The Gumball Dilemma
Problem Statement
In this problem we were given three statements and one questions per statement. We were told to experiment these statements and add our own sets of data. This would help us be more organized and find patterns that could expand the gumball dilemma. Here are the statements and questions we had to answer:
2. The next day, Ms Hernandez and her twins pass a gumball machine with three colors: red, white, blue.
a) Assuming that these gumballs also cost one penny, what is the most Ms Hernandez will have to spend for her twins to get matching gumballs?
3. Here comes Mr Hodges with his triplets past the three-color gumball machine described in question 2. His triplets also want the same color gumball?
a) What is the most Mr. Hodges might have to spend?
Procedure:
In this problem I thought that the starting point was the most important. The way you start out a problem usually influences your final solution in a way, so I wanted to make sure I did it right. I started out by answering the questions that I was presented with, which were all about what the maximum price to pay and why it’s the maximum price to pay. After solving the questions I organized them using different variables and started asking myself some questions: Is there a pattern when you add either more kids or more colored gumballs? and Is there an equation that will help me find this information? I decided to experiment more by keeping the same value of kids and changing gumballs to see a pattern in cost and then I tried changing the number of kids with the same amount of gumballs.
Solution
Here are the answers to the initial questions with a key below.
W = White Gumball, R = Red Gumball, B = Blue Gumball
This is a table of how I further expanded this experiment.
In this problem we were given three statements and one questions per statement. We were told to experiment these statements and add our own sets of data. This would help us be more organized and find patterns that could expand the gumball dilemma. Here are the statements and questions we had to answer:
- Ms Hernandez comes across a gumball machine when she is out with her twins. The twins each want a gumball and, of course, they want the same color gumball. The machine has white gumballs and red gumballs. Each gumball costs one penny so Ms. Hernandez decides to keep on putting pennies into the machine until she gets two of the same color gumballs
2. The next day, Ms Hernandez and her twins pass a gumball machine with three colors: red, white, blue.
a) Assuming that these gumballs also cost one penny, what is the most Ms Hernandez will have to spend for her twins to get matching gumballs?
3. Here comes Mr Hodges with his triplets past the three-color gumball machine described in question 2. His triplets also want the same color gumball?
a) What is the most Mr. Hodges might have to spend?
Procedure:
In this problem I thought that the starting point was the most important. The way you start out a problem usually influences your final solution in a way, so I wanted to make sure I did it right. I started out by answering the questions that I was presented with, which were all about what the maximum price to pay and why it’s the maximum price to pay. After solving the questions I organized them using different variables and started asking myself some questions: Is there a pattern when you add either more kids or more colored gumballs? and Is there an equation that will help me find this information? I decided to experiment more by keeping the same value of kids and changing gumballs to see a pattern in cost and then I tried changing the number of kids with the same amount of gumballs.
Solution
Here are the answers to the initial questions with a key below.
W = White Gumball, R = Red Gumball, B = Blue Gumball
- W, R, W or R ==> 3 cents
- W, R, B, W or R or B ==> 4 cents
- W, R, B, W, R, B, W or R or B ==> 7 cents
This is a table of how I further expanded this experiment.
Looking at this table setup helped me find my equation. I looked at this and realized that when you multiply the # of kids by the # of colors your product will always be somewhat close to the total cost. For example 2 kids and 4 colors, maximum cost = 5¢, now if you multiply 2 and 4 you get 8 which is 3 numbers off of 5 but I knew I could find an equation that works around it. I used those two numbers as my base (starting point) and came up with a formula right off the bat, 2*4-(4-1) = 5. The variables I used to create this equation were k # of kids, c # of colors, ¢ cost,
k*c-(c-1) = ¢ is the equation I got. Thankfully this equation worked out for the data points I recorded.
Reflection
Habits of a Mathematician -
k*c-(c-1) = ¢ is the equation I got. Thankfully this equation worked out for the data points I recorded.
Reflection
Habits of a Mathematician -
- Look for Patterns - I used this habit mainly when I was trying to find an equation to my problem I used my table to help me look for patterns and even though I didn't quite find a “pattern” I did find a relationship between the numbers and the maximum costs.
- Start Small - I used this when I was finding the answers to the initial questions. I tried looking at each individual piece of data I was given and worked from there, trying to put together bigger and bigger amounts of information with each small chunk of information.