My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Reply
 
LinkBack Thread Tools Display Modes
January 8th, 2012, 04:23 PM   #1
Newbie
 
Joined: Jan 2012

Posts: 5
Thanks: 0

Challenging maths puzzle! Give it a go!! Calling all aspirin

Hey you guys! I'm a maths fanatic in my senior year of high school and I was working on an interesting maths „puzzle“ I stumbled upon today and, since I had extra time this evening, I thought it would be a good idea to post the said puzzle online just to see what you guys think of it, to compare my answers and see if there are some other approaches to solving the problems.
No, I actually DID complete all of the questions in the puzzle, I'm not trying to get someone else to do the work for me, I'm just double-checking my answers and seeing if there are any different approaches to solving the puzzle than mine
Okay, here goes the puzzle. It's more of an investigation really, about a trip students take, prices of the hotel… Anyway, you'll see (and appreciate I took time to write this all up):
---------------------------------------
THE TRIP
Students of the class of 2011 went to a travelling agency to ask about the conditions for a 4-day-long trip which they are planning. They decided on a trip which cost $1000 per person, the option when each student went to the trip individually (no group discounts).
However, they do not need luxurious, single-bedded rooms the aforementioned cost covers. Furthermore, they heard that it is possible to get a cheaper price if more students went on the trip. There are 31 students in their class and all of the students would like to go, but some cannot afford it. So far, 20 students gathered $14000. A few more students consider paying for the trip if the price isn't too high, but some students are not able to pay anything at all.
Finally, they wanted to know exactly what the travelling agencies can offer them. They inquired the agency which said it will review their demands and get back to them.
Next day, they received an offer from the travelling agency: The price is $1120 per person, and each extra person costs $20 less.
(e.g. 1 person = 1120, 2 people = 1120 + 1100 = 2220, 3 people = 1120 + 1100 + 1080 = 1300 etc. Etc.)
Questions:
1. What is the price of the trip if only one person goes on it? How many students need to go on the trip for the price to equal the price of each student going individually ($1000 per person, mentioned in introduction)?
2. What is the price per student if 20 people go on the trip? What is the total price then ?
3. What is the price per student if 22 people go on the trip? What is the total price then ?
4. How many students can go on the trip with the already collected $14000?
5. How much would they pay if all 31 students would go on the trip? What would be the price per student then?
6. How many more students can go on the trip for the same price as for 31 students? What is the price per student then?
7. All of the students of the class of 2011 want to go on the trip. Suggest an option of payment which would allow all of the students of the class to go on the trip. Keep in mind the various factors we already mentioned (some students cannot pay etc.)
The boss of the travelling agency, upon seeing the offer given to the students, was pleased so he asked the agent in charge of the class of 2011 to explain to him how he figured out the optimal offer. This is what he answered:
„I calculated the lowest price at which 31 students can go on the trip and the agency isn't losing money. My calculations led me to a number of $15500: this was calculated by the formula = (A-Bx)x = (where A is the price, B is the discount per person), where A and B have several different values , and I chose the most optimal of those values.
8. Calculate a few different values for A and B, and comment on how they affect the price.
9. For each of the values you chose, calculate how many more students can go on the trip for a price of $15500? How does that number relate to the number 31 (number of students)?
10. Determine A and B so that less than 31 students can go on the trip for $15500?
11. Make an offer for a class of 36 students , if the total price of the trip is $13500 and if maximum 29 students can afford to pay for the trip.
--------------------------------------------------
If anything sounds wrong, it will usually start to make sense as you begin to solve the question, I had the same problem. Anyway, have a go at it, and please, do not hesitate to ask me if I made something unclear or I made an error somewhere. I'll check back at this page multiple times today!
Happy mathematicating!
Cheers!
Johnatan
mobilefreak10 is offline  
 
January 8th, 2012, 05:25 PM   #2
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Re: Challenging maths puzzle! Give it a go!! Calling all asp

Quote:
1. What is the price of the trip if only one person goes on it? How many students need to go on the trip for the price to equal the price of each student going individually ($1000 per person, mentioned in introduction)?
a) $1120

b)









We could also use the fact that for each additional student, the price per student goes down $10, and it takes 12 additional students to get to $1000 per student for a total of 13 students.

With 13 students going, the price is $13000, or $1000 per person.

Quote:
2. What is the price per student if 20 people go on the trip? What is the total price then ?




Quote:
3. What is the price per student if 22 people go on the trip? What is the total price then ?
With 2 more than part 3, we deduct $20 from $930 to get $910.



Quote:
4. How many students can go on the trip with the already collected $14000?






We take the root for which 1 < n < 32 and round down to get n = 14.

Quote:
5. How much would they pay if all 31 students would go on the trip? What would be the price per student then?




Quote:
6. How many more students can go on the trip for the same price as for 31 students? What is the price per student then?






We find that 82 students can also go for $25420. That's an additional 51 students.



