SOLUTION: Hello, I have a problem, that I dont know what am I doing wrong with it.The problem is like this: A vendor sends an invoice that totals $6115.68, this invoice contains 2 types o

Algebra ->  Rate-of-work-word-problems -> SOLUTION: Hello, I have a problem, that I dont know what am I doing wrong with it.The problem is like this: A vendor sends an invoice that totals $6115.68, this invoice contains 2 types o      Log On


   

Question 823594: Hello, I have a problem, that I dont know what am I doing wrong with it.The problem is like this:
A vendor sends an invoice that totals $6115.68, this invoice contains 2 types of products that have different prices.
Product 1: $1.65 per case
Product 2: $2.46 per case.
How many cases of each product is the vendor charging for?
I just cant figure out how to do this, can someone please explain me?




Found 2 solutions by josgarithmetic, KMST:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
The question lacks some data for a specific answer.

Let x and y be the number of cases for product 1 and product 2 in that order.

Accounting for money, 1.65x%2B2.46y=6115.68.
You may expect these simple restrictions.
x%3E=1 and y%3E=1, because you must have whole numbers for x and y; negative numbers cannot happen.

Additionally, the cost of either x or y must not reach or go beyond 6115.68 dollars. This means
1.65x<6115.68
x<6115.68/1.65
x%3C3706
and
2.46y<6115.68
y<6115.68/2.46
y%3C2485

With this, you may have several possible solutions. Having any further more specific data may give you a much more specific answer.
graph%28300%2C300%2C-1%2C3000%2C-1%2C3000%2C-%281.65%2F2.46%29x%2B6115.68%2F2.46%29
The y-intercept is at 2485.77

Do you have ANY case count data to include with the description?

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
x= number of cases of product 1.
y= number of cases of product 2.
1.65x= cost in $ for x units of product 1.
2.46y= cost in $ for y units of product 2.
1.65x%2B2.46y= total cost for x units of product 1, and y units of product 2.
according to the vendor's invoice, that is $6115.68, so
1.65x%2B2.46y=6115.68 is our equation.
We have two variables and only one equation.
In general that would mean an infinite number of answers.
However, in this case, only integer, non-negative solutions would work.
That limits the number of solutions.
1.65x%2B2.46y=6115.68
165x%2B246y=611568
55x%2B82y=203856
Once we find one solution, we can increase the x found by 82 while decreasing the y found by 55.
That would change both terms by 82%2A55=4510 units and in opposite directions, so that they would still add up to 203856 .
Since 203856 divided by 4510 yields a quotient of 45 with a remainder of 906 , there are at least 45 solutions, and maybe 46 .

If x=y , 55x%2B82x=203856 transforms into
137x=203856 , x=203856%2F137 , and x=1488 .
So one of the possible solutions is x=y=1488 .
That is a non-negative integer, so it is allowed to be x and y.
That solution means 1488 units of each product.
It could be that there were 55 less units of product 2,
meaning 1488-55=1433 units of product 2,
but 82 more units of product 1,
meaning 1488%2B82=1570 .
That would decrease the term 82y by 82%2A55=4510 ,
but would increase term 55x by 82%2A55=4510 .
The total would still be 203856 , with x=1570 and y=1433 .
We could repeat the procedure a few more times, as long as the decreasing y is still a nonn-egative number.
Since 1488 divided by 55 has a quotient of 45 with a remainder of 3, we can do that decreasing y by 55 a total of 27 times until we get to y=3 .
That means that besides ordered pair (x,y)=(1488,1488), There are 27 other solutions with lesser values of y, going all the way to x=3702 , y=3 .
On the other hand, we could decrease x by 82, while increasing y by 55.
That would give us another 18 solutions, with x decreasing all the way to 12 because 1488+divided+by+%7B%7B%7B82 yields a quotient of 18 and a remainder of 12.
There is a total of 1%2B27%2B18=46 solutions:
12 units of product 1 with 2478 units of product 2,
94=12%2B82 units of product 1 with 2423=2478-55 units of product 2,
176 units of product 1 with 2368 units of product 2,
258 units of product 1 with 2313 units of product 2,
and so on, all the way to
3702=12%2B45%2A82 units of product 1 with 3=2368-45%2A55 units of product 2.
I guess we could write them as
x%5Bn%5D=12%2B%28n-1%29%2A82 paired with y%5Bn%5D=2478-%28n-1%29%2A55
foe n integer, such that 1%3C=n%3C=46