Question 78775: Shivam & Hakim went to cedar point for rides. Shivam used 20 tickets to ride the roller coaster, corkscrew 3 times and the Blue Streak twice. Hetal used 12 tickets to ride the corkscrew twice and the Bluestreak once. Howmany tickets were needed to ride Corkscrew and howmany tickets were needed for Blue streak?
Answer by mathdoc314(58)   (Show Source):
You can put this solution on YOUR website! The words in the word problem are too long and confuse my mind.
I will use abbreviations
Shivam is S
Hakim is H
Rollercoaster ride is R
Corkscrew ride is C
Blue Streak ride is B
Please note that these are not numbers yet. Just abbreviations.
Now here is the story much shorter with the abbreviations
I see S rode R one time and C 3 times and B two times
I see S used 20 tickets
I see H went on C two times and B one time
I see H used 12 tickets
Questions for reader now
- Does this shorter story have the same information as the
original question?
- What number do you use when the answer is 'none'? (Zero!!)
- What information in this shorter version is missing?
- Is the information that is missing important for solving the problem?
- What is the problem, anyway?
- How many times did H ride on R?
The problem is: How many tickets are needed to ride on B and how many tickets are needed to ride on R?
There is an unstated assumption here that the Cedar Park amusement park does not economically favor either Hakim or Shivam. Both of them pay a certain fixed price for a ride each time they ride it, and there are no coupons and no discount for riding a second time or a third time, etc. Word problems often have unstated assumptions like this. It is part of the cultural bias. What if you have never been to an amusement park? What if you do not have the kind of inductive reasoning or intuitive leaping ability that would allow you to conceive of these assumptions? Especially for people who are more sociable and less nerdy, who would naturally assume they could make friends with the ride person and get a free ticket? Anyway,
In doing the algebra we need to use these symbols, or variables:
R = number of tickets for the roller coaster, each time, every time, for S or H
C = number of tickets for the corkscrew, each time, every time, for S or H
B = number of tickets for the Blue Streak, each time, every time, for S or H
Now we can make equations to solve:
1 R + 3 C + 2 B = 20 - this is total tickets for Hakim.
0 R + 2 C + 1 B = 12 - this is the total tickets for Shivam.
In algebra there is a confusing pattern where a single variable like
1 R is written as just R, 1 B is written just as B, and 0 R is written
as an empty space
R + 3C + 2B = 20
2C + B = 12
One problem I am noticing in this problem is that there are three unknown variables, but only two equations to relate them. In some situations this would mean we would have more than one answer to the problem. However in this word problem we are dealing with tickets which are represented by whole numbers 0,1,2,3... instead of fractions (or other unwhole numbers).
Also I think the amusement park does not give you tickets to reward you for going on any of the rides - so C, R, and B are positive numbers, not negative.
C >= 0
R >= 0
B >= 0
Also there is an unstated assumption that the number of tickets used for any of the rides will be a whole number, not a fraction.
The two equations are
0 R + 2 C + 1 B = 12 (H - Hakim)
1 R + 3 C + 2 B = 20 (S - Shivam)
The H equation can be divided by 2
2 C + B = 12
possible solutions:
(C=0, B=12) (C=1, B=10) (C=2, B=8) (C=3, B=6) (C=4, B=4) (C=5, B=2) (C=6, B=0)
The 3rd equation can be subtracted from the second equation
1 R + 3 C + 2 B = 20 (S)
1 R + 1 C + 0 B = 4 (subtract this)
2 C + 2 B = 16 (what I got by subtracting)
or 1 C + 1 B = 8 (divided that by two)
Possible solutions
(C=1, B=7), (C=2, B=6) (C=3, B=5), (C=4, B=4)
The common solution in these possibilities is (C=4, B=4)
What is R if C=4 and B=4?
1 R + 3 C + 2 B is 20 (Shivam's tickets)
1 R + 3*4 + 2*4 = 20
R = 20 - 12 - 8
R = 0
Let's check that Hakim's tickets work out
0 R + 2 C + 1 B =? 12
0 + 2*4 + 1*4 = 12 Check!
So I think the answer is
C=4 Corkscrew costs 4 tickets
B=4 Blue Thunder costs 4 tickets
R=0 Roller coaster costs 0 tickets - you can ride for free!
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