Sticky Gum Dilemma
For this pow we were given a problem called the Gum-ball Dilemma. This pow had 3 different questions. The first question told us that Ms.Hernandez is with her twins and they pass a gum-ball machine. The twins want a gum-ball but they want the same color. The machine has only 2 colors, red and white. Each gum-ball costs 1 penny. Ms. Hernandez keeps putting pennies in the machine until she gets 2 of the same color. Why is 3 cents the most money she will have to spend?
I explained that if there are only 2 colors in the gum-ball machine then its only ganna take the most of 3 different times because lets say you put 1 penny in and you get a red gum-ball. Then you put another in and you get white, well no matter what the next color is your ganna have to of the same color. So if you have red and white then you get a white again then there's 2 of the same color.
The second question is somewhat the same. There is Ms. Hernandez and the twins passing a gum-ball machine but this time theirs 3 different colors. Red, White and Blue. Assuming that each gum-ball still cost only 1 penny, what is the most that she will have to spend to get the twins matching colors? This one was a little more tricky to get but not tricky enough. I came up with 4 cents is the most she will have to spend because if you get all different colors like red, blue and white then the last penny you put in will get you another gum-ball, doesn't matter what color cause you already have one of everything. So let's say you have white,blue and red then you get another blue, well then there are now 2 blues.
Now the last question is a little more difficult, Mr. Hodges is out with his triplets and pass a gum-ball machine. The triplets want the same color gum. There are still 3 different colors and each gum-ball still cost 1 penny. What is the most Mr. Hodges might have to spend to get 3 of the same color gum-ball? Now that there are 3 kids and 3 different gum the price will go up a little more. The answer is 7 cents, because if there are 3 different gum-balls and 3 kids then as you are putting pennies into the machine eventually you will be forced to get the same color. I showed an example that demonstrates the situation. It shows 3 circles each one having the letter R,B and W. I put a total of 6 circles, each letter was in 2 circles. That shows the gum-balls and the colors. Then I put a gum-ball that had each letter inside it proving that no matter what the color is you will have 3 of the same color. So just a little review if you have 3 different colors but you have 2 of each color, the next color will have to give you a 3 of the same color.
I explained that if there are only 2 colors in the gum-ball machine then its only ganna take the most of 3 different times because lets say you put 1 penny in and you get a red gum-ball. Then you put another in and you get white, well no matter what the next color is your ganna have to of the same color. So if you have red and white then you get a white again then there's 2 of the same color.
The second question is somewhat the same. There is Ms. Hernandez and the twins passing a gum-ball machine but this time theirs 3 different colors. Red, White and Blue. Assuming that each gum-ball still cost only 1 penny, what is the most that she will have to spend to get the twins matching colors? This one was a little more tricky to get but not tricky enough. I came up with 4 cents is the most she will have to spend because if you get all different colors like red, blue and white then the last penny you put in will get you another gum-ball, doesn't matter what color cause you already have one of everything. So let's say you have white,blue and red then you get another blue, well then there are now 2 blues.
Now the last question is a little more difficult, Mr. Hodges is out with his triplets and pass a gum-ball machine. The triplets want the same color gum. There are still 3 different colors and each gum-ball still cost 1 penny. What is the most Mr. Hodges might have to spend to get 3 of the same color gum-ball? Now that there are 3 kids and 3 different gum the price will go up a little more. The answer is 7 cents, because if there are 3 different gum-balls and 3 kids then as you are putting pennies into the machine eventually you will be forced to get the same color. I showed an example that demonstrates the situation. It shows 3 circles each one having the letter R,B and W. I put a total of 6 circles, each letter was in 2 circles. That shows the gum-balls and the colors. Then I put a gum-ball that had each letter inside it proving that no matter what the color is you will have 3 of the same color. So just a little review if you have 3 different colors but you have 2 of each color, the next color will have to give you a 3 of the same color.
This is the table I created that was required for the first 3 questions. I explained a little work on the side of this table just so that if you still don't understand theres is a visual. Basically all I wrote down on the side was the number of kids and gum-balls. The gum-balls had letters in them, depending on which colors were available in the machine. Then I drew a gum-ball and it had the letters that would have to make the other gum-balls doubled. So if you look at the first one it has 2 kids and 2 gum-balls then there a gum-ball with the letter R or W because that was the next possible outcome that would give the twins the exact color gum.
After we solved those 3 problems we were then told to make our own table and make up a number of kids and different amount of gum-balls, so I just kept continuing the pattern. So I started adding one on to the amount of kids. This is the table I created. So as I finished I looked back and looked fro anytime of pattern I could fine. Eventually I found that the amount of cents is increasing by 4 each time. With this pattern I can go up to higher numbers and it would be much easier to solve for knowing that I just have to add 4 each time.As you can see instead of having 3 gum-balls and have it increasing 1 each time. I just went up to 4 and kept in remaining the same for each one. For this table I created I was able to find an equation that represents it. (K*G)-3=C
This is a correct function that works for each table expression: Just to prove that it works, I will demonstrate an example, I will explain the 3rd one down in the table: If you have 7 kids and you multiply it by the number of gum-balls which is 4 that would equal 28. Then if you take 28 and subtract it by 3, it then gives you 25 which is the answer for how many pennies the parent would at most have to spend.When I came up with this function I was just playing around with the 3 numbers in each slot of the table and later on I just ended up finding the function. |