Solve 3 variables with 2 equations

[abenedet]

Here is the problem:

Total sales are $5125 and units sold are 1653. Units can be sold either on sale or not on sale. The price on sale or feature sale price (FSP) was $2.25. How many units were sold on sale?

I can derive 2 formulas, but I have 3 variables to solve. Here's how far I have gotten:

Regular price = RSP
Units sold on Sale = Promoted volume (PV)
Units sold at regular = non-promoted volume (NPV)
Total volume = TV

Equation 1: TV = PV + NPV
Equation 2: Total Sales = (RSP x NPV) + (FSP x PV) or $5125 = (RSP x NPV) + 2.25PV

I cannot determine another equation to help solve this. Any help will be greatly appreciated.

 

[Mark44]

Here is the problem:

Total sales are $5125 and units sold are 1653. Units can be sold either on sale or not on sale. The price on sale or feature sale price (FSP) was $2.25. How many units were sold on sale?

I can derive 2 formulas, but I have 3 variables to solve. Here's how far I have gotten:

Regular price = RSP
Units sold on Sale = Promoted volume (PV)
Units sold at regular = non-promoted volume (NPV)
Total volume = TV

Equation 1: TV = PV + NPV
Equation 2: Total Sales = (RSP x NPV) + (FSP x PV) or $5125 = (RSP x NPV) + 2.25PV

I cannot determine another equation to help solve this. Any help will be greatly appreciated.

Homework Statement

Let's see: you have RSP, PV, NPV, TV, and Total Sales. I think you have way too many variables that aren't important and not enough of the variables you are actually looking for. Plus, there is some information given that you didn't use. The way I read this is an item sells for its normal price or its sale price. What are they looking for in this problem?

Hint: One equation should give the total number of items, and the other should give the total revenue for the two types of items (regular price items and on-sale items).

 

[abenedet]

Let's see: you have RSP, PV, NPV, TV, and Total Sales. I think you have way too many variables that aren't important and not enough of the variables you are actually looking for. Plus, there is some information given that you didn't use. The way I read this is an item sells for its normal price or its sale price. What are they looking for in this problem?

They are looking for how many items were sold at the sale price. Also, the only known variables I have are total volume, FSP, and total Sales.

Hint: One equation should give the total number of items, and the other should give the total revenue for the two types of items (regular price items and on-sale items).

Yes, these are the equations that I was able to derive, but I then got stuck because I have 3 variables to solve in these two equations in order to derive how many items were sold at sale price:

Equation 1: TV = PV + NPV
Equation 2: Total Sales = (RSP x NPV) + (FSP x PV) or $5125 = (RSP x NPV) + 2.25PV

Given these two equations (or any other equations that you can think of), how can I solve for PV?

 

[Mark44]

They are looking for how many items were sold at the sale price. Also, the only known variables I have are total volume, FSP, and total Sales.

If a number is known, it's not a variable and shouldn't be represented as a variable.

For example, total volume is known, total sales are known, and sale price is known.

After eliminating all "variables" that aren't actually variable, I get two equations in three unknowns, as well. Are you sure you have provided all the information in this problem?

Yes, these are the equations that I was able to derive, but I then got stuck because I have 3 variables to solve in these two equations in order to derive how many items were sold at sale price:

Equation 1: TV = PV + NPV
Equation 2: Total Sales = (RSP x NPV) + (FSP x PV) or $5125 = (RSP x NPV) + 2.25PV

Given these two equations (or any other equations that you can think of), how can I solve for PV?

 

[abenedet]

Yes, you are correct...my terminology was misleading. There are 3 unknowns. Sorry for the confusion.

And yes, this is all the information that I have.

Thanks

 

[Mark44]

I used variables of x, y, and FP.
x = # of items sold at full price
y = # of items sold at sale price of $2.25

Altogether 1653 items were sold, so it's possible to get one equation in two variables from this information. From this equation, you can solve for y, the number of items sold at the sale price, in terms of the full price.

It's reasonable to assume that both x and y are greater than 0, and that both x and y have integer values. This means that the full price has to be larger than $3.10, which yields a negative number of items sold.

 

[D H]

One solution to the problem (there are other solutions) is arrived at by assuming that all but one item were sold at the sale price. This would mean the full price is $1408, a price that makes the variations in the cost of an airplane seat look mild.

If one assumes that the full price expressed in cents is a positive integer, the problem becomes a fairly simple second order Diophantine problem.