I'll get back to the rest later if someone else doesn't finish.
MarkFL is offline  
January 8th, 2012, 06:47 PM   #3
Senior Member
 
Joined: Apr 2011
From: USA

Posts: 782
Thanks: 1

Re: Challenging maths puzzle! Give it a go!! Calling all asp

This is too much like real cost accounting work. :P
Erimess is offline  
January 9th, 2012, 09:51 AM   #4
Newbie
 
Joined: Jan 2012

Posts: 5
Thanks: 0

Re: Challenging maths puzzle! Give it a go!! Calling all asp

@MarkFL
Real interesting stuff you got going there. My answer for question 6. was wrong an you corrected it! Thanks for your time! Please do go on with the questions, I would very much like to see how you approach the rest of the questions! Cheers!
mobilefreak10 is offline  
January 9th, 2012, 01:22 PM   #5
Senior Member
 
Joined: Oct 2011
From: Belgium

Posts: 522
Thanks: 0

Re: Challenging maths puzzle! Give it a go!! Calling all asp

Quote:
Originally Posted by mobilefreak10
@MarkFL
Real interesting stuff you got going there. My answer for question 6. was wrong an you corrected it! Thanks for your time! Please do go on with the questions, I would very much like to see how you approach the rest of the questions! Cheers!
Is not really the "challenging maths puzzle" as promised in the title.
wnvl is offline  
January 10th, 2012, 09:00 AM   #6
Newbie
 
Joined: Jan 2012

Posts: 5
Thanks: 0

Re: Challenging maths puzzle! Give it a go!! Calling all asp

Please do go on with the question, MarkFL, I'd be thrilled to see your answers to the rest of the questions, especially No. 11!
mobilefreak10 is offline  
January 10th, 2012, 12:15 PM   #7
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 14,581
Thanks: 1038

Re: Challenging maths puzzle! Give it a go!! Calling all asp

Le general case:

n = number of students
f = first student cost
r = reduction

Total = n(f) - r[n(n - 1) /2]

Total = 31(1120) - 20(31*30/2) = 25420

Repeating what WilliamNicholasVictorLadouceur said: where's the challenge?
Denis is offline  
January 10th, 2012, 12:23 PM   #8
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Re: Challenging maths puzzle! Give it a go!! Calling all asp

Quote:
7. All of the students of the class of 2011 want to go on the trip. Suggest an option of payment which would allow all of the students of the class to go on the trip. Keep in mind the various factors we already mentioned (some students cannot pay etc.)
Well, as we found in part 5, the cost for 31 students is $25420. If those who can pay are willing to equally cover those who cannot, then the cost to those who can pay, with n students not paying would be:



Quote:
The boss of the traveling agency, upon seeing the offer given to the students, was pleased so he asked the agent in charge of the class of 2011 to explain to him how he figured out the optimal offer. This is what he answered:
"I calculated the lowest price at which 31 students can go on the trip and the agency isn't losing money. My calculations led me to a number of $15500: this was calculated by the formula = (A-Bx)x = (where A is the price, B is the discount per person), where A and B have several different values , and I chose the most optimal of those values."

8. Calculate a few different values for A and B, and comment on how they affect the price.
We find that:





With B and x fixed, we find the price ?P = 31?A.

With A and x fixed, we find ?P = -31²?B.

Quote:
9. For each of the values you chose, calculate how many more students can go on the trip for a price of $15500? How does that number relate to the number 31 (number of students)?




Taking the larger root:



We see that an increase in A allows more students to go, while an increase in B allows fewer students to go.

Now I'm going to let someone else finish.
MarkFL is offline  
January 15th, 2012, 04:47 AM   #9
Newbie
 
Joined: Jan 2012

Posts: 5
Thanks: 0

Re: Challenging maths puzzle! Give it a go!! Calling all asp

Aww, c'mon MarkFL, I really wanted to see your offer for question 11! I came up with some crappy, half-baked offer for #11 and I was counting on you to set me straight! Pretty please Please please please it's just two more itsy bitsy questions, it must be a piece of cake for you! How much more dignity must I lose begging you to answer them?
mobilefreak10 is offline  
January 15th, 2012, 05:07 AM   #10
Math Team
 
Joined: Oct 2011
From: Ottawa Ontario, Canada

Posts: 14,581
Thanks: 1038

Re: Challenging maths puzzle! Give it a go!! Calling all asp

Quote:
Originally Posted by mobilefreak10
How much more dignity must I lose begging you to answer them?
Dignity is worthless: does not buy groceries...
Denis is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
aspirin, calling, challenging, give, maths, puzzle



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Lockers Maths Puzzle miansons Complex Analysis 1 November 25th, 2012 07:11 AM
An Aspirin a Day n1ml Calculus 11 December 17th, 2011 12:14 PM
Maths dont know what a number is-maths ends in meaningless bas Number Theory 3 March 22nd, 2011 10:26 AM
Two likes give a negative, two unlikes give positive - WHY? sharp Elementary Math 3 October 27th, 2010 03:36 PM
Challenging maths puzzle! Give it a go!! Calling all aspirin mobilefreak10 Applied Math 1 December 31st, 1969 04:00 PM





Copyright © 2019 My Math Forum. All rights reserved.