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Subject: "math problem no answer in sight"     Previous Topic | Next Topic
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Conferences The CTK Exchange Middle school Topic #80
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Dmom
guest
Feb-27-03, 12:04 PM (EST)
 
"math problem no answer in sight"
 
   I beleive this is a problem from Benjamin Banneker
you must spend $100 and buy 100 animals
Cows $5.oo each
Goats $1.00 each
Chickens .05 cents each
You must buy some of each animal
I have worked many anwsers that are 99 & 100 but not 100 & 100
I would love to know the anwser
thanks


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  Subject     Author     Message Date     ID  
math problem no answer in sight Dmom Feb-27-03 TOP
  RE: math problem no answer in sight MrToad Feb-27-03 1
  RE: math problem no answer in sight CSpeed0001 Feb-27-03 2
  RE: math problem no answer in sight Ben Feb-27-03 3
     RE: math problem no answer in sight Ben Feb-28-03 4
     RE: math problem no answer in sight malaka Jan-07-04 14
  RE: math problem no answer in sight Whymme Apr-12-03 5
  RE: math problem no answer in sight SnowBoB Jun-25-03 6
  RE: math problem no answer in sight g. sambasivan Jun-27-03 7
     RE: math problem no answer in sight Puppkid Dec-15-03 8
         RE: math problem no answer in sight Flip Dec-17-03 9
             RE: math problem no answer in sight CSpeed0001 Dec-27-03 11
             RE: math problem no answer in sight malaka Jan-07-04 12
         RE: math problem no answer in sight Econ Jan-28-04 19
  RE: math problem no answer in sight Carlos Garcia Dec-24-03 10
     RE: math problem no answer in sight malaka Jan-07-04 13
  RE: math problem no answer in sight ann Jan-12-04 15
     RE: math problem no answer in sight alexb Jan-12-04 16
  RE: math problem no answer in sight Chris H Jan-15-04 17
  RE: math problem no answer in sight gilberto Jan-25-04 18
     RE: math problem no answer in sight Vantixs Oct-07-06 20
         RE: math problem no answer in sight alexb Oct-07-06 21
  RE: math problem no answer in sight cow problem Mar-03-07 22
  RE: math problem no answer in sight Augusto Campos Mar-21-07 23
  RE: math problem no answer in sight Martin Jan-13-08 24

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MrToad
guest
Feb-27-03, 03:01 PM (EST)
 
1. "RE: math problem no answer in sight"
In response to message #0
 
   The unique answer is:

19 cows = $95
1 goat = $1
80 chickens = $4

To see this, note that you have two equations in two unknowns:

(1) A+B+C=100
(2) 5A+B+0.05C=100

where A = # of cows, B = # of goats, and C = # of chickens.

If you multiply (2) by 20 and then subtract (1), you get:

(3) 99A+19B=1900

Now, A, B, and C are positive integers (I believe that it's implicit in the question that one can't buy, for example, 3/4 of a goat - otherwise, there are infinitely many solutions). Note that 19 divides 1900 and 19B, and so it must divide 99A. But, 19 is prime and does not divide 99, so it divides A. On the other hand, A cannot be greater than 20, because, from (2), 5A+(something positive)=100. Therefore A=19. From (3), one sees then that B=1, and then, from (1), one gets C=80.


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CSpeed0001
Member since Feb-19-03
Feb-27-03, 03:01 PM (EST)
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2. "RE: math problem no answer in sight"
In response to message #0
 
   I was able to get 19 cows, 1 goat, and 80 chickens.

From the information that we have, we can generate 2 equations each with three unknowns,

x+y+z=100
5x+y+.05z=100

and three (actually six) inequalities.

0<x<100
0<y<100
0<z<100
(Saying that we can't have negative numbers, or numbers bigger than 100. Actually, in this problem, one statement forces the other to be true.)

Using whatever methods we have to solve linear equations (I used matrix algebra and a really powerful calculator), we get three equations down to two equations:

x=(19/80)z
y=-(99/80)z+100

From here we can see that z has to be 80, or some product(?) of it (i.e. 160, 240, 320, etc). This is true because all of our numbers have to be whole numbers (unless the animals are dead meat). Z also has to be less than 100, because we can't buy a negative number of animals. That means z=80. From there, we can solve the rest of the equations by plugging 80 in for z.


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Ben
guest
Feb-27-03, 03:01 PM (EST)
 
3. "RE: math problem no answer in sight"
In response to message #0
 
   Assuming that you have to have a whole number of each animal (and I think that that is a fair assumption) then it is not possible.

