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