**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 and dinner are the same for both plans,
how much does one dinner cost?**

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

**I would use the Four
Step Plan for Problem Solving here.**

**Explore:****What do you know? What do you want to find out?**

**We know that Plan A
costs $326 for:**

**4 nights lodging and
3 dinners > 4n + 3d = 326**

(**n** represents the number
of nights, while **d** represents the cost of the dinner)

**We also know that Plan
B costs $413 for:**

**5 nights lodging and
4 dinners > 5n + 4d = 413**

**We want to find out
the cost of 1 night of lodging (hoping and assuming they are the
same) and 1 dinner**

**First, we need to subtract the two equations in order to show the cost of 1 night plus the cost of 1 dinner, which totals $87: (see subtraction #1 under Solve)****Second, we need to isolate the d in order to solve for the cost of 1 dinner. To do this, we need to multiply both sides of Plan B's equation by 4 (this will eventually get rid of n and leave d all alone...see the multiplication #2 under Solve)****Third, we need to continue the isolation of d by multiplying both sides of Plan A's equation by -5:(see the multiplication #3 under Solve)****Finally, we need to subtract the two equations we now have, leaving only d, the cost of the dinner!!! Let's hope it works!! (see the subtraction #4 under Solve)**

**Solve:****SOLVE the problem.**

**#1****5n + 4d = 413**__- 4n + 3d = 326__**n + d = 87****#2****4(5n + 4d = 413)****20n + 16d = 1652**

**#3**

**-5(4n
+ 3d = 326)**

**-20n
- 15d = -1630**

**#4**

** 20n
+ 16d = 1652**

__-
20n - 15d = -1630__

**d =
22**

**Examine:****How does your plan relate to your solved problem? Is it reasonable? Do you need to make corrections?**

- $22 for the cost of dinner is quite reasonable, especially since this is a ski resort!!! :-) Also, re-solving the equations using $22 as the price of dinner in each Plan, gave the cost of lodging as $65, respectively. We may also isolate n in order to solve for the cost of lodging by multiplying Plan A's equation by - 4 and Plan B's equation by 3...we would have the following:
- Plan A - 4(4n + 3d = 326), which yields -16n - 12d = -1304
- Plan B 3(5n + 4d = 413), which yields 15n + 12d = 1239
- Once these are subtracted, we have:
- -16n - 12d = -1304
__15n + 12d = 1239__- - x = - 65
- so x = 65
- I do believe this will work!!!

- Return