What is a systematic method to solve this Diophantine equation?

[the_Doctor111]

Homework Statement

Suppose you spend $12.30 on chocolate bars and chips. If chocolate bars cost $1.20 and each bag of chips cost $2.50, how many bags of chips did you buy?

*Both the number of chocolate bars and bags of chips must be positive.

Homework Equations

12.30 = 1.20x + 2.50y

The Attempt at a Solution

What is the method i go about solving this problem besides guess and check. I assume it has something to do with the euclidian algorithm since that's what I have been learning.
Cheers.

 

[happysauce]

Euclidean algorithm?

 

[epenguin]

Well I think I got it, start with see why the chip bags have got to be an odd number and after that the chocolate bars have got to be...

Hopefully this leads to something interesting (because I can't say it starts that way )

 

[AlephZero]

Notice that all three numbers are "nearly" multiples of 1.2.

You can use that fact to solve the problem without guessing, or chugging through Euclid's algorithm.

 

[HallsofIvy]

The first thing I would do it is multiply 12.30 = 1.20x + 2.50y by 10 to get 12x+ 25y= 123.

The only "Euclidean algorithm" needed is to note that 25- 2(12)= 1. Multiplying that equation by 123, 12(-246)+ 25(123)= 123. That is, one solution is x= -246 and y= 123. That is not the solution because -246 is not positive. But x= -246+ 25k and y= 123-12k is also a solution for ay integer, k: 12(-246+ 25k)+ 25(123- 12k)= 12(-246)+ (12)(25k)+ 25(123)- 25(12k)= 123 since the two terms in k cancel.

So you want to find an integer, k, such that x= -245+ 25k> 0 and y= 123- 12k> 0.

 

[the_Doctor111]

Thanks heap guys! :!)

 

[SammyS]

Thanks heap guys! :!)

By the Way: Welcome to PF, Doctor111 !