Question

help?

Answer 1

Happy to help you, ashlee, but please present your question first (instead of "help?").

Answer 2

At least write down the topic. Don't just write help.

Answer 3

Owen made 100 sandwiches which
she sold for exactly $100. She sold
caviar sandwiches for $5.00 each, the
bologna sandwiches for $2.00, and
the liverwurst sandwiches for 10 cents.
How many of each type of sandwich
did she make?

Answer 4

100 sandwiches only guyz

Answer 5

5x + 2y + .1z = 100
x + y + z = 100

Answer 6

eliminate z first

Answer 7

then u will get a linear diaphontine equation
which is easy to solve

Answer 8

A possible answer can be:
18 Caviar Sandiwiches = $90
4 Bologna Sandwiches = $8
20 Liverwurst Sandwiches = $2

This all equals $100. There can be many more solutions to this problem but this is one of them.

Answer 9

is this q from number theory ?
if so, u can diaphontine method :)

Answer 10

18 Caviar Sandiwiches = $90
4 Bologna Sandwiches = $8
20 Liverwurst Sandwiches = $2

This is nota correct answer : count the sandwiches; they dont add up to 100

Answer 11

your right

Answer 12

is this from number theory ?

Answer 13

yeaah

Answer 14

good :) u familiar wid solving diaphontine equations right ?

Answer 15

kind of

Answer 16

5x + 2y + .1z = 100
50x + 20y + z = 1000

x + y + z = 100
z = 100-x-y

50x + 20y + 100-x-y = 1000
49x + 19y = 900

since gcd(49, 19) = 1, which divides 900, there will be oly 1 single solution

Answer 17

lets find the actual solution. u wanto try ?

Answer 18

ganeshie8 wrote the two equations we need to get started: 5x + 2y + .1z = 100
x + y + z = 100. Note that this is (to be continued)

Answer 19

ohhh

Answer 20

We could solve the 2nd equation for z: z = 100-x-y.
We could then substitute this expression for z into the first equation.
Unfortunately, that leaves us with ONE equation in TWO unknowns.

Answer 21

Making the indicated substitution, we get \[5x+2y+0.1(100-x-y)+100.\]
All we can do here is to come up with an equation for y in terms of x or x in terms of y.
anyone want to brainstorm on what to do next?

Answer 22

we can solve a single equation wid two unknows using diaphontine method

Answer 23

to solve 49x + 19y = 900 :
step1 : find x and y such that 49x + 19y = 1 using euclid algorithm in reverse

Answer 24

@ganeshie8 : I have 43 years experience in teaching math and have yet to see the term "diaphontine method." It's unfair to expect that students here will understand it. Unless you're prepared to define and explain this method, I'd suggest you stick with explanations aimed at high school and early college students.

Answer 25

is that a challenge ? :o

Answer 26

wow thats really nice :) ashlee is doing number theory course; diaphontine equations are there in every number theory syllabus

Answer 27

OK, I'll take your word for this as I have no personal experience with number theory. if your focus is simply on ashlee22, fine.

Answer 28

yess... this method is oly for those who take number theory :)

Answer 29

I've come up with the exact same equation as you: 49x - 19y = 900. I'd be interested in seeing how you arrive at a solution for x, y and z that takes into account the facts that x+y+z=100, and that x, y and z must all be zero or greater, but certainly not greater than 100. Is there really a unique solution? Again, we're dealing with one equation in 2 unknowns, but with domain constraints.

Answer 30

yes il put the solution in a minute,
the key thing to observe is that x, y, z are integers (natural)

Answer 31

to solve 49x + 19y = 900 :
step1 : find x and y such that 49x + 19y = 1 using euclid algorithm in reverse

Find gcd(49, 19) :
49 = 19*2 + 11
19 = 11*1 + 8
11 = 8*1 + 3
8 = 3*2 + 2
3 = 2*1 + 1

reverse euclid :
1 = 3 - 2
= 3 - [8-3*2]
= 3 - 8 + 3*2
= 3*3 - 8
= 3*(11-8) - 8
= 3*11 - 4*8
....
= 49(7) + 19(-18)

Answer 32

step2 : multiply 900 both sides to the above relation

49(7) + 19(-18) = 1
49(6300) + 19(-16200) = 900

so, x = 6300, y = -16200 provide one solution to the diaphontine equation in question;

Answer 33

all other solutions can be produced by :-
x = 6300 + 19t
y = -16200 - 49t
t is any integer.

Answer 34

next, to solve our problem :-
x > 0
y > 0

6300 + 19t > 0
-16200 - 49t > 0

solving, t = -331

Answer 35

So x, y and z are necessarily integer? Of course, if we're counting how many of each kind of sandwich were made. Still wondering how you determine the actual, integer values of x, y and z subject to the restraints mentioned.

Answer 36

yess they must be integers; then oly we can solve it using diaphantine :)

plug t=-331 for x, y :-
x = 11
y = 19
z = 100-11-19 = 70

Answer 37

Cool. Nice meeting two students on a math-tutoring site who know 'way more than I do about number theory! :) Good going.

Answer 38

<3

Answer 39

I think I just learned that I don't ever want to take a number theory course :-)

Answer 40

lol number theory is beautiful, its not that scary as i made it look wid this question :)