Since chickens are $0.05 or 5 cents each in order to get a whole number of dollars you must have a multiple of 20 of them. That is you must have either 20, 40, 60 or 80 of them. The actual problem boils down to 2 equations, using H for cows, G for goats and C for Chickens

1) Fro money 5H + G + 0.05C = 100

2) From Animal numbers H + G + C = 100

Taking each of the 4 values of C we can have in turn then

For C=20

1) 5H + G + 1 = 100 which equals 5H + G = 99
2) H + G + 20 = 100 which equals H + G = 80

(1) - (2) Gives (5H + G) - (H + G) = 99 - 80

or (5-1)H + (1-1)G = 19
or 4H = 19
or H = 19/4

This is not a whole number therefore can not be a solution.

Performing the same calculation for C=40, C=60 and C=80 all gives the same result H is not whole. Therefore the problem has no solution.


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Ben
Member since Nov-18-02
Feb-28-03, 10:40 AM (EST)
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4. "RE: math problem no answer in sight"
In response to message #3
 
   >Performing the same calculation for C=40, C=60 and C=80 all
>gives the same result H is not whole. Therefore the problem
>has no solution.

Doh, I really got it wrong, sorry. I put it down to trying to do the calculations after a day numbing my brain into submission peforming DVT VBI tests.

Anyway for the case C=80

1) 5H + G + 4 = 100 which equals 5H + G = 96
2) H + G + 80 = 100 which equals H + G = 20

(1) - (2) Gives (5H + G) - (H + G) = 96 - 20

or (5-1)H + (1-1)G = 76
or 4H = 76
or H = 76/4 = 19

Put H=19 and C=80 into H + G + C = 100 and you get

19 + G + 80 = 100
pr
G = 1

Same as everyone else.

In my stupor last night I made the mistake

96 - 20 = 86

Once More Doh


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malaka
guest
Jan-07-04, 08:56 PM (EST)
 
14. "RE: math problem no answer in sight"
In response to message #3
 
   when you change C to 40 and so on, your equations change, so it doesnt yields H=19/4 (what you call the same result) for all the values of C

c=40
h=38/4=19/2

c=60
h=57/4

c=80
h=76/4=19

which gives you with, 80 chickens, 19 cows and 1 goat

80*0.05=4 euros
19*5=95 euros
1*1=1 euros

80+19+1=100 animals
4+95+1=100 euros

as somebody stated before, so it does have a solution


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Whymme
guest
Apr-12-03, 03:57 PM (EST)
 
5. "RE: math problem no answer in sight"
In response to message #0
 
   And yet another way to compute it:

First, leave the goats out. They are just used to fill up the number of animals until you reach 100.

Then we're left with:

5X+0.05Y = X+Y, where X = cows and Y = chicken
5X+0.05Y-X-Y = 0 --> subtract X+Y from both sides
4X-0.95Y = 0
80X-19Y=0 --> multiply by 20

It's easy to see that the only solution possible is where X=19 and Y=80, so that you get

80x19 = 19x80

Whymme


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SnowBoB
guest
Jun-25-03, 11:48 AM (EST)
 
6. "RE: math problem no answer in sight"
In response to message #0
 
   Do you have to buy each one of the animals? If you don't you could do 5,10,15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95,$100! yeah! Oh, but that would'nt be 100 animals. we'll, i can't do it all here but try my e-mail- peabody_booboos@yahoo.com


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g. sambasivan
guest
Jun-27-03, 03:56 PM (EST)
 
7. "RE: math problem no answer in sight"
In response to message #0
 
   fairly simple solution

19 cows for 95$
1 goat for 1 $
80 chickens 4$

100 animals and 100$

is there any alternate solution; i think not.

sambasivan


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Puppkid
guest
Dec-15-03, 09:14 AM (EST)
 
8. "RE: math problem no answer in sight"
In response to message #7
 
   I got it!

You get: 80 chickens + 14 cows + 16 goats = 100 animals.

I did it the long way....simple adding....no equation. Here:


1.00 (20 chickens)
+10.00 (2 cows)
________
$11.00 (22)
+ 10.00 (2 cows)
_______
$21.00 (24)
+ 1.00 (1 goat)
_______
$22.00 (25)
+ 1.00 (20 chickens)
_______
$23.00 (35)
+ 10.00 (2 cows)
_______
$33.00 (37)
+ 3.00 (3 goats)
_______
$36.00 (40)
+ 10.00 (2 cows)
_______
$46.00 (42)
+ 1.00 (20 chickens)
_______
$47.00 (62)
+ 10.00 (2 cows)
_______
$57.00 (64)
+ 5.00 (5 goats)
_______
$62.00 (69)
+ 10.00 (2 cows)
_______
$73.00 (71)
+ 10.00 (2 cows)
_______
$83.00 (73)
+ 1.00 (20 chickens)
_______
$93.00 (93)
+ 7.00 (7 goats)
_______
$100.00 (100 animals)



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Flip
guest
Dec-17-03, 02:40 PM (EST)
 
9. "RE: math problem no answer in sight"
In response to message #8
 
   Time for a new calculator, Puppkid:

80 chickens @ $0.05 = $4
14 cows @ $5 = $70
16 goats @ $1 = $16

$4 + $70 + $16 = $90 - you're still $10 short...

