### Supper with Skiing

A ski resort offers two package plans. Plan
A gives one person four nights lodging and three dinners for$326.
Plan B gives one person five nights lodging and four dinners for
$413. Assuming that the costs per night for lodging anddinner
are the same for both plans, how much does one dinner cost?

(Source: Mathematics Teaching in the Middle
School, Mar-Apr 1996).

### The Solution

We can set this problem up as a system of equations, and then
solve for the uknown variables. We are given two distinct situations,
the first is a skiing package offering four nights of lodging
and three dinners for $326. If we allow l= cost of lodging per
night and d= cost of each dinner, then we can write the following
equation:

**4l + 3d = 326**
Using the same idea, we can write an equation for plan B, which
porvides five nights of lodging and four dinners for $413:

**5l + 4d = 413**
Next we can solve for one of the variables in one of the equations,
and use that solution to plug into the other equation and then
solve.

**5l +4d = 413**
**5l = 413-4d**
**l = (413-4d)/5**
Now that we have solved for l in terms of d, we can plug the
solution for l into the first equation to solve for d:

## 4*[(413-4d)/5] + 3d = 326

**330.4 - 3.2d + 3d = 326**
**330.4 - 0.2d = 326**
**4.4 = 0.2d**
**22 = d**
Now we have solved for the price of each dinner,
and we can use that solution to find the price of each nights
lodging by simply plugging the price of dinner into the original
equations that we have. Solving for the price of lodging in both
equations should produce the same answer:

**Plan A**

4l + 3(22) = 326
**4l = 260**
**l = 65**
**Plan B**
**5l + 4(22) = 413**
**5l = 325**
**l = 65**
**Thus we get a final solution telling us
that each night of lodging costs $65.00 and each dinner costs
$22.00**

**Return**