Tuesday, October 23, 2007

The Math Wizzes Strike Out

I love math, and I'm good at it. Daniel has a math talent, too. But today our brains were not functioning in high gear.

Dan's been working through the Algebra 1 Honors Book, and today's lesson was about solving problems with multiple variables. You know, you're given a word problem and need to write the equation(s) and then solve. Knowing how to translate a question into an equation is actually the most important part of this. In math books you can sometimes fiddle around to find the right answer, but in real life the numbers aren't always tidy, so the proper set-up is vital.

Anyway, problem #3 had four variables, and so needed four equations. Daniel tried it, then called for help because the answer was obviously wrong. I helped and it was still wrong. We tried again and it was still wrong. We looked at the solution (thank goodness this answer key SHOWS THE WORK!), and realized we had made the same careless error several times. A truly humbling and time consuming experience. Math was over for the day!

Read on only if you really enjoy algebra:

"Super Suites Hotel has four rates for their 250 basic rooms. Senior citizens pay $35 a night. Businesses pay $45, and coupon-holders pay $40. The standard rate is $50 per night if none of these other rates apply. On New Year's Eve, the hotel's room rates brought in $8640 in income.

The number of rooms sold to senior citizens that night was 10 fewer than the number of standard rooms. The number of business rooms was 8 fewer than the number of coupon-holder rooms sold. Also, the sale of coupon-holder rooms was 10 times less than the number of standard rooms sold. How many rooms were empty on New Year's Eve at Super Suites?"

If you're interested in solving this, put your four starter equations and final answer in the comment section. I'll put the correct answers there later. As a start, we used S for senior, B for business, C for coupon-holder, and R for regular (standard). Our mistake was getting senior and standard partially mixed up--beware!

5 comments:

Frau Nelson said...

I used to looove math. Sadly, it has been too long and I am no longer sure how to begin. Thanksgiving tutor? :)

Pam P said...

Hi. Porter here. Before I start, I just want you guys to know that I did this problem without ANY help (except Mr. Calculator, of course).

Anyway, my starting equations were: S=R-10 (rooms), B=C-8 (rooms)
, and C=R/10 (rooms). I guessed and checked (with Mr. Calculator's help, of course), and finally came up with the final, final answer of 48 EMPTY ROOMS!!!!!!!!!!!!!!!!!!!!

Better luck next time!:):):)

Dad said...

Before I start, I just want you guys to know that I did this without any help from Porter or any of our other 17 brilliant grandkids. Same equations as Porter were substituted in the equation 35S+45B+40C+50R=8640 to get 935C=9350 or C=10. Put that into Porter's equations to get the other numbers. Good job, Porter -- but I'm not so sure about the guessing part :-)

Barbara said...

Good work, people! Yes, your equations and solutions are correct:
35S + 45B + 40C + 50R = 8640
S = R - 10
B = C - 8
R = 10C (or C = R/10)
When you make the substitutions, the number of occupied rooms = 202, so there are 48 empty rooms. Congratulations!

David said...

its easier than that. you set up the four equations like so:

35s+45b+40c+50r=8640
s=r-10
b=c-8
c=10r

Now you put it into a 4x5 matrix where the first column is s, then b, then c, then r, then the answer, and you get this:

[35 45 40 50 8640]
[A] = [1 0 0 -1 -10 ]
[0 1 -1 0 -8 ]
[0 0 1 -10 0 ]

Now just enter that into a calculator and perform reduced row echelon form on it:

rref([A])

You now have the answer matrix:

[1 0 0 0 0 ]
[B] = [0 1 0 0 92 ]
[0 0 1 0 100]
[0 0 0 1 10 ]

So we have the answers:

s=0
b=92
c=100
r=10

Voila!