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 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

• Plan: Estimate your answer. Make a list of data that pertains to the problem and see if it would be reasonable. If the problem does not make sense, try to solve a simpler problem...or try solving it in another way!!
• 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!!!

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