2 1/2 = 100
> ...


My teacher would have scolded you for not showing inverting 5/2 and stating that the two's cancel leaving 195/1*1/5=195/5.  We were not allowed to do that in our heads for written problems.

Actually you are not using the rules of algebra to solve it fractionally you are using the rules of arithmetic and can solve the problem without ever stating it as an algebraic equation and with no knowledge at all of algebra per se.  My dad never saw an algebra text while in school and could solve many of the problems I got in beginning algebra.


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