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Question

Marie goes buy three items X, Y, and Z. Unit price on each item as follow: X=$6 each; Y=$3 each; Z=$0.1 each. Marie has to buy all three items in a total of 100 units, and must spend a $100 no less no more. How many units of each item Marie can buy?

Answer 1

X + Y + Z = 100

6X + 3Y + 0.1Z = 100

 

Three variables need three equations. You present only two.

Answer 2

Kenneth is correct that the requirement for non-negative integers brings you to a Diophantine equation, which is why you don't need a third equation.  Mark is correct that Euclidean Algorithm is the standard method to solve a Diophantine, although in this case, it simplifies enough to not quite require that (although you should be familiar with the Euclidean Algorithm for slightly harder problems).  Either way, you really shouldn't require an excel or a computer program to solve this.  Our two equations are:

 

x + y + z = 100

6x + 3y + .1z = 100

 

If we solve the first equation for z, we get

 

z = 100 - x - y

 

If we multiply the second equation by 10 (to get all integer coefficients) and substitute in z, we get

 

60x + 30y + 100 - x - y = 1000

59x + 29y = 900

 

So first, modulo 29, we have

 

59x ≡ 900 (mod 29)

x ≡ 1 (mod 29)

 

So x = 1 or 30 or 59, etc.

 

Considering modulo 59, we have

 

29 y ≡ 15 (mod 59)

 

Noting that 2*29 was close to 59, let's multiply both sides by 2:

 

58y ≡ 30 (mod 59)

-y ≡ 30 (mod 59)

y ≡ -30 ≡ 29 (mod 59)

 

So y ≡ 29 (mod 59)

 

If we try x = 1, y = 29, z = 100 - x - y = 70, we discover that 

 

6(1) + 3(29) + .1(70) = 100

 

If you have any other number theory questions or would like to discuss the Euclidean Algorithm, please let me know.

Answer 3

When you purchase things in USA, there is usually sales tax to compute (which consists of state tax and local tax). Thus coming out to be exactly $100 of purchases is most difficult.

 

If we choose to ignore that, then the this problem should be examined carefully.

Name of item   Unit cost of item    Number of units purchased

   X                      6                             x

   Y                      3                             y

   Z                     0.10                         z

 

I take it that x+y+z must equal 100 [this is the 100 units condition]

and 6x + 3y + 0.1z must equal 100 [this is the condition that exactly $100 is to be spent].

 

So there are 2 equations but three unknowns. We do know, furthermore, that the values for x, y and z are required to be non-negative integers. This situation is an example of Diophantine equations, not normally considered in conventional high school mathematics.

 

A method of solution is to write a computer program (or perhaps an Excel spreadsheet) to experiment with all possible values of x, y and z, to find a solution.  Good luck with that!