... but anyway: why does everyone have to re-prove the solution that has already been given (and is unique within the constraints of the problem)?

I am particularly astounded by the post that declares the problem impossible after two other people have already posted a solution!!


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CSpeed0001
guest
Dec-27-03, 11:00 PM (EST)
 
11. "RE: math problem no answer in sight"
In response to message #9
 
   Sometimes, it takes a while for what we post to actually make it to the webpage. All three of us posted on the same day, so we didn't see the other two posts when we posted.

The guy who said that there wasn't a solution hadn't the earlier two posts.

--CS


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malaka
guest
Jan-07-04, 08:56 PM (EST)
 
12. "RE: math problem no answer in sight"
In response to message #9
 
   80+14+16=100? i think not...


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Econ
guest
Jan-28-04, 08:43 AM (EST)
 
19. "RE: math problem no answer in sight"
In response to message #8
 
   80+14+16=110 animals;)


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Carlos Garcia
guest
Dec-24-03, 02:34 PM (EST)
 
10. "RE: math problem no answer in sight"
In response to message #0
 
   I got 100-100 with 25 chickens, 70 goats, and 5 cows. It equaled to 100-100.


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malaka
guest
Jan-07-04, 08:56 PM (EST)
 
13. "RE: math problem no answer in sight"
In response to message #10
 
   how?

25+0.05=1,25 that's not a good starting point

it has to be an integer, all the numbers must be integers


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ann
guest
Jan-12-04, 02:24 PM (EST)
 
15. "RE: math problem no answer in sight"
In response to message #0
 
   I solved this problem for you.

My purchase:10 cows

40 goats

200 chickens
My total is 100. Please inform me of the correct amswer.

Thanks,
ladinobam@yahoo.com


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alexb
Charter Member
2158 posts
Jan-12-04, 02:35 PM (EST)
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16. "RE: math problem no answer in sight"
In response to message #15
 
   >My total is 100. Please inform me of the correct amswer.

Just check your total. It's that simple.


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Chris H
guest
Jan-15-04, 01:00 PM (EST)
 
17. "RE: math problem no answer in sight"
In response to message #0
 
   How about :-
80 chickens = $4 (I assume you meant 5 cents not .05 of a cent ;) )
1 goat = $1
19 Cows = $95


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gilberto
guest
Jan-25-04, 09:46 AM (EST)
 
18. "RE: math problem no answer in sight"
In response to message #0
 
   Dmom,


The number of chickens should be integer(multiples of 20).
You have 4 choices for the chickens:
x= number of cows
y= number of goats

20 chickens= $1.00
($5)x + ($1)y = $99.00
x + y = 20 no integer solutions

40 chickens = $2.00
($5)x + ($1)y = $98.00
x + y = 60 no integer solutions

60 chickens = $3.00
($5)x + ($1)y = $97.00
x + y = 40 no integer solutions

80 chickens = $4.00
(5)x + ($1)y = $96.00
x + y = 20 x=19 and y=1

80 chickens, 19 cows and 1 goat is the solution.


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Vantixs
guest
Oct-07-06, 07:44 AM (EST)
 
20. "RE: math problem no answer in sight"
In response to message #18
 
   why would the chickens have to be mulitpiles of 20?


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alexb
Charter Member
2158 posts
Oct-07-06, 07:48 AM (EST)
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21. "RE: math problem no answer in sight"
In response to message #20
 
   >why would the chickens have to be mulitpiles of 20?

Otherwise the total cost would include fractions of dollar.


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cow problem
guest
Mar-03-07, 08:14 PM (EST)
 
22. "RE: math problem no answer in sight"
In response to message #0
 
   20 chickens = 1.00
40 = 2.00
60 = 3.00
80 = 4.00
100 =5.00
so why did you evean put thie here it's INPOSABLE!!!duah
unless you could have 100 chickens and then spend over 100 dollors, don't waste youre time.


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Augusto Campos
guest
Mar-21-07, 07:22 AM (EST)
 
23. "RE: math problem no answer in sight"
In response to message #0
 
   I founded the following answer:
80 Chickens ($4)
19 Cows ($95)
1 Goat ($1)


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Martin
guest
Jan-13-08, 08:24 AM (EST)
 
24. "RE: math problem no answer in sight"
In response to message #0
 
   Cow 19 5 95
Goat 1 1 1
Chicken 80 0.05 4

100 100


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