What is a systematic method to solve this Diophantine equation?

In summary, the problem involves spending $12.30 on chocolate bars and chips, with chocolate bars costing $1.20 each and chips costing $2.50 per bag. The task is to determine the number of bags of chips bought, with both the number of chocolate bars and bags of chips being positive. The Euclidean algorithm can be used to solve this problem by noting that all three numbers are nearly multiples of 1.2. After multiplying the initial equation by 10, it can be manipulated to find a solution that satisfies the given conditions.
  • #1
the_Doctor111
6
0

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.
 
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  • #2
Euclidean algorithm?
 
  • #3
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 :biggrin:)
 
  • #4
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.
 
  • #5
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.
 
  • #6
Thanks heap guys! :!)
 
  • #7
the_Doctor111 said:
Thanks heap guys! :!)
By the Way: Welcome to PF, Doctor111 !
 

1. What is a Diophantine equation?

A Diophantine equation is a type of algebraic equation in which only integer solutions are sought. The solutions to these equations are referred to as Diophantine solutions.

2. What is a systematic method to solve a Diophantine equation?

One of the most commonly used systematic methods to solve a Diophantine equation is the method of substitution. This involves substituting different values for the variables in the equation until a solution is found.

3. How is a Diophantine equation different from a regular algebraic equation?

Unlike regular algebraic equations, Diophantine equations only have integer solutions. This means that any solutions to the equation must be whole numbers and cannot include decimals or fractions.

4. Can all Diophantine equations be solved?

No, not all Diophantine equations have solutions. In fact, some Diophantine equations have been proven to have no solutions at all. It is still an open question in mathematics whether there is a general method for determining if a Diophantine equation has solutions.

5. How are Diophantine equations used in real-world applications?

Diophantine equations have various applications in fields such as cryptography, coding theory, and number theory. They are also used in certain types of optimization problems, such as finding the most efficient way to pack a certain number of objects into a container